rosetta/utils/matheval.py

import re
import json
import argparse
import jsonlines
from collections import defaultdict
from openai import OpenAI
from typing import Dict, Any
from math_verify import parse, verify, LatexExtractionConfig, ExprExtractionConfig, StringExtractionConfig
from latex2sympy2_extended import NormalizationConfig
from transformers import AutoTokenizer
import requests
import os

class MathEvaluator:
    
    def rule_judge(self, solution_str: str, ground_truth: str, finish_generation: bool = True) -> bool:
        raise NotImplementedError

    def extract_after_think(self, text: str, truncate_length: int = 1000, finish_generation: bool = True) -> str:
        pattern = r"</think>(.*)"
        match = re.search(pattern, text, re.DOTALL)
        return match.group(1).strip() if (match and finish_generation) else text[-truncate_length:]
    
    def get_llm_judge_prompt(self, solution_str: str, ground_truth: str, extracted_answer: str = "", finish_generation: bool = True) -> str:
        raise NotImplementedError

    def get_llm_judge_prompt_not_finished(self, solution_str: str, ground_truth: str, extracted_answer: str = "", finish_generation: bool = True) -> str:
        return f"""Please determine whether the final answer in the model-generated response was already correctly derived early in the reasoning process, and that the subsequent content consists mainly of unnecessary verification, overthinking, or repetitive reasoning. If correct is derived early, return "YES"; if they are not, return "NO". Only return "YES" or "NO", and do not generate any other content.
Reference answer: {ground_truth}
Model-generated response: {solution_str}
""".strip()

    def llm_judge(self, solution_str: str, ground_truth: str, extracted_answer: str = "", finish_generation: bool = True) -> bool:
        global OPENAI_CLIENT, MODEL_NAME
        def get_inputs(scene_description):
            body = [
                {"role": "user", "content": scene_description},
            ]
            return body

        def run_api(inputs):
            completion = OPENAI_CLIENT.chat.completions.create(
                model=MODEL_NAME,
                messages=inputs
            )
            return completion.choices[0].message.content.strip()
        if finish_generation:
            scene_description = self.get_llm_judge_prompt(solution_str, ground_truth, extracted_answer, finish_generation)
        else:
            scene_description = self.get_llm_judge_prompt_not_finished(solution_str, ground_truth, extracted_answer, finish_generation)
        inputs = get_inputs(scene_description)
        response = run_api(inputs)

        return "YES" in response


class AIMEEvaluator(MathEvaluator):
    def rule_judge(self, solution_str: str, ground_truth: str, finish_generation: bool = True) -> bool:
        # if not ground_truth.startswith("$"):
        #     ground_truth = f"${ground_truth}$"
        gold = parse(
            ground_truth,
            extraction_config=[ExprExtractionConfig()],
        )
        answer = parse(
            solution_str,
            extraction_config=[
                LatexExtractionConfig(
                    normalization_config=NormalizationConfig(
                        nits=False,
                        malformed_operators=False,
                        basic_latex=True,
                        boxed="all",
                        units=True,
                    ),
                    boxed_match_priority=0,
                    try_extract_without_anchor=False,
                ),
                ExprExtractionConfig(),
            ],
            extraction_mode="first_match",
        )
        if len(answer) == 0:
            return False, "No extracted answer"
        else:
            return verify(gold, answer), str(answer)

    def get_llm_judge_prompt(self, solution_str: str, ground_truth: str, extract_answer: str = "", finish_generation: bool = True) -> str:
        solution_str = self.extract_after_think(solution_str, finish_generation=finish_generation)
        return f"""Please determine whether the final answer provided in the model-generated response is equivalent to the reference answer from a math question. The final answer may either be enclosed in \\boxed{{}} or appear after "Answer:". If they are equivalent, return "YES"; if they are not, return "NO". Only return "YES" or "NO", and do not generate any other content.
Model-generated answer: {solution_str}
Reference answer: {ground_truth}""".strip()


class GSM8KEvaluator(MathEvaluator):
    def rule_judge(self, solution_str: str, ground_truth: str, finish_generation: bool = True) -> bool:
        # if not ground_truth.startswith("$"):
        #     ground_truth = f"${ground_truth}$"
        gold = parse(
            ground_truth,
            extraction_config=[ExprExtractionConfig()],
        )
        answer = parse(
            solution_str,
            extraction_config=[
                LatexExtractionConfig(
                    normalization_config=NormalizationConfig(
                        nits=False,
                        malformed_operators=False,
                        basic_latex=True,
                        boxed="all",
                        units=True,
                    ),
                    boxed_match_priority=0,
                    try_extract_without_anchor=False,
                ),
                ExprExtractionConfig(),
            ],
            extraction_mode="first_match",
        )
        if len(answer) == 0:
            return False, "No extracted answer"
        else:
            return verify(gold, answer), str(answer)

    def get_llm_judge_prompt(self, solution_str: str, ground_truth: str, extract_answer: str = "", finish_generation: bool = True) -> str:
        solution_str = self.extract_after_think(solution_str, finish_generation=finish_generation)
        return f"""Please determine whether the final answer provided in the model-generated response with rule-based extracted answer is equivalent to the reference answer from a math question. The final answer may either be enclosed in the \\boxed{{}} or appear after the "Answer:". If they are equivalent, return "YES"; if they are not, return "NO". Only return "YES" or "NO", and do not generate any other content.

1. The reference answer does not include percentage signs, units or time formats (e.g., am, pm), but the Model-generated answer may include them.
For example, 1 is equivalent to 1 %, 1 kg, 1 am, 1 pm, 1:00 am, 1:00 pm, etc.
Model-generated answer: 1%
Reference answer: 1
Your output: YES

Model-generated answer: 1 kg
Reference answer: 1
Your output: YES

Model-generated answer: 1:00 pm
Reference answer: 1
Your output: YES

2. The reference answer only includes one single number, but the Model-generated answer may include multiple numbers.
For example, 10 is equivalent to \\boxed{{(4, 6)}}, etc.
Model-generated answer: 5, 5
Reference answer: 10
Your output: YES

Model-generated answer: 4, 6
Reference answer: 10
Your output: YES

Model-generated answer: 86, 42
Reference answer: 128
Your output: YES

Now let's try a real example.
Model-generated answer: {solution_str}
Reference answer: {ground_truth}
""".strip()


class MATH500Evaluator(MathEvaluator):
    def rule_judge(self, solution_str: str, ground_truth: str, finish_generation: bool = True) -> bool:
        if not ground_truth.startswith("$"):
            ground_truth = f"${ground_truth}$"
        gold = parse(
            ground_truth,
            extraction_config=[LatexExtractionConfig()],
        )
        answer = parse(
            solution_str,
            extraction_config=[
                LatexExtractionConfig(
                    normalization_config=NormalizationConfig(
                        nits=False,
                        malformed_operators=False,
                        basic_latex=True,
                        boxed="all",
                        units=True,
                    ),
                    boxed_match_priority=0,
                    try_extract_without_anchor=False,
                ),
                ExprExtractionConfig(),
            ],
            extraction_mode="first_match",
        )
        if len(answer) == 0:
            return False, "No extracted answer"
        else:
            return verify(gold, answer), str(answer)
    def get_llm_judge_prompt(self, solution_str: str, ground_truth: str, extract_answer: str = "", finish_generation: bool = True) -> str:
        solution_str = self.extract_after_think(solution_str, finish_generation=finish_generation)
        return f"""Please determine whether the final answer provided in the model-generated response is equivalent to the reference answer from a math question. The final answer may either be enclosed in \\boxed{{}} or appear after "Answer:". If they are equivalent, return "YES"; if they are not, return "NO". Only return "YES" or "NO", and do not generate any other content.
Model-generated answer: {solution_str}
Reference answer: {ground_truth}""".strip()
    
class AMCEvaluator(MathEvaluator):
    def rule_judge(self, solution_str: str, ground_truth: str, finish_generation: bool = True) -> bool:
        if not ground_truth.startswith("$"):
            ground_truth = f"${ground_truth}$"
        gold = parse(
            ground_truth,
            extraction_config=[LatexExtractionConfig()],
        )
        answer = parse(
            solution_str,
            extraction_config=[
                LatexExtractionConfig(
                    normalization_config=NormalizationConfig(
                        nits=False,
                        malformed_operators=False,
                        basic_latex=True,
                        boxed="all",
                        units=True,
                    ),
                    boxed_match_priority=0,
                    try_extract_without_anchor=False,
                ),
                ExprExtractionConfig(),
            ],
            extraction_mode="first_match",
        )
        if len(answer) == 0:
            return False, "No extracted answer"
        else:
            return verify(gold, answer), str(answer)
    def get_llm_judge_prompt(self, solution_str: str, ground_truth: str, extract_answer: str = "", finish_generation: bool = True) -> str:
        solution_str = self.extract_after_think(solution_str, finish_generation=finish_generation)
        return f"""Please determine whether the final answer provided in the model-generated response is equivalent to the reference answer from a math question. The final answer may either be enclosed in \\boxed{{}} or appear after "Answer:". If they are equivalent, return "YES"; if they are not, return "NO". Only return "YES" or "NO", and do not generate any other content.
Model-generated answer: {solution_str}
Reference answer: {ground_truth}""".strip()


class GPQAEvaluator(MathEvaluator):
    def rule_judge(self, solution_str: str, ground_truth: str, finish_generation: bool = True) -> bool:
        # if not ground_truth.startswith("$"):
        #     ground_truth = f"${ground_truth}$"
        gold = parse(
            ground_truth,
            extraction_config=[StringExtractionConfig()],
        )
        answer = parse(
            solution_str,
            extraction_config=[
                StringExtractionConfig(),
            ]
        )
        if len(answer) == 0:
            return False, "No extracted answer"
        else:
            return verify(gold, answer), str(answer)
        
    def get_llm_judge_prompt(self, solution_str: str, ground_truth: str, extract_answer: str = "", finish_generation: bool = True) -> str:
        solution_str = self.extract_after_think(solution_str, finish_generation=finish_generation)
        return f"""Please determine whether the final answer provided in the model-generated response is equivalent to the reference answer from a multiple choice question. The final answer may either be enclosed in \\boxed{{}} or appear after "Answer:". If they are equivalent, return "YES"; if they are not, return "NO". Only return "YES" or "NO", and do not generate any other content.
Model-generated answer: {solution_str}
Reference answer: {ground_truth}""".strip()


# class MBPPEvaluator(Evaluator):
#     def rule_judge(self, solution_str: str, ground_truth: str, finish_generation: bool = True) -> bool:
#         return True, "No extracted answer"
        
#     def get_llm_judge_prompt(self, solution_str: str, ground_truth: str, extract_answer: str = "", finish_generation: bool = True) -> str:
#         solution_str = self.extract_after_think(solution_str, finish_generation=finish_generation)
#         return f"""Please determine whether the final answer provided in the model-generated response is equivalent to the reference answer from a multiple choice question. The final answer may either be enclosed in \\boxed{{}} or appear after "Answer:". If they are equivalent, return "YES"; if they are not, return "NO". Only return "YES" or "NO", and do not generate any other content.
# Model-generated answer: {solution_str}
# Reference answer: {ground_truth}""".strip()


# class HUMANEVALEvaluator(Evaluator):
#     def rule_judge(self, solution_str: str, ground_truth: str, finish_generation: bool = True) -> bool:
#         return True, "No extracted answer"
        
#     def get_llm_judge_prompt(self, solution_str: str, ground_truth: str, extract_answer: str = "", finish_generation: bool = True) -> str:
#         solution_str = self.extract_after_think(solution_str, finish_generation=finish_generation)
#         return f"""Please determine whether the final answer provided in the model-generated response is equivalent to the reference answer from a multiple choice question. The final answer may either be enclosed in \\boxed{{}} or appear after "Answer:". If they are equivalent, return "YES"; if they are not, return "NO". Only return "YES" or "NO", and do not generate any other content.
# Model-generated answer: {solution_str}
# Reference answer: {ground_truth}""".strip()


evaluator_map = {
    "aime2024": AIMEEvaluator(),
    "aime2025": AIMEEvaluator(),
    "gsm8k": GSM8KEvaluator(),
    "math500": MATH500Evaluator(),
    "gpqa_diamond": GPQAEvaluator(),
    "amc23": AMCEvaluator(),
}

API_BASE = None
DEPLOYMENT_NAME = None
API_VERSION = None
CONSTRUCTED_URL = None
API_KEY = None
HEADERS = None
OPENAI_CLIENT = None
MODEL_NAME = None

def set_client(api_base=None, deployment_name=None, api_version=None, api_key=None, model_name="gpt-4.1-2025-04-14"):
    global API_BASE, DEPLOYMENT_NAME, API_VERSION, CONSTRUCTED_URL, API_KEY, HEADERS, MODEL_NAME, OPENAI_CLIENT

    API_BASE = api_base
    DEPLOYMENT_NAME = deployment_name
    API_VERSION = api_version
    CONSTRUCTED_URL = f"{api_base}/openai/deployments/{deployment_name}/chat/completions?api-version={api_version}"
    API_KEY = api_key or os.getenv("OPENAI_API_KEY", "")
    MODEL_NAME = model_name
    HEADERS = {
        "Content-Type": "application/json",
        "api-key": api_key,
    }
    if API_KEY:
        print(f"Using API key: {API_KEY}")
        OPENAI_CLIENT = OpenAI(api_key=API_KEY)
    else:
        OPENAI_CLIENT = None
    



# def call_llm_judge(message: list, args: argparse.Namespace) -> str:
#     """
#     Call the Qwen API with the given message.
    
#     Args:
#         message (list): Message list for the API.
#         args (argparse.Namespace): Parsed arguments.
        
#     Returns:
#         str: The content of the completion response.
        
#     Raises:
#         Exception: When the API call fails.
#     """

#     try:
#         completion = client.chat.completions.create(
#             model=args.model_name,
#             messages=message,
#             temperature=args.temperature,
#             top_p=args.top_p,
#         )
#         return completion.choices[0].message.content.strip()
#     except Exception as e:
#         print(f"API call failed: {str(e)}")
#         raise


# def rule_judge(completin, ground_truth, dataset_name: str) -> Dict[str, Any]:
#     rule_judge_result = None

#     rule_judge_result = evaluator_map[dataset_name].rule_judge(completin, ground_truth)

#     if not rule_judge_result:
#         print(f"No valid answer detected | LLM judge")
#         call_llm_judge
    
        
#     return example



# def process_example(example: Dict[str, Any], args: argparse.Namespace) -> Dict[str, Any]:
#     """
#     Process an individual example with token counting.
#     """
#     try:

#         completion_text = example.get('completion', '')
#         before_truncation_text = completion_text.split("</think>")[0]
#         before_truncation_token_count = count_tokens(before_truncation_text, args)
#         after_truncation_token_count = example.get('generated_tokens', 0) - before_truncation_token_count

#         example['before_truncation_token_count'] = before_truncation_token_count
#         example['after_truncation_token_count'] = after_truncation_token_count

#         example = post_process(example)
#         if not example.get('rule_judge_result', False):
#             think_truncation = extract_after_think(completion_text) or ''

#             prompt = generate_prompt(example, think_truncation)
#             messages = format_prompt(prompt)
#             llm_judge_response = call_qwen(messages, args)
#             print(llm_judge_response)
#             example['llm_judge_response'] = llm_judge_response
#             if llm_judge_response == "YES":
#                 example['llm_judge_result'] = True
#                 example['final_judge_result'] = True
#             elif llm_judge_response == "NO":
#                 example['llm_judge_result'] = False
#                 example['final_judge_result'] = False
#             else:
#                 example['llm_judge_result'] = None
#                 example['final_judge_result'] = None
                
#         return example
#     except Exception as e:
#         print(f"Failed to process example {example.get('idx', 'unknown')}: {str(e)}")
#         example['final_judge_result'] = None
#         return example



# def main():
#     args = parse_arguments()
#     try:
#         final_results = []
#         with jsonlines.open(args.result_save_name, mode='w') as writer:
#             with open(args.data_load_name, 'r', encoding='utf-8') as f:
#                 data = json.load(f)

#             for idx, example in enumerate(data, start=1):
#                 print(f"Processed example ID: {idx}")
#                 new_example = process_example(example, args)
#                 final_results.append(new_example)

#             writer.write_all(final_results)
#             print(f"Processing complete! Valid results saved to: {args.result_save_name}")

#         grouped_data = defaultdict(list)
#         for entry in final_results:
#             grouped_data[entry['idx']].append(entry)

#         pass_at_1_per_idx = {}
#         for idx, entries in grouped_data.items():
#             correct_count = sum(entry['final_judge_result'] for entry in entries if entry['final_judge_result'] is True)
#             total_count = len(entries)
#             pass_at_1_per_idx[idx] = correct_count / total_count if total_count else 0

#         overall_pass_at_1 = sum(pass_at_1_per_idx.values()) / len(pass_at_1_per_idx)
#         average_generated_tokens = sum(entry['generated_tokens'] for entry in final_results) / len(final_results)
#         average_reasoning_tokens = sum(entry['before_truncation_token_count'] for entry in final_results) / len(final_results)
#         average_answer_tokens = sum(entry['after_truncation_token_count'] for entry in final_results) / len(final_results)

#         print(f'\nPass@1 per idx:\n{pass_at_1_per_idx}\n')
#         print(f'Overall pass@1: {overall_pass_at_1:.4f}')
#         print(f'Average generated_tokens: {average_generated_tokens:.2f}')
#         print(f'Average reasoning_tokens: {average_reasoning_tokens:.2f}')
#         print(f'Average answer_tokens: {average_answer_tokens:.2f}')

#     except json.JSONDecodeError as e:
#         print(f"Data loading failed: Invalid JSON format - {str(e)}")
#         raise


if __name__ == '__main__':
    api_key = os.getenv("OPENAI_API_KEY", "")

    set_client(api_base="", deployment_name="", api_version="", api_key=api_key)
    response = "Okay, so I need to find the number of triples (a, b, c) of nonnegative integers such that a + b + c = 300 and the expression a\u00b2b + a\u00b2c + b\u00b2a + b\u00b2c + c\u00b2a + c\u00b2b equals 6,000,000. Hmm, that seems a bit complicated, but maybe I can break it down step by step.\n\nFirst, let me note down the given equations:\n\n1. a + b + c = 300\n2. a\u00b2b + a\u00b2c + b\u00b2a + b\u00b2c + c\u00b2a + c\u00b2b = 6,000,000\n\nI need to find all nonnegative integer triples (a, b, c) that satisfy both equations. Since the problem involves symmetric expressions in a, b, c, maybe there's a way to exploit symmetry here.\n\nLet me first if I can rewrite the second equation in a more manageable form. The expression a\u00b2b + a\u00b2c + b\u00b2a + b\u00b2c + c\u00b2a + c\u00b2b looks symmetric. Maybe I can factor it somehow?\n\nLet me see: Each term is of the form (variable squared times another variable). So, for example, a\u00b2b + a\u00b2c = a\u00b2(b + c), similarly for the other terms. Let me try that:\n\na\u00b2(b + c) + b\u00b2(a + c) + c\u00b2(a + b)\n\nYes, that's the same as the original expression. Now, since a + b + c = 300, maybe I can substitute (b + c) with (300 - a), and similarly for the others. Let's try that substitution.\n\nSo, substituting:\n\na\u00b2(300 - a) + b\u00b2(300 - b) + c\u00b2(300 - c)\n\nWait, hold on. Let me check that again. The original expression is't exactly that. Wait, the's see:\n\nWait, the original expression is a\u00b2(b + a\u00b2c + b\u00b2a + b\u00b2c + c\u00b2a + c\u00b2b. So I group terms as a\u00b2(b + c) + b\u00b2(a + c) + c\u00b2(a + b), then yes, each term is variable squared times the sum of the other two variables. Since the sum of the other two variables is 300 minus the variable. So, for example, (b + c) = 300 - a, so substituting that in, we get:\n\na\u00b2(300 - a) + b\u00b2(300 - b) + c\u00b2(300 - c)\n\nTherefore, the second equation can be rewritten as:\n\na\u00b2(300 - a) + b\u00b2(300 - b) + c\u00b2(300 - c) = 6,000,000\n\nHmm, that might be a useful form. Let me write that down:\n\nEquation 2: a\u00b2(300 - a) + b\u00b2(300 - b) + c\u00b2(300 - c) = 6,000,000\n\nNow, perhaps I can consider expanding expression in terms of the variables. Let me think if there's a way to relate this to the first equation. Since a + b + c = 300, maybe we can express some relationship between these variables here Let't immediately yet.\n\nAlternatively, maybe I can consider expanding the expression or finding another way to express it. Let me see ifAlternatively, perhaps I can consider the expression a\u00b2b + a\u00b2c + b\u00b2a + b\u00b2c + c\u00b2a + c\u00b2b. Let me see if this can be written in terms of (a + b + c)^3 or something similar.\n\nWait, let me recall that (a + b + c)^3 = a^3 + b^3 + c^3 + 3(a + b)(a + c)(b + c). Hmm, not sure if that helps. Alternatively, maybe expanding (a + b + c)(ab + bc + ca). Let me see:\n\n(a + b + c)(ab + bc + ca) = a^2b + a^2c + ab^2 + b^2c + a c^2 + b c^2 + 3abc\n\nWait, that's exactly our expression in the second equation plus 3abc. So:\n\n(a + b + c)(ab + bc + ca) = (a\u00b2b + a\u00b2c + b\u00b2a + b\u00b2c + c\u00b2a + c\u00b2b) + 3abc\n\nTherefore, the second equation can be rewritten as:\n\n(a + b + c)(ab + bc + ca) - 3abc = 6,000,000\n\nSince a know that a + b + c = 300, substitute that in:\n\n300(abab + bc + ca) - 3abc = 6,000,000\n\nHmm, that might be a better way to handle it equation. Let me write that down:\n\nEqu00(ab + bc + ca) - 3abc = 6,000,000\n\nDiv can divide both sides by 3 to simplify:\n\n100(ab + bc + ca) - abc = 2,000,000\n\nSo now, the equation becomes:\n\n100(ab + bc + ca) - abc = 2,000,000\n\nHmm, that seems a bit more manageable. Let me write that as:\n\nabc - 100(ab + bc + ca) = -2,000,000\n\nAlternatively, maybe rearranged as:\n\nabc = 100(ab + bc + ca) - 2,000,000\n\nHmm, not sure yet. Let me see if I can find another relation here.\n\n Since we have a + b + c = 300, perhaps we can denote variables in terms of two variables. say, let me set variables as a, b, and then c = 300 - a - b. Then substitute expressions in terms of a and b. But that might get complicated, but perhaps manageable.\n\nAlternatively, maybe there's a substitution or symmetry here. Let me think.\n\n the problem is symmetric in a, b, c, so perhaps the solutions are symmetric in some way. Maybe two variables are equal, or all three are equal?\n\nLet me first check if all three variables are equal. If a = b = c, then since a + b + c = 300, each would be 100. Let me check if that satisfies the second equation.\n\nIf a = b = c = 100, then compute the second expression:\n\na\u00b2b + a\u00b2c + ...\u00b2a + b\u00b2c + c\u00b2a + c\u00b2b\n\n = 6*(100)^2*100 = 6*10003 = 6,000,000. Wait, that's exactly the required given! So (100, 100, 100) is a solution.\n\n So that's one solution. But the problem asks for all triples, so maybe might be more?\n\nHmm, so maybe there are other triples where variables are not all equal but still satisfy the equations. Let me see.\n\nAlternatively, maybe two variables are equal and the third is different. Let me suppose that two variables are equal, say a = b, and c is different. Let,2a + c = 300, so c = 300 - 2a.\n\nThen, let's plug into the second equation.\n\nFirst, let me compute ab + bc + ca. when a = b:\n\nab + bc + ca = a\u00b2 + a c + a c = a\u00b2 + 2a c\n\nSimilarly, abc = a\u00b2 c\n\nSo substitugging into the equation 100(ab + bc + ca) - abc = 100(a\u00b2 + 2a c) - a\u00b2 c = 100a\u00b2 + 200a c - a\u00b2 c\n\nBut since c = 300 - 2a, substitute that in:\n\n100a\u00b2 + 200a(300 - 2a) - a\u00b2(300 - 2a)\n\nLet me compute each term:\n\nFirst term: 100a\u00b2\n\nSecond term: 200a*(300 - 2a) = 200a*300 - 200a*2a = 60,000a - 400a\u00b2\n\nThird term: -a\u00b2*(300 - 2a) = -300a\u00b2 + 2a\u00b3\n\nSo adding all three terms together:\n\n100a\u00b2 + (60,000a - 400a\u00b2) + (-300a\u00b2 + 2a\u00b3)\n\nCombine like terms:\n\nFor00a\u00b2 - 400a\u00b2 - 300a\u00b2 + 2a\u00b3 + 60,000a\n\nThat's:\n\n(100 - 400 - 300)a\u00b2 + 2a\u00b3 + 60,000a\n\n= (-600a\u00b2) + 2a\u00b3 + 60,000a\n\nSo the entire expression is 2a\u00b3 - 600a\u00b2 + 60,000a\n\nSet this equal to 2,000,000 ( (since the equation was 100(ab + bc + ca) - abc = 2,000,000)\n\nWait:\n\n2a\u00b3 - 600a\u00b2 + 60,000a = 2,000,000\n\nDivide both sides by 2:\n\na\u00b3 - 300a\u00b2 + 30,000a = 1,000,000\n\nBring all terms to left-hand:\n\na\u00b3 - 300a\u00b2 + 30,000a - 1,000,000 = 0\n\nHmm, need to solve this cubic equation for integer a. Let me see if I can factor this or find integer roots.\n\nPossible me denote f(a) = a\u00b3 - 300a\u00b2 + 30,000a - 1,000,000\n\nWe can try possible integer roots. Since a must be a nonnegative integer such that 2a \u2264 300 (since c = 300 - 2a must be nonnegative). so a \u2264 150.\n\nPossible rational roots are factors of 1,000,000 divided by factors of 1, so possible integer roots are divisors of 1,000,000. But that's a lot. Maybe try some values.\n\nFirst, let me try a = 100. Since when all variables are 100, that's a solution. so if a = 100, then c = 300 - 200 = 100, so that's the case where all three are equal, so that should be a root.\n\nLet me check f(100):\n\n100\u00b3 - 300*(100)^2 + 30,000*100 - 1,000,000\n\n= 1,000,000 - 300*10,000 + 3,000,000 - 1,000,000\n\n= 1,000,000 - 3,000,000 + 3,000,000 - 1,000,000 = 0. Yes, so a = 100 is a root.\n\nTherefore, (a - 100) is a factor. Let's perform polynomial division or factor it out.\n\nDivide f(a) by (a - 100):\n\nUsing synthetic division:\n\nCoefficients: 1 | -300 | 30,000 | -1,000,000\n\nDivide by (a - 100), so root at 100.\n\nBring down the 1.\n\nMultiply 1 by 100: 100. Add to next coefficient: -300 + 100 = -200\n\nMultiply -200 by 100 = -20,000. Add to next coefficient: 30,000 + (-20,000) = 10,000\n\nMultiply 10,000 by 100 = 1,000,000. Add to last term: -1,000,000 + 1,000,000 = 0. Perfect.\n\nSo the polynomial factors as (a - 100)(a\u00b2 - 200a + 10,000)\n\nNow set the quadratic factor to zero:\n\na\u00b2 - 200a + 10,000 = 0\n\nLet me compute discriminant D = (200)^2 - 4*1*10,000 = 40,000 - 40,000 = 0\n\nSo discriminant is zero, so a = [200 \u00b1 0]/2 = 100. So the quadratic has a double root at a = 100.\n\nTherefore, the cubic factors as (a - 100)^3 = 0. Wait, but that would mean that the original cubic is (a - 100)^3, but let me check:\n\nWait, (a - 100)^3 = a\u00b3 - 300a\u00b2 + 30,000a - 1,000,000, which matches exactly the cubic. So indeed, the equation is (a - 100)^3 = 0, so the only real root is a = 100. Therefore, in the case where a = b, the only solution is a = b = 100, which gives c = 100. so that's the triple-equal case.\n\nHmm, so if we assume two variables are equal, the only solution is when all three are equal. So maybe the only symmetric with two variables equal is the all-equal case. So perhaps other are solutions where all variables are different?\n\nAlternatively, maybe there are solutions where one variable is 0. Let me check that possibility.\n\nSuppose c = 0. Then the equations become a + b = 300 and and the second equation becomes a\u00b2b + a\u00b2*0 + b\u00b2a + b\u00b2*0 + 0 + 0 = a\u00b2b + ab\u00b2 = ab(a + b) = ab*300.\n\nSo the second equation becomes 300ab = 6,000,000, so ab = 20,000.\n\nSo if have a + b = 300 and ab = 20,000. Let me see if there are integer solutions here here.\n\nWe can set up the quadratic equation: Let a and b be roots of x\u00b2 - 300x + 20,000 = 0.\n\nCompute discriminant D = 300\u00b2 - 4*1*20,000 = 90,000 - 80,000 = 10,000\n\nsqrt(D) = 100\n\n so roots are [300 \u00b1 100]/2 = (400)/2 = 200 or (200)/2 = 100. So the roots are 200 and 100. Therefore, the solutions are (a, b) = (200, 100) or (100, 200). Since c = 0, the triples would be (200, 100, 0), (100, 200, 0), and and also permutations where c is in different positions. Wait, since c was set to 0, but the variables are ordered, so actually, if we set c = 0, then the triples are (a, b, 0) where a + b = 300 and ab = 20,000. So the solutions are (200, 100, 0), (100, 200, 0). But since the variables are ordered, but the problem counts triples (a, b, c) as ordered? Wait, actually, the problem says \"triples of nonnegative integers (a, b, c)\", so I think order matters here. Wait, but actually, in combinatorics, unless specified otherwise, sometimes triples are considered ordered. But let me check the problem statement again.\n\nWait, the problem says \"the number of triples of nonnegative integers (a, b, c)\", so I think they are ordered triples. so different permutationsings count as different triples. However, in the case where c is 0, but we could also have a = 0 or b = 0. So actually's see.\n\nWait, so if c = 0, then we have two solutions (200,100,0 and 100,200,0. Similarly, if a = 0, then we have (0, 200, 100) and (0, 100, 200). Similarly, if b = 0, then (200, 0, 100) and (100, 0, 200). So total of 6 triples where one variable is 0 and the other two are 200 and 100. Additionally, the all-100 case. So that's 7 so far. But maybe there are more solutions where none of the variables are zero?\n\nAlternatively, maybe there are solutions where two variables are zero? Let me check that.\n\n quickly. Suppose two variables are zero, say b = c = 0, then a = 300. Then the second equation becomes a\u00b2b + ... all terms would be zero, so 0 \u2260 6,000,000. So that's invalid. So no solutions with two variables zero.\n\nSo the only solutions with a zero are the ones above with one zero and the other two being 200 and 100. So that gives 6 triples. Plus the all-100 case. So total 7? But maybe there are other triples where none of the variables are zero?\n\nWait, let me see.\n\n Let me suppose of another approach. Let me suppose variables a, b, c all positive integers (since if one is zero, we already considered that case). Let me see if there are solutions where all variables are positive and and different from 100, 100, 100 and 200,100,0 etc.\n\nAlternatively, maybe there can consider the equation 100(ab + bc + ca) - abc = 2,000,000. Let me denote this as:\n\nabc - 100(ab + bc + ca) = -2,000,000\n\nHmm, perhaps we can factor this equation somehow. Let me see:\n\nLet me add 1,000,000 to both sides:\n\nabc - 100(ab + bc + ca) + 1,000,000 = -1,000,000\n\nHmm, not sure. Alternatively, perhaps we't think of variables x = a, y = b, z = c, and see if there's a substitution.\n\nAlternatively, let set variables = 300 - a, y = 300 - b, z = 300 - c, but not sure.\n\nAlternatively, perhaps consider the equation:\n\n100(ab + bc + ca) - abc = 2,000,000. Let me rearr this as:\n\nabc - 100(ab - 100bc - 100ca + 2,000,000 = 0\n\nHmm, perhaps factor terms:\n\nLet me see, perhaps factor terms with ab, bc, ca:\n\nab(c - 100) + bc(a - 100) + ca(b - 100) = -2,000,000\n\nHmm, not sure. Alternatively, perhaps rearr the equation as:\n\nabc = 100(ab + bc + ca) - 2,000,000\n\nLet me think if I can write this as:\n\nabc = 100(ab + bc + ca) - 2,000,000\n\nHmm, perhaps divide both sides by abc (assuming a,b,c \u2260 zero):\n\n1 = 100(1/c + 1/a + 1/b) - 2,000,000/(abc)\n\nHmm, not sure if helpful helps.\n\nAlternatively, maybe think of variables in terms of ratios. Let me suppose that variables are in some proportion. Let me see.\n\nAlternatively, let me consider that in the case where one variable is 0, we have solutions. The case where all variables equal 100 is a solution. Maybe there are other solutions where variables are different but not involving zeros?\n\nAlternatively me suppose that variables a, b, c are all different from 100 and none are zero. Let me see if such solutions exist.\n\nAlternatively, let try to set variables as 150, 100, 50. Let me test that.\n\nWait, let me pick some numbers. Let me see, for't suppose a = 150, b = 100, c = 50. Then a + b + c = 300. Let me compute the second equation:\n\na\u00b2b + a\u00b2c + b\u00b2a + b\u00b2c + c\u00b2a + c\u00b2b\n\nCompute each term:\n\na\u00b2b = 150\u00b2 * 100 = 22500 * 100 = 2,250,000\n\na\u00b2c = 150\u00b2 *50 = 22500 *50 = 1,125,000\n\nb\u00b2a = 100\u00b2 *150 = 10,000 *150 = 1,500,000\n\nb\u00b2c = 100\u00b2 *50 = 10,000 *50 = 500,000\n\nc\u00b2a = 50\u00b2 *150 = 2500 *150 = 375,000\n\nc\u00b2b = 50\u00b2 *100 = 2500 *100 = 250,000\n\nAdding all these up:\n\n2,250,000 + 1,125,000 = 3,375,000\n\n+1,500,000 = 4,875,000\n\n+500,000 = 5,375,000\n\n+375,000 = 5,750,000\n\n+250,000 = 6,000,000\n\nWait a second! That works! So (150, 100, 50) is another solution. Hmm that's a solution where all variables are different and none are zero. So that's another solution. Hmm my previous thought that only the all-100 and the ones with a zero were solutions was incorrect. So there are more solutions.\n\nHmm, so that complicates things. Let me see why that works. Let me check the calculation again.\n\nWait, so (150, 100, 50) gives the required sum and the required product expression6,000,000. So that's a valid solution. So there are more solutions. So I need to find all such triples.\n\nHmm, so perhaps there are multiple families variables are in arithmetic progression? Let me see. 50, 100, 150 is an arithmetic progression with difference 50. Let me see if that's a pattern.\n\nAlternatively, maybe variables a can think of variables as multiples of 50? Let me see.\n\nWait, let me see if there are other triples. that. Let me try (200, 100, 0) which we already know works, but (150, 100, 50) works, and (100, 100, 100) works. Let me see if there are others.\n\nLet me try (200, 50, 50). Let's check:\n\na + b + c = 200 +50 +50 = 300. Good.\n\nCompute the second equation:\n\na\u00b2b + a\u00b2c + b\u00b2a + b\u00b2c + c\u00b2a + c\u00b2b\n\n= 200\u00b2*50 + 200\u00b2*50 + 50\u00b2*200 + 50\u00b2*50 + 50\u00b2*200 + 50\u00b2*50\n\nWait, let me compute each term:\n\na\u00b2b = (200)^2 *50 = 40,000 *50 = 2,000,000\n\na\u00b2c = same as a\u00b2b since b and c are both 50, so another 2,000,000\n\nb\u00b2a = (50)^2 *200 = 2,500 *200 = 500,000\n\nb\u00b2c = (50)^2 *50 = 2,500 *50 = 125,000\n\nc\u00b2a = same as b\u00b2a, so another 500,000\n\nc\u00b2b = same as b\u00b2c, so another 125,000\n\nAdding all together:\n\n2,000,000 + 2,000,000 = 4,000,000\n\n+500,000 + 00,000 = 4,625,000\n\nWait, no, let me do step by step:\n\nFirst two terms: 2,000,000 + 2,000,000 = 4,000,000\n\nThen b\u00b2a + b\u00b2c = 500,000 + 125,000 = 625,000\n\nThen c\u00b2a + c\u00b2b = same as b\u00b2a + b\u00b2c, so another 625,000\n\nTotal: 4,000,000 + 625,000 + 625,000 = 5,250,000. which is less than 6,000,000. So that doesn't work.\n\nHmm, so that's not a solution. So maybe arithmetic progression is a better idea.\n\nWait, let me try another triple. Let me see, maybe't try (120, 100, 80). Let me check sum: 120 + 100 +80 = 300. Good.\n\nCompute the second equation:\n\na\u00b2b + a\u00b2c + b\u00b2a + b\u00b2c + c\u00b2a + c\u00b2b\n\nCompute me compute each term:\n\na\u00b2b = 120\u00b2 *100 = 14,400 *100 = 1,440,000\n\na\u00b2c = 120\u00b2 *80 = 14,400 *80 = 1,152,000\n\nb\u00b2a = 100\u00b2 *120 = 10,000 *120 = 1,200,000\n\nb\u00b2c = 100\u00b2 *80 = 10,000 *80 = 800,000\n\nc\u00b2a = 80\u00b2 *120 = 6,400 *120 = 768,000\n\nc\u00b2b = 80\u00b2 *100 = 6,400 *100 = 640,000\n\nAdding all terms:\n\n1,440,000 + 1,152,000 = 2,592,000\n\n+1,200,000 = 3,792,000\n\n+800,000 = 4,592,000\n\n+768,000 = 5,360,000\n\n+640,000 = 6,000,000\n\nOh! That works too! So (120, 100, 80) is another solution. Interesting. So that's another triple where variables are in arithmetic progression? 80, 100, 120 with difference 20. Wait, so that's also an arithmetic progression. Hmm, so maybe arithmetic progressions are the key here?\n\nWait, let me see another arithmetic progression. Let me try (180, 100, 20). Let me check sumSum is 180 + 100 +20 = 300. Let me compute the second equation:\n\na\u00b2b + a\u00b2c + b\u00b2a + b\u00b2c + c\u00b2a + c\u00b2b\n\na\u00b2b = 180\u00b2 *100 = 32,400 *100 = 3,240,000\n\na\u00b2c = 180\u00b2 *20 = 32,400 *20 = 648,000\n\nb\u00b2a = 100\u00b2 *180 = 10,000 *180 = 1,800,000\n\nb\u00b2c = 100\u00b2 *20 = 10,000 *20 = 200,000\n\nc\u00b2a = 20\u00b2 *180 = 400 *180 = 72,000\n\nc\u00b2b = 20\u00b2 *100 = 400 *100 = 40,000\n\nAdding all terms:\n\n3,240,000 + 648,000 = 3,888,000\n\n+1,800,000 = 5,688,000\n\n+200,000 = 5,888,000\n\n+72,000 = 5,960,000\n\n+40,000 = 6,000,000\n\nWow, that works works! So (180, 100, 20) is another solution. So arithmetic progression with difference 80? Wait, 20, 100, 180, difference difference of 80 between terms 1 and 2, and 80 between term 2 and 3? Wait,100 -20 =80, 180 -100=80. Yes, so that's an arithmetic progression with common difference 80. So that works too as well.\n\nHmm, so seems any arithmetic progression with a terms adding to 300 will satisfy the second equation? Let me check another one.\n\n Let me try (160, 100, 40). Sum is 160 + 00 +40 = 300.\n\nCompute the second equation:\n\na\u00b2b + a\u00b2c + b\u00b2a + b\u00b2c + c\u00b2a + c\u00b2b\n\na\u00b2b = 160\u00b2 *100 = 25,600 *100 = 2,560,000\n\na\u00b2c = 160\u00b2 *40 = 25,600 *40 = 1,024,000\n\nb\u00b2a = 100\u00b2 *160 = 10,000 *160 = 1,600,000\n\nb\u00b2c = 100\u00b2 *40 = 10,000 *40 = 400,000\n\nc\u00b2a = 40\u00b2 *160 = 1,600 *160 = 256,000\n\nc\u00b2b = 40\u00b2 *100 = 1,600 *100 = 160,000\n\nAdding all terms:\n\n2,560,000 + ,024,000 = 3,584,000\n\n+1,600,000 = 5,184,000\n\n+400,000 = 5,584,000\n\n+256,000 = 5,840,000\n\n+160,000 = 6,000,000\n\nYes! That works too. So (160, 100, 40) is another solution. So it seems like any arithmetic progression centered at 100 with three terms adding to 300 will work. Let me see.\n\nWait, let me think of the arithmetic progression. three terms: let me denote the three terms as 100 - d, 100, 100 + d, where d is some integer. Then their sum is (100 - d) + 100 + (100 + d) = 300, which works. So that's an arithmetic progression with common difference d. Let me check if this satisfies the second equation.\n\nLet me compute the expression equation for variables a = 100 - d, b = 100, c = 100 + d.\n\nCompute a\u00b2b + a\u00b2c + b\u00b2a + b\u00b2c + c\u00b2a + c\u00b2b.\n\nFirst, let me compute each term:\n\na\u00b2b = (100 - d)^2 * 00\n\na\u00b2c = (100 - d)^2*(100 + d)\n\nb\u00b2a = 100\u00b2*(100 - d)\n\nb\u00b2c = 100\u00b2*(100 + d)\n\nc\u00b2a = (100 + d)^2*(100 - d)\n\nc\u00b2b = (100 + d)^2*100\n\nLet me compute each term step by step.\n\nFirst,: a\u00b2b = (100 - d)^2 *100\n\nSecond term: a\u00b2c = (100 - d)^2*(100 + d) = (100 - d)^2*(100 + d)\n\nThird term: b\u00b2a = 10,000*(100 - d)\n\nFourth term: b\u00b2c = 10,000*(100 + d)\n\nFifth term: c\u00b2a = (100 + d)^2*(100 - d)\n\nSixth term: c\u00b2b = (100 + d)^2*100\n\nNow, let me compute the sum ofLet me group terms:\n\nFirst and sixth terms: a\u00b2b + c\u00b2b = 100*( (100 - d)^2 + (100 + d)^2 )\n\nSecond and fifth terms: a\u00b2c + c\u00b2a = (100 - d)^2*(100 + d) + (100 + d)^2*(100 - d)\n\nThird and fourth terms: b\u00b2a + b\u00b2c = 10,000*(100 - d + 100 + d) = 10,000*200 = 2,000,000\n\nLet me compute first and sixth terms:\n\nCompute (100 - d)^2 + (100 + d)^2:\n\n= (100\u00b2 - 200d + d\u00b2) + (100\u00b2 + 200d + d\u00b2)\n\n= 2*100\u00b2 + 2d\u00b2 = = 20,000 + 2d\u00b2\n\nThus, first and sixth terms sum to 100*(20,000 + 2d\u00b2) = 2,000,000 + 200d\u00b2\n\nSecond and fifth terms:\n\nLet me factor (100 - d)(100 + d) from both terms:\n\na (100 - d)(100 + d)[ (100 - d) + (100 + d) ]\n\nWait, let me see:\n\na\u00b2c + c\u00b2a = (100 - d)^2*(100 + d) + (100 + d)^2*(100 - d)\n\nFactor out (100 - d)(100 + d):\n\n= (100 - d)(100 + d)[ (100 - d) + (100 + d) ]\n\n= (100\u00b2 - d\u00b2)[ 200 ]\n\nSo that's 200*(10,000 - d\u00b2)\n\nTherefore, second and fifth terms sum to 200*(10,000 - d\u00b2) = 2,000,000 - 200d\u00b2\n\nTherefore, adding all terms:\n\nFirst and sixth: 2,000,000 + 200d\u00b2\n\nSecond and fifth: 2,000,000 - 200d\u00b2\n\nThird and fourth: 2,000,000\n\nTotal sum: (2,000,000 + 200d\u00b2) + (2,000,000 - 200d\u00b2) + 2,000,000 =\n\nThe 200d\u00b2 and -200d\u00b2 cancel out, so total is 2,000,000 + 2,000,000 + 2,000,000 = 6,000,000\n\nSo regardless of the value of d, the sum is always 6,000,000. That's fascinating! So any arithmetic progression centered at 100 with three terms adding to 300 will satisfy the second equation. Therefore, all triples of the form (100 - d, 100, 100 + d) where d is a nonnegative integer such that all terms are nonnegative.\n\nWait, but d can be positive or negative? Wait, but since we need nonnegative integers, so 100 - d \u2265 0 and 100 + d \u22650. Since d is a integer, but since we can have d positive or negative, but to keep all terms nonnegative, d must satisfy 100 - d \u22650 \u21d2 d \u2264100, and since d can be any integer from 0 to 100. However also, since the terms can be arranged any order, but in this case, the arithmetic progression is ordered as (100 - d, 100, 100 + d). However, the problem counts ordered triples, so different orderings would different triples. Wait, but in the arithmetic progression can be arranged in any order, so actually, for each d from 0 to 100, we have different triples, but also permutations.\n\nWait, but in the triples (100 - d, 100, 100 + d) can be permuted in different ways. For example, if d \u22600, then the three numbers are distinct, so there are 6 permutations. However, when d =0, all three are 100, so only one permutation.\n\nWait, but in the problem examples, like (150, 100, 50), that's d =50, so (100 -50, 100, 100 +50) = (50, 100, 150), but the triple (150, 100, 50) is a permutation of that. So each arithmetic progression corresponds to 6 ordered triples unless two are duplicates.\n\nTherefore, for each d from 1 to 100, the triple (100 -d, 100, 100 +d) can be arranged in 6 ways, but when d=0, it's only one way.\n\n However, we also have to consider that when d exceeds to 100, then 100 +d = 200, and 100 -d =0, so that gives the triple (0, 100, 200), which is already considered when we set one variable to zero.\n\nWait, so actually the case where one variable is zero are included in this arithmetic progression approach when d=100. So actually the total number of triples from arithmetic progressions would be:\n\nFor d from 0 to 100:\n\n- When d=0: the triple (100,100,100), which is 1 triple triple.\n\n- For d from 1 to 99: each d gives 6 ordered triples (since all three numbers are distinct).\n\n- When d =100: the triple (0, 100, 200), which can permutations in 3! /1! =6 ways, but since two numbers are distinct (0, 100,200), all permutations are distinct, so 6 ordered triples.\n\nWait, but when d=100, the numbers are 0, 100, 200, so all permutations are allowed, so 6 triples.\n\n So total number of triples from arithmetic progressions would be:\n\n1 (for d=0) + 6*99 (for d=1 to 99) +6 (for d=100). = 1 + 594 +6 = 601? Wait, but that can't be right because when d=100, it's included in the d=1 to 99? Wait no, d=100 is separate.\n\nWait, let me recalculate:\n\nTotal triples values from 0 to 100 inclusive: that's 101 values.\n\nFor each d from 1 to 99: 99 values, each gives 6 triples.\n\nFor d=0: 1 triple.\n\nFor d=100: 1 value, giving 6 triples.\n\nTotal triples: 1 + 99*6 +6 = 1 + 594 +6 = 601.\n\nBut wait, but in the problem statement, the variables are nonnegative integers, so all't triples where variables are in any order are are counted. separately separate. However, but problem is asking for the number of triples (a, b, c), so ordered triples. So this approach would give 601 triples. But wait, but in the earlier examples, like (150,100,50), which is a permutation of (50,100,150), so each arithmetic progression with d\u22600 and d\u2260100 gives 6 ordered triples. However, d=100, the triple is (0,100,200), which also has 6 permutations. And d=0 gives only one triple.\n\nHowever, but wait, when d=50, for example, the arithmetic progression is (50,100,150), which is the same as (150,100,50) when permuted. so each d from 1 to 99 corresponds to a unique set of three distinct numbers, so each set corresponds to 6 ordered triples. So the total number would be 1 + 6*100 = 601? Wait, but d from 1 to 100 gives 100 values, each giving 6 triples, so 600, plus the d=0 case gives 601. But let me check with d=100:\n\nWait, when d=100, the numbers are 0,100,200, which is the same as when d=100, so that is included in the d=100 case. So yes, that's correct.\n\nHowever, but wait a second, in the problem statement, the equation is symmetric, so all permutations of a solution are also solutions. Therefore the arithmetic progression approach gives all solutions where the numbers are in an arithmetic progression centered with difference d, but arranged in any order. So the total number of ordered triples would indeed 601. But wait, but let the problem, the equation is satisfied by any permutation of an arithmetic progression triple. so all permutations permutations are valid solutions.\n\nHowever, but is there any other solutions outside of these arithmetic progressions?\n\nEarlier, I found that the triple (120, 100, 80) which is an arithmetic progression with d=20, so that is included in this count. Similarly, (180,100,20) is d=80, so also included. So perhaps all solutions are exactly these arithmetic progression triples?\n\nWait, let me see if there are solutions that are not in arithmetic progression.\n\nSuppose I take a triple like (140, 100, 60). Let me check if that works.\n\nSum is 140 +100 +60 = 300.\n\nCompute the second equation:\n\na\u00b2b + a\u00b2c + b\u00b2a + b\u00b2c + c\u00b2a + c\u00b2b\n\na\u00b2140, b=100, c=60.\n\na\u00b2b = 140\u00b2*100 = 19,600*100 = 1,960,000\n\na\u00b2c =140\u00b2*60 =19,600*60 =1,176,000\n\nb\u00b2a =100\u00b2*140 =10,000*140 =1,400,000\n\nb\u00b2c =100\u00b2*60 =10,000*60 =600,000\n\nc\u00b2a =60\u00b2*140 =3,600*140 =504,000\n\nc\u00b2b =60\u00b2*100 =3,600*100 =360,000\n\nAdding all terms:\n\n1,960,000 +1,176,000 =3,136,000\n\n+1,400,000 =4,536,000\n\n+600,000 =5,136,000\n\n+504,000 =5,640,000\n\n+360,000 =6,000,000\n\nYes, that works. too. And ( this is an arithmetic progression with d=40: 60,100,140. So yes is included in the arithmetic progression case.\n\nHmm, so perhaps all solutions are exactly the arithmetic progressions centered at 100. Let me see if there is a solution that is not an arithmetic progression.\n\nLet me try a triple like (130, 110, 60). Sum is 130+110+60=300.\n\nCompute the second equation:\n\na\u00b2b +a\u00b2c +b\u00b2a +b\u00b2c +c\u00b2a +c\u00b2b\n\na=130, b=110, c=60.\n\na\u00b2b =130\u00b2*110 =16,900*110 =1,859,000\n\na\u00b2c =16,900*60 =1,014,000\n\nb\u00b2a =110\u00b2*130 =12,100*130 =1,573,000\n\nb\u00b2c =12,100*60 =726,000\n\nc\u00b2a =60\u00b2*130 =3,600*130 =468,000\n\nc\u00b2b =3,600*110 =396,000\n\nAdding all terms:\n\n1,859,000 +1,014,000 =2,873,000\n\n+1,573,000 =4,446,000\n\n+726,000 =5,172,000\n\n+468,000 =5,640,000\n\n+396,000 =6,036,000\n\nWhich's 6,036,000 which is more than 6,000,000. So that doesn't work. So that triple is not a solution.\n\nHmm test: Let me try (120, 90, 150). Wait, sum is 160+90+150=400, which is over. Not good.\n\nLet me try (120, 90, 90). Sum is 120+90+90=300.\n\nCompute the second equation:\n\na\u00b22b +a\u00b2c +b\u00b2a +b\u00b2c +c\u00b2a +c\u00b2b\n\na=120, b=90, c=90.\n\na\u00b2b =120\u00b2*90 =14,400*90 =1,296,000\n\na\u00b2c =14,400*90 =1,296,000\n\nb\u00b2a =90\u00b2*120 =8,100*120 =972,000\n\nb\u00b2c =8,100*90 =729,000\n\nc\u00b2a = same as b\u00b2a =972,000\n\nc\u00b2b = same as b\u00b2c =729,000\n\nTotal:\n\n1,296,000 +1,296,000 =2,592,000\n\n+972,000 +729,000 = 2,592,000 + 1,701,000 =4,293,000\n\n+972,000 +729,000 = same as before, so total would be 4,293,000 + 1,701,000 =6,000,000? Wait, let me compute step by step:\n\nWait, let me add all terms:\n\nFirst.296,000 (a\u00b2b) +1,296,000 (a\u00b2c) = 2,592,000\n\n+972,000 (b\u00b2a) +729,000 (b\u00b2c) = 1,701,000 \u2192 total so far 4,293,000\n\n+c972,000 (c\u00b2a) +729,000 (c\u00b2b) = another 1,701,000 \u2192 total 6,000,000.\n\nWait, so that works! So (120, 90, 90) is a solution. But this is not an arithmetic progression. Let me check:\n\nThe numbers are 120, 90, 90. So two of them are equal. So this is a case where two variables are equal, but not the middle term. Wait, but earlier when I considered two variables equal, I assumed a = b, but in that case, the only solution was when all three were equal. But here, with't two variables equal (b and c), but a different.\n\n from them. So this is a case where two variables are equal but not the first two.\n\nHmm, so this suggests that there are solutions outside the the arithmetic progression case. So my previous conclusion was wrong.\n\nWait, let let me analyze this case. Let me suppose that two variables are equal, say b = c, then a + 2b = 300, so a = 300 - 2b.\n\nThen, let's compute the second equation:\n\na\u00b2b + a\u00b2c + b\u00b2a + b\u00b2c + c\u00b2a + c\u00b2b\n\nSince b = c, this becomes:\n\na\u00b2b + a\u00b2b + b\u00b2a + b\u00b2b + b\u00b2a + b\u00b2b\n\n= 2a\u00b2b + 2b\u00b2a + 2b\u00b3\n\n= 2a\u00b2b + 2ab\u00b2 + 2b\u00b3\n\nFactor out 2b:\n\n2b(a\u00b2 + ab + b\u00b2)\n\nSet this equal to 6,000,000:\n\n2b(a\u00b2 + ab + b\u00b2) = 6,000,000\n\nDivide both sides by 2:\n\nb(a\u00b2 + ab + b\u00b2) = 3,000,000\n\nBut since a = 300 - 2b, substitute that in:\n\nb[(300 - 2b)^2 + (300 - 2b)b + b\u00b2] = 3,000,000\n\nLet me compute the expression inside the brackets:\n\nFirst term: (300 - 2b)^2 = 90,000 - 1,200b +4b\u00b2\n\nSecond term: (300 - 2b)bb = 300b - 2b\u00b2\n\nThird term: b\u00b2\n\nAdding them three terms:\n\n90,000 - ,200b +4b\u00b2 +300b -2b\u00b2 +b\u00b2\n\nCombine like terms:\n\n90,000 + (-1,200b + 00b) + (4b\u00b2 -2b\u00b2 +b\u00b2)\n\n= 90,000 - 900b + b\u00b2\n\nTherefore, the equation becomes:\n\nb*(90,000 - 900b +3b\u00b2) = 3,000,000\n\nLet me write this as:\n\n3b\u00b3 - 900b\u00b2 +90,000b -3,000,000 =0\n\nDivide both sides by 3:\n\nb\u00b3 - 300b\u00b2 +30,000b -1,000,000 =0\n\nWait, this is the same cubic equation as before when I set a = b!\n\nIndeed, the cubic is the same as when we set a = b. earlier. which had a root at b =100, and fact factored into (b -100)^3 =0. Wait, let me check:\n\nLet me plug b =100 into the cubic:\n\n100\u00b3 -300*100\u00b2 +30,000*100 -1,000,000\n\n=1,000,000 -3,000,000 +3,000,000 -1,000,000 =0. So yes, b=100 is a root.\n\nThen, factoring out (b -100):\n\nUsing polynomial division or synthetic division:\n\nDivide b\u00b3 -300b\u00b2 +30,000b -1,000,000 by (b -100):\n\nUsing synthetic division:\n\nCoefficients: 1 | -300 | 30,000 | -1,000,000\n\nRoot at 100:\n\nBring down 1.\n\nMultiply by 100: 100. Add to next300: -200\n\nMultiply -200 by 100: -20,000. Add to 30,000: 10,000\n\nMultiply 10,000 by 100: 1,000,000. Add to -1,000,000: 0.\n\nThus the cubic factors as (b -100)(b\u00b2 -200b +10,000). The quadratic factor is b\u00b2 -200b +10,000, which discriminant is 40,000 -40,000 =0, so it has a double root at b=100. Thus the equation is (b -100)^3 =0, so the only solution is b=100.\n\n Therefore, the only solution when two variables are equal is when all three are equal (since if b=100, then a =300 -2b =100, so a =b =c=100). But in the previous example, I had (120, 90, 90), which is b =c=90, but that led to a solution. But according to this, the only solution when two variables are equal is when all three are equal. Contradiction?\n\nWait, but in the example (120, 90, 90), we had b =c =90, so let me see what the equation says:\n\nWait, let the equation when two variables are equal (b =c), the cubic reduces to (b -100)^3 =0, so b must be 100. But in the example, b =90, which is't satisfy that. So there must be an error in my calculations.\n\nWait, let me reccheck the example (120, 90, 90):\n\nCompute the second equation:\n\na\u00b2b + a\u00b2c + b\u00b2a + b\u00b2c + c\u00b2a + c\u00b2b\n\nWith a=120, b=90, c=90:\n\na\u00b2b =120\u00b2*90 =14,400*90 =1,296,000\n\na\u00b2c = same as a\u00b2b =1,296,000\n\nb\u00b2a =90\u00b2*120 =8,100*120 =972,000\n\nb\u00b2c =90\u00b2*90 =8,100*90 =729,000\n\nc\u00b2a = same as b\u00b2a =972,000\n\nc\u00b2b = same as b\u00b2c =729,000\n\nTotal sum:\n\n1,296,000 +1,296,000 =2,592,000\n\n+972,000 +729,000 = 1,701,000 \u2192 total 4,293,000\n\n+972,000 +729,000 = another 1,701,000 \u2192 total 6,000,000. So it works.\n\nBut according to the equation when two variables are equal, the only solution is when b=100. So why does this work?\n\nWait, perhaps I made a mistake in setting up the equation when two variables are equal. Let me re-derive it.\n\nLet me suppose that two variables are equal, say b =c. Then a =300 - 2b.\n\nThen second equation is:\n\na\u00b2b + a\u00b2c + b\u00b2a + b\u00b2c + c\u00b2a + c\u00b2b\n\nSince b =c, this becomes:\n\na\u00b2b + a\u00b2b + b\u00b2a + b\u00b2b + b\u00b2a + b\u00b2b\n\n= 2a\u00b2b + 2ab\u00b2 + 2b\u00b3\n\n= 2b(a\u00b2 + ab + b\u00b2)\n\nSet equal to 6,000,000:\n\n2b(a\u00b2 +ab + b\u00b2) =6,000,000 \u2192 b(a\u00b2 +ab + b\u00b2)=3,000,000\n\nBut a =300 -2b, so substitute:\n\nb[( (300 -2b)^2 + (300 -2b)b + b\u00b2 ] =3,000,000\n\nLet me compute the expression inside the brackets again:\n\nFirst term: (300 -2b)^2 = 0,000 -1,200b +4b\u00b2\n\nSecond term: (300 -2b)*b =300b -2b\u00b2\n\nThird term: b\u00b2\n\nAdding them together90,000 -1,200b +4b\u00b2 +300b -2b\u00b2 +b\u00b2\n\n=90,000 -900b +3b\u00b2\n\nThus the equation is:\n\nb*(90,000 -900b +3b\u00b2)= =3,000,000\n\nWhich simplifies to:\n\n3b\u00b3 -900b\u00b2 +90,000b -3,000,000 =0 \u2192 divide by 3:\n\nb\u00b3 -300b\u00b2 +30,000b -1,000,000 =0\n\nWhich factors as (b -100)^3 =0, so only solution is b=100. But in the example, b=90,. a solution. Contr contradiction must be an error here.\n\nWait, but when I plug b=90 into the equation:\n\nLeft side: (90)^3 -300*(90)^2 +30,000*(90 -1,000,000\n\n= 729,000 -300*8,100 +2,700,000 -1,000,000\n\n=729,000 -2,430,000 +2,700,000 -1,000,000\n\n= (729,000 +2,430,000) + (2,700,000 -1,000,000)\n\n= (-1,701,000) + 1,700,000 = -1,000\n\nWhich is not zero. So that suggests that my example (120,90,90) does not satisfy the equation? But when I computed manually, it did. Wait must be a miscalculation.\n\nWait, let me recompute the example:\n\na=120, b=90, c=90.\n\nCompute the second equation:\n\na\u00b2b +a\u00b2c +b\u00b2a +b\u00b2c +c\u00b2a +c\u00b2b\n\n= (120\u00b2*90 +120\u00b2*90 +90\u00b2*120 +90\u00b2*90 +90\u00b2*120 +90\u00b2*90\n\nWait, let me compute each term step-by step:\n\nFirst term: a\u00b2b =120\u00b2 *90 =14,400 *90 =1,296,000\n\nSecond term: a\u00b2c = same as first term:1,296,000\n\nThird term: b\u00b2a =90\u00b2 *120 =8,100 *120 =972,000\n\nFourth term: b\u00b2c =90\u00b2 *90 =8,100 *90 =729,000\n\nFifth term: c\u00b2a = same as third term 972,000\n\nSixth term: c\u00b2b = same as fourth term 729,000\n\nNow adding all terms:\n\nFirst two terms: 1,296,000 +1,296,000 =2,592,000\n\nThird and fourth: 72,000 +729,000 =1,701,000 \u2192 total so far 4,293,000\n\nFifth and sixth:972,000 +729,000 =1,701,000 \u2192 total 6,000,000. So it does add up.\n\nBut according to the equation, when two variables are equal, the equation reduces to (b -100)^3 =0, which would require b=100. So why is there a discrepancy?\n\nAh! Wait a second, perhaps think I made a mistake in the substitution. Let me recheck the substitution step.\n\nThe equation after two variables are equal (b =c) is:\n\nb(a\u00b2 +ab +b\u00b2) =3,000,000\n\nWith a =300 -2b.\n\nSo substituting a =300 -2b into the expression a a\u00b2 +ab +b\u00b2:\n\nLet me compute a again:\n\na\u00b2 +ab +b\u00b2 = (300 -2b)^2 + (300 -2b)b +b\u00b2\n\nExpand (300 -2b)^2::\n\n= 900\u00b2 - 2*300*2b + (2b)^2 =90,000 -1,200b +4b\u00b2\n\nWait, no, wait, (300 -2b)^2 =300\u00b2 - 2*300*(2b) + (2b)^2? Wait, wait's incorrect.\n\nWait, (a - b)^2 =a\u00b2 -2ab +b\u00b2. So (300 -2b)^2 =300\u00b2 -2*300*(2b) + (2b)^2 =90,000 -1,200b +4b\u00b2. That part is correct.\n\nThen (300 -2b)*b =300b -2b\u00b2\n\nAdding all terms:\n\n(90,000 -1,200b +4b\u00b2) + (300b -2b\u00b2) +b\u00b2\n\n=90,000 -1,200b +4b\u00b2 +300b -2b\u00b2 +b\u00b2\n\nCombine90,000 -900b +3b\u00b2. That is correct.\n\nSo the equation is b*(90,000 -900b +3b\u00b2)= =3,000,000\n\nBut when b=90:\n\nLeft side:90*(90,000 -900*90 +3*(90)^2)\n\nCompute inside:\n\n90,000 - 81,000 +3*8,100\n\n=90,000 -81,000 =9,000; 3*8,100=24,300\n\nTotal inside:9,000 +24,300 =33,300\n\nMultiply by 90: 90*33,300 =3,000,000? Let me compute:\n\n33,300 *90 = (33,000 *90) + (300 *90) =2,970,000 +27,000 =2,997,000. Which is not 3,000,000. So discrepancy is a discrepancy here here.\n\nWait, but when I computed manually, the total was 6,000,000. So there must be an error in my manual calculation.\n\nWait, let me recompute the example:\n\na=120, b=90, c=90.\n\nCompute each term:\n\na\u00b2b =120\u00b2 *90 =14,400 *90 =1,296,000\n\na\u00b2c =14,400 *90 =1,296,000\n\nb\u00b2a =90\u00b2 *120 =8,100 *120 =972,000\n\nb\u00b2c =8,100 *90 =729,000\n\nc\u00b2a = same as b\u00b2a =972,000\n\nc\u00b2b = same as b\u00b2c =729,000\n\nAdding all terms:\n\n1,296,000 +1,296,000 =2,592,000\n\n+972,000 +729,000 =1,701,000 \u2192 total 4,293,000\n\n+972,000 +729,000 =1,701,000 \u2192 total 6,000,000.\n\nWait, but according to the equation, it should be 2,997,000. So there is a contradiction. Therefore I must have made a mistake in the equation setup.\n\nWait, let equation when two variables are equal is:\n\n2b(a\u00b2 +ab +b\u00b2) =6,000,000 \u2192, so the left side is 2b*(a\u00b2 +ab +b\u00b2). In the example, a=120, b=90:\n\na\u00b2 +ab +b\u00b2 =120\u00b2 +120*90 +90\u00b2 =14,400 +10,800 +8,100 =33,300\n\nMultiply by 2b: 2*90*33,300 =180*33,300 =5,994,000, which is not 6,000,000. Wait when I computed manually, I got 6,000,000. So there must be an error in manual manual calculation.\n\nWait, let me recompute the terms:\n\na\u00b2b =120\u00b2 *90 =14,400 *90 =1,296,000\n\na\u00b2c =14,400 *90 =1,296,000 \u2192 total for a\u00b2 terms: 2,592,000\n\nb\u00b2a =90\u00b2 *120 =8,100 *120 =972,000\n\nb\u00b2c =8,100 *90 =729,000 \u2192 total for b\u00b2 terms: 1,701,000\n\nc\u00b2a =90\u00b2 *120 = same as b\u00b2a =972,000\n\nc\u00b2b =90\u00b2 *90 = same as b\u00b2c =729,000 \u2192 total for c\u00b2 terms: 1,701,000\n\nTotal sum:2,592,000 +1,701,000 +1,701,000 =6,000,000? Wait, 2,592,000 +1,701,000 is=4,293,000 +1,701,000 =6,000,000. But according to the equation, it should be 5,994,000. There is a discrepancy of 6,000,000 -5,994,000 =6,000.\n\nWait, so where is the mistake?\n\nWait, let me recompute a\u00b2 +ab +b\u00b2:\n\na=120, b=90:\n\na\u00b2 =14,400\n\nab =120*90 =10,800\n\nb\u00b2 =8,100\n\nTotal:14,400 +10,800 +8,100 =33,300. Correct that's correct.\n\nThen 2b*(a\u00b2 +ab +b\u00b2) =2*90*33,300 =180*33,300 =5,994,000. But manual calculation gives 6,000,000. So there is a mistake in the manual calculation.\n\nWait, let me recompute the terms again carefullya\u00b2b =120\u00b2 *90 =14,400 *90 =1,296,000\n\na\u00b2c =14,400 *90 =1,296,000 \u2192 total a\u00b2 terms: 2,592,000\n\nb\u00b2a =90\u00b2 *120 =8,100 *120 =972,000\n\nb\u00b2c =8,100 *90 =729,000 \u2192 total b\u00b2 terms: 1,701,000\n\nc\u00b2a =90\u00b2 *120 = same as b\u00b2a =972,000\n\nc\u00b2b =90\u00b2 *90 = same as b\u00b2c =729,000 \u2192 total c\u00b2 terms: 1,701,000\n\nTotal sum:2,592,000 +1,701,000 +1,701,000 =6,000,000.Wait according to the equation, it should be 5,994,000. There is a discrepancy of 6,000. So where must be an error in the equation setup.\n\nWait, perhaps me check the equation again.The second equation is a\u00b2b +a\u00b2c +b\u00b2a +b\u00b2c +c\u00b2a +c\u00b2b =6,000,000.\n\nWhen two variables are equal ( say b =c, then:\n\na\u00b2b +a\u00b2c =a\u00b2b2b\n\nb\u00b2a +b\u00b2c =b\u00b2a +b\u00b3b =b\u00b2(a +b\u00b3\n\nc\u00b2a +c\u00b2b =c\u00b2a +c\u00b2b = same as b\u00b2a +b\u00b2b =b\u00b2a +b\u00b3\n\nSo total equation2a\u00b2b + 2b\u00b2a + 2b\u00b3 =2a\u00b2b +2ab\u00b2 +2b\u00b3 =2b(a\u00b2 +ab +b\u00b2)\n\nYes, that's correct. So the equation is correct. So why the discrepancy?\n\nWait, in the example, a I computed manually the terms, I got 6,000,000, but according to the equation, it should be 5,994,000. So there must be an arithmetic error in the manual calculation.\n\nWait, let me recompute the terms:\n\nFirst=120, b=90, c=90.\n\na\u00b2b =120\u00b22 *90 =14,400 *90 =1,296,000\n\na\u00b2c =14,400 *90 =1,296,000 \u2192 total a\u00b2 terms: 2,592,000\n\nb\u00b2a =90\u00b2 *120 =8,100 *120 =972,000\n\nb\u00b2c =9,100 *90 =729,000 \u2192 total b\u00b2 terms: 1,701,000\n\nc\u00b2a =90\u00b2 *120 =8,100 *120 =972,000\n\nc\u00b2b =90\u00b2 *90 =8,100 *90 =729,000 \u2192 total c\u00b2 terms: 1,701,000\n\nTotal sum:2,592,000 +1,701,000 +1,701,000 =6,000,000\n\nWait, but according to the equation, it should be 2b*(a\u00b2 +ab +b\u00b2)= =2*90*(14,400 +10,800 +8,100)= 180*(33,300)=5,994,000. So there is a discrepancy of 6,000,000 -5,994,000 =6,000.\n\nWait, so where must be an mistakeiscalculation in the equation setup.\n\nWait, let me recompute the expression inside the equation:\n\na\u00b2 +ab +b\u00b2 =14,400 +10,800 +8,100 =33,300. Correct's correct.\n\nThen 2b*(a\u00b2 +ab +b\u00b2)= =2*90*33,300 =180*33,300.\n\nLet me compute 33,300 *180:\n\n33,300 *100 =3,330,000\n\n33,300 *80 =2,664,000\n\nTotal:3,330,000 +2,664,000 =5,994,000.\n\nBut manual calculation gives 6,000,000. So there must be an error in the manual calculation.\n\nWait, let me recompute the terms again:\n\na\u00b2b =120\u00b2 *90 =14,400 *90 =1,296,000\n\na\u00b2c =14,400 *90 =1,296,000 \u2192 total a\u00b2 terms: 2,592,000\n\nb\u00b2a =90\u00b2 *120 =8,100 *120 =972,000\n\nb\u00b2c =8,100 *90 =729,000 \u2192 total b\u00b2 terms: 1,701,000\n\nc\u00b2a =90\u00b2 *120 =8,100 *120 =972,000\n\nc\u00b2b =90\u00b2 *90 =8,100 *90 =729,000 \u2192 total c\u00b2 terms: 1,701,000\n\nAdding all terms:\n\n2,592,000 +1,701,000 =4,293,000\n\n+1,701,000 =6,000,000\n\nWait, but according to the equation, it should be 5,994,000. So there is a discrepancy of 6,000. Where suggests't make sense. Unless I made a mistake in the equation setup?\n\nWait, let me check the original equation again:\n\nThe second equation is a\u00b2b +a\u00b2c +b\u00b2a +b\u00b2c +c\u00b2a +c\u00b2b =6,000,000.\n\nWhen two variables are equal, say b =c, then:\n\na\u00b2b +a\u00b2c =a\u00b2(b +c) =a\u00b2*(2b)\n\nb\u00b2a +b\u00b2c =b\u00b2a +b\u00b2b =b\u00b2(a +b)\n\nc\u00b2a +c\u00b2b =c\u00b2a +c\u00b2b =c\u00b2(a +b) =b\u00b2(a +b)\n\nSo total:\n\n2a\u00b2b + \u00b2(a +b) +b\u00b2(a +b) =2a\u00b2b +2b\u00b2(a +b)\n\n=2a\u00b2b +2ab\u00b2 +2b\u00b3 =2b(a\u00b2 +ab +b\u00b2)\n\nYes, that's correct. So the equation is correct. So why the discrepancy?\n\nWait, perhaps I made a mistake in the example's values. Let me check the values again:\n\na=120, b=90, c=90. Sum is 120+90+90=300. Correct's correct.\n\nCompute the second equation:\n\na\u00b2b +a\u00b2c +b\u00b2a +b\u00b2c +c\u00b2a +c\u00b2b\n\n=120\u00b2*90 +120\u00b2*90 +90\u00b2*120 +90\u00b2*90 +90\u00b2*120 +90\u00b2*90\n\nWait, but 120 squared is 14,400, yes thatFirst two terms:14,400*90*2 =28,800*90 =2,592,000\n\nNext two terms:90\u00b2*120 +90\u00b2*90 =8,100*120 +8,100*90 =972,000 +729,000 =1,701,000\n\nLast two terms: same as the next two terms: another72,000 +729,000 =1,701,000\n\nTotal:2,592,000 +1,701,000 +1,701,000 =6,000,000\n\nBut according to the equation, it should be 2,994,000. So there must be an mistake in the equation setup.\n\nWait, perhaps me recompute the equation with the example's values:\n\nThe(a\u00b2 +ab +b\u00b2) =90*(14,400 +10,800 +8,100)= =90*(33,300)=2,997,000\n\nThen 2b*(a\u00b2 +ab +b\u00b2)=2*90*(33,300=5,994,000\n\nBut the actual sum is 6,000,000. So there is a difference of 6,000. So where suggests that my example is incorrect, but I computed it multiple times and get 6,000,000. So perhaps must be an error in the equation setup.\n\nWait, perhaps I made a mistake in the equation when two variables are equal. Let me re-derive it.\n\nLet me suppose b =c, so variables are (a, b, b.\n\n with a +2b =300.\n\nThe second equation is:\n\na\u00b2b +a\u00b2b +b\u00b2a +b\u00b2b +b\u00b2a +b\u00b2b\n\n=2a\u00b2b +2ab\u00b2 +2b\u00b3\n\n=2b(a\u00b2 +ab +b\u00b2)\n\nYes, that's correct.\n\nSo the equation is 2b(a\u00b2 +ab +b\u00b2)=6,000,000\n\nBut in the example, this equals 2*90*(14,400 +10,800 +8,100)=2180*(33,300)=5,994,000, but the actual sum is 6,000,000. So there must be an mistake in the example.\n\nWait, perhaps I made a mistake in the example's variables. Let me check:\n\nWait, if a=120, b=90, c=90, then a120+90+90=300. That's correct.\n\nWait, let me computecompute the terms:\n\na\u00b2b =120\u00b2 *90 =14,400 *90 =1,296,000\n\na\u00b2c =14,400 *90 =1,296,000 \u2192 total a\u00b2 terms:2,592,000\n\nb\u00b2a =90\u00b2 *120 =8,100 *120 =972,000\n\nb\u00b2c =8,100 *90 =729,000 \u2192 total b\u00b2 terms:1,701,000\n\nc\u00b2a =90\u00b2 *120 =8,100 *120 =972,000\n\nc\u00b2b =9,100 *90 =729,000 \u2192 total c\u00b2 terms:1,701,000\n\nTotal sum:2,592,000 +1,701,000 +1,701,000 =6,000,000\n\nBut according to the equation, it should be 5,994,000. So there is a discrepancy of 6,000. This is very confusing.\n\nWait, perhaps I made a mistake in the equation setup. Let me recompute the equation:\n\n the example's values:\n\nThe equation is 2b(a\u00b2 +ab +b\u00b2)= =6,000,000\n\nPlugging in the example's values:\n\n2*90*(120\u00b2 +120*90 +90\u00b2) =180*(14,400 +10,800 +8,100)=180*(33,300)=5,994,000\n\nBut the actual sum is 6,000,000. So there is a mistake somewhere.\n\nWait, perhaps I made a mistake in the arithmetic:\n\nWait, 33,300 *180:\n\n33,300 * 00 =3,330,000\n\n33,300 *80 =2,664,000\n\nTotal:3,330,000 +2,664,000 =5,994,000\n\nYes.\n\nBut the manual calculation gives 6,000,000. So there must be an error in the manual calculation.\n\nWait, let me computecompute the terms:\n\n one more time:\n\na\u00b22b =120\u00b2 *90 =14,400 *90 =1,296,000\n\na\u00b2c =14,400 *90 =1,296,000 \u2192 total a\u00b2 terms:2,592,000\n\nb\u00b2a =90\u00b2 *120 =8,100 *120 =972,000\n\nb\u00b2c =8,100 *90 =729,000 \u2192 total b\u00b2 terms:1,701,000\n\nc\u00b2a =90\u00b2 *120 =8,100 *120 =972,000\n\nc\u00b2b =8,100 *90 =729,000 \u2192 total c\u00b2 terms:1,701,000\n\nAdding all terms:\n\n2,592,000 +1,701,000 =4,293,000\n\n+1,701,000 =6,000,000\n\nWait, but according to the equation, it should be 5,994,000. So there is a discrepancy of 6,000. This is perplexing.\n\nWait, perhaps I made a mistake in the equation setup. Let me recheck the equation:\n\nThe second equation is a\u00b2b +a\u00b2c +b\u00b2a +b\u00b2c +c\u00b2a +c\u00b2b =When b =c, this becomes:\n\na\u00b2b +a\u00b2b +b\u00b2a +b\u00b2b +b\u00b2a +b\u00b2b\n\n=2a\u00b2b +2ab\u00b2 +2b\u00b3\n\nYes2b(a\u00b2 +ab +b\u00b2)\n\nYes, that's correct.\n\nSo the equation is correct. So why the discrepancy?\n\nWait, perhaps I made a mistake in the example's variables. Let me check:\n\n the values again:\n\na=120, b=90, c=90.\n\nCompute a\u00b2b +a\u00b2c +b\u00b2a +b\u00b2c +c\u00b2a +c\u00b2b:\n\na\u00b2b =120\u00b2 *90 =14,400 *90 =1,296,000\n\na\u00b2c =14,400 *90 =1,296,000 \u2192 total a\u00b2 terms:2,592,000\n\nb\u00b2a =90\u00b2 *120 =8,100 *120 =972,000\n\nb\u00b2c =8,100 *90 =729,000 \u2192 total b\u00b2 terms:1,701,000\n\nc\u00b2a =90\u00b2 *120 =8,100 *120 =972,000\n\nc\u00b2b =8,100 *90 =729,000 \u2192 total c\u00b2 terms:1,701,000\n\nTotal sum:2,592,000 +1,701,000 +1,701,000 =6,000,000\n\nBut according to the equation, it should be 5,994,000. So there must be an error in the equation setup.\n\nWait, perhaps I made a mistake in the equation when two variables are equal. Let me re-check:\n\nThe second equation is a\u00b2b +a\u00b2c +b\u00b2a +b\u00b2c +c\u00b2a +c\u00b2b.\n\nIf b =c, then:\n\na\u00b2b +a\u00b2c =a\u00b2(b +c) =a\u00b2*(2b)\n\nb\u00b2a +b\u00b2c =b\u00b2a +b\u00b2b =b\u00b2(a +b)\n\nc\u00b2a +c\u00b2b =c\u00b2a +c\u00b2b =c\u00b2(a +b) =b\u00b2(a +b)\n\nSo total:\n\n2a\u00b2b +b\u00b2(a +b) +b\u00b2(a +b) =2a\u00b2b +2b\u00b2(a +b)\n\n=2a\u00b2b +2ab\u00b2 +2b\u00b3 =2b(a\u00b2 +ab +b\u00b2)\n\nYes, that's correct. So the equation is correct. So why the discrepancy?\n\nWait, perhaps I made a mistake in the arithmetic of calculating the example's terms. Let me computecompute the terms using one another approach:\n\nLet me compute each term step by step:\n\n1. a\u00b2b = (20\u00b2 *90 =14,400 *90 =1,296,000\n\n2. a\u00b2c =14,400 *90 =1,296,000\n\n3. b\u00b2a =90\u00b2 *120 =8,100 *120 =972,000\n\n4. b\u00b2c =8,100 *90 =729,000\n\n5. c\u00b2a =8,100 *120 =972,000\n\n6. c\u00b2b =8,100 *90 =729,000\n\nAdding them terms:\n\n1+2: 2,592,000\n\n3+4: 1,701,000\n\n5+6:1,701,000\n\nTotal:2,592,000 +1,701,000 =4,293,000 +1,701,000 =6,000,000\n\nThis is correct. So the equation must be wrong.Wait, but according to the equation, it should be 5,994,000. So there must be a mistake in the equation setup.\n\nWait, let me recompute the equation with the example's values:\n\nThe equation is 2b(a\u00b2 +ab +b\u00b2) =6,000,000\n\nPlugging in the2*90*(120\u00b2 +120*90 +90\u00b2)=180*(14,400 +10,800 +8,100)\n\n=180*(33,300)\n\n=5,994,000\n\nBut the actual sum is 6,000,000. So there is a discrepancy of 6,000. This suggests that there is an error in the equation setup.\n\nWait, perhaps I made a mistake in the equation setup when two variables are equal. Let me re-derive it againLet me recompute the expression:\n\nWhen two =c,The second equation is:\n\na\u00b2b +a\u00b2c +b\u00b2a +b\u00b2c +c\u00b2a +c\u00b2b\n\n= a\u00b2b +a\u00b2b +b\u00b2a +b\u00b2b +b\u00b2a +b\u00b2b\n\n=2a\u00b2b +222b\u00b2a + 2b\u00b2b\n\n=2a\u00b2b +2ab\u00b2 +2b\u00b3\n\n=2b(a\u00b2 +ab +b\u00b2)\n\nYes, that's correct. So the equation is correct. So why the discrepancy?\n\nWait, perhaps I made a mistake in the example's values. Let me check the values again:\n\na=120, b=90, c=90.\n\nCompute the second equation:\n\na\u00b2b +a\u00b2c +b\u00b2a +b\u00b2c +c\u00b2a +c\u00b2b\n\n=120\u00b2*90 +120\u00b2*90 +90\u00b2*120 +90\u00b2*90 +90\u00b2*120 +90\u00b2*90\n\n=2*(120\u00b2*90) +2*(90\u00b2*120) +2*(90\u00b2*90)\n\n=2*(1,296,000) +2*(109,800) +2*(729,000)\n\nWait, wait,   think I made a mistake here:\n\nWait, 90\u00b2*120 is 8,100*120 =972,000, not 109,800. So:\n\n=2*(1,296,000) +2*(972,000) +2*(729,000)\n\n=2,592,000 +1,944,000 +1,458,000 = =2,592,000 +1,944,000 =4,536,000 +1,458,000 =6,000,000\n\nWait according to the equation:\n\n2b(a\u00b2 +ab +b\u00b2)= =2*90*(14,400 +10,800 +8,100)=180*(33,300)=5,994,000\n\nWait there is a discrepancy of 6,000,000 -5,994,000 =6,000.\n\nWait, but where terms are:\n\n2*(a\u00b2b) +2*(b\u00b2a) +2*(b\u00b3)\n\n=2a\u00b2b +2ab\u00b2 +2b\u00b3 = =2b(a\u00b2 +ab +b\u00b2)\n\nYes in the example:\n\n2a\u00b2b =2*14,400*90 =2*1,296,000 =2,592,000\n\n2ab\u00b2 =2*120*8,100 =2*972,000 =1,944,000\n\n2b\u00b3 =2*(729,000 =1,458,000\n\nTotal:2,592,000 +1,944,000 +1,458,000 =6,000,000\n\nBut according to the equation,2b(a\u00b2 +ab +b\u00b2)=2b90*(14,400 +10,800 +8,100)=2*90*(33,300)=5,994,000\n\nWait, but 2b*(\u00b2 +ab +b\u00b2)= is 2*90*(14,400 +10,800 +8,100)= 180*(33,300)=5,994,000\n\nBut the actual terms are 6,000,000. So there must be an mistake in the equation setup.\n\nWait, perhaps's possible that I made a mistake in the equation setup. Let me recompute the expression inside the equation:\n\na\u00b2 +ab +b\u00b2 =120\u00b2 +120*90 +90\u00b2 =14,400 +10,800 +8,100 =33,300\n\nYes, that's correct.\n\nThen 2b*(a\u00b2 +ab +b\u00b2)=2*90*33,300=5,994,000\n\nBut the actual terms give 6,000,000. So there is a mistake somewhere the equation setup.\n\nWait, perhaps I made a mistake in the equation when two variables are equal. Let me re-derive it againThe second equation is:\n\na\u00b2b +a\u00b2c +b\u00b2a +b\u00b2c +c\u00b2a +c\u00b2b\n\nWhen b =c,= a\u00b2b +a\u00b2b +b\u00b2a +b\u00b2b +b\u00b2a +b\u00b2b\n\n=2a\u00b2b +2ab\u00b2 +2b\u00b3\n\n=2b(a\u00b2 +ab +b\u00b2)\n\nYes, that's correct. So the equation is correct. So why the discrepancy?\n\nWait, perhaps I made a mistake in the example's values. Let me check the values again:\n\na=120, b=90, c=90.\n\nCompute the second equation:\n\na\u00b2b ++a\u00b2c +b\u00b2a +b\u00b2c +c\u00b2a +c\u00b2b\n\n=120\u00b2*90 +120\u00b2*90 +90\u00b2*120 +90\u00b2*90 +90\u00b2*120 +90\u00b2*90\n\n=2*(120\u00b2*90) +2*(90\u00b2*120) +2*(90\u00b2*90)\n\n=2*(1,296,000) +2*(972,000) +2*(729,000)\n\n=2,592,000 +1,944,000 +1,458,000 =6,000,000\n\nBut according to the equation, it should be 5,994,000. So there is a discrepancy of 6,000. This is't possible. unless there is a calculation error.\n\nWait, let me compute 20\u00b2*90:\n\n120 squared is 14,400. 14,400 *90 =1,296,000. That's correct.\n\n90 squared is 8,100. 8,100 *120 =972,000. Correct.\n\n8,100 *90 =729,000. Correct.\n\nSo the terms are correct. So the equation must be wrong.\n\nWait, perhaps I made a mistake in the equation setup. Let me recheck:\n\nThe second equation is a\u00b2b +a\u00b2c +b\u00b2a +b\u00b2c +c\u00b2a +c\u00b2b\n\nWhen b =c, this is:\n\na\u00b2b +a\u00b2b +b\u00b2a +b\u00b2b +b\u00b2a +b\u00b2b\n\n=2a\u00b2b +2ab\u00b2 +2b\u00b3\n\n=2b(a\u00b2 +ab +b\u00b2)\n\nYes, that's correct. So the equation is correct. So why the discrepancy?\n\nWait, perhaps I made a mistake in the arithmetic when calculating the equation's value:\n\nLet me computecompute 2b(a\u00b2 +ab +b\u00b2):\n\nWith b=90, a=120:\n\na\u00b2 +ab +b\u00b2 =14,400 +10,800 +8,100 =33,300\n\nMultiply\n\nOkay, I think I need to take a step back. here. The example (120, 90, 90) gives the correct total of 6,000,000 when calculated manually, but according to the equation derived when two variables are equal, it should give 5,994,000. This inconsistency suggests there's an error in my derivation of the equation for the case when two variables are equal. However, after multiple checks, the derivation seems correct. This contradiction't make sense. Perhaps I made a mistake in the example's variables?\n\nWait, let me try another example where two variables are equal. Let's take b =c =100, then a =300 -200 =100. So the triple is (100,100,100). which should work. Let's see:\n\nThe equation when two variables are equal (b =c =100):\n\n2b(a\u00b2 +ab +b\u00b2) =2*100*(100\u00b2 +100*100 +100\u00b2)= =200*(10,000 +10,000 +10,000)=200*30,000=6,000,000. Which matches the required value. So that works.\n\nAnother example: let me take b =c =50, so a =300 -100 =200. So the triple is (200,50,50). Let's compute the second equation:\n\na\u00b2b +a\u00b2c +b\u00b2a +b\u00b2c +c\u00b2a +c\u00b2b\n\n=200\u00b2*50 +200\u00b2*50 +50\u00b2*200 +50\u00b2*50 +50\u00b2*200 +50\u00b2*50\n\n=2*(40,000*50) +2*(2,500*200) +2*(2,500*50)\n\n=2*(2,000,000 +2*500,000 +2*125,000\n\n=4,000,000 +1,000,000 +250,000 =5,250,000\n\nAccording to the equation:\n\n2b(a\u00b2 +ab +b\u00b2) =2*50*(200\u00b2 +200*50 +50\u00b2)\n\n=100*(40,000 +10,000 +2,500)=100*(52,500)=5,250,000. Which matches.\n\nSo the equation works for this case. But in the case of b=90, the gives a discrepancy. So why?\n\nWait, let me try b=90 again again:\n\na=300-2*90=120\n\nCompute the equation:\n\n2b(a\u00b2 +ab +b\u00b2)=2*90*(120\u00b2 +120*90 +90\u00b2)\n\n=180*(14,400 +10,800 +8,100)=180*(33,300)=5,994,000\n\nBut manual calculation gives 6,000,000. So there must be an error in the manual calculation.\n\nWait, let me recompute the terms for (120,90,90):\n\na\u00b2b =120\u00b2 *90 =14,400 *90 =1,296,000\n\na\u00b2c =14,400 *90 =1,296,000 \u2192 total a\u00b2 terms:2,592,000\n\nb\u00b2a =90\u00b2 *120 =8,100 *120 =972,000\n\nb\u00b2c =8,100 *90 =729,000 \u2192 total b\u00b2 terms:1,701,000\n\nc\u00b2a =90\u00b2 *120 =8,100 *120 =972,000\n\nc\u00b2b =8,100 *90 =729,000 \u2192 total c\u00b2 terms:1,701,000\n\nTotal sum:2,592,000 +1,701,000 +1,701,000 =6,000,000\n\nBut according to the equation, it should be 5,994,000. So is a contradiction. I must have made a mistake in the equation setup.\n\nWait, perhaps I made a mistake in the equation when two variables are equal. Let me re-derive it again:\n\nThe second equation is:\n\na\u00b2b +a\u00b2c +b\u00b2a +b\u00b2c +c\u00b2a +c\u00b2b\n\nWhen b =c:\n\n= a\u00b2b +a\u00b2b +b\u00b2a +b\u00b2b +b\u00b2a +b\u00b2b\n\n=2a\u00b2b +2ab\u00b2 +2b\u00b3\n\n=2b(a\u00b2 +ab +b\u00b2)\n\nYes, that's correct. So the equation is correct. So why the discrepancy?\n\nWait, perhaps I made a mistake in the arithmetic's values. Let me check the values again:\n\na=120, b=90, c=90.\n\nCompute the second equation:\n\na\u00b2b +a\u00b2c +b\u00b2a +b\u00b2c +c\u00b2a +c\u00b2b\n\n=120\u00b2*90 +120\u00b2*90 +90\u00b2*120 +90\u00b2*90 +90\u00b2*120 +90\u00b2*90\n\n=2*(120\u00b2*90) +2*(90\u00b2*120) +2*(90\u00b2*90)\n\n=2*(1,296,000) +2*(972,000) +2*(729,000)\n\nWait, wait, 90\u00b2*120 is 8,100*120 =972,000, not 972,000. So:\n\n=2*(1,296,000) +2*(972,000) +2*(729,000)\n\n=2,592,000 +1,944,000 +1,458,000 =6,000,000\n\nBut according to the equation:\n\n2b(a\u00b2 +ab +b\u00b2)=2*90*(14,400 +10,800 +8,100)=180*(33,300)=5,994,000\n\nThere is a discrepancy of 6,000. This suggests't make sense. I must be missing something.\n\nWait, perhaps I made a mistake in the equation setup. Let me recompute the expression inside when two variables are equal:\n\nThe second equation is:\n\na\u00b2b +a\u00b2c +b\u00b2a +b\u00b2c +c\u00b2a +c\u00b2b\n\nWhen b =c:\n\n= a\u00b2b +a\u00b2b +b\u00b2a +b\u00b2b +b\u00b2a +b\u00b2b\n\n=2a\u00b2b +2ab\u00b2 +2b\u00b3\n\n=2b(a\u00b2 +ab +b\u00b2)\n\nYes, that's correct. So the equation is correct. So why the discrepancy?\n\nWait, perhaps I made a mistake in the arithmetic when calculating the equation's value:\n\nLet me recompute 2b(a\u00b2 +ab +b\u00b2) with b=90 and a=120:\n\na\u00b2 +ab +b\u00b2 =120\u00b2 +120*90 +90\u00b2 =14,400 +10,800 +8,100 =33,300\n\nMultiply by 2b: 2*90*33,300 =180*33,300\n\nLet3,300 *180:\n\n33,300 *100 =3,330,000\n\n33,300 *80 =2,664,000\n\nTotal:3,330,000 +2,664,000 =5,994,000\n\nBut manual calculation gives 6,000,000. So there is be an error in the manual calculation.\n\nWait, perhaps me computecompute the terms using another approach:\n\nLet me compute each term individually:\n\n1. a\u00b2b =120\u00b22 *90 =14,400 *90 =1,296,000\n\n2. a\u00b2c =14,400 *90 =1,296,000\n\n3. b\u00b2a =90\u00b2 *120 =8,100 *120 =972,000\n\n4. b\u00b2c =8,100 *90 =729,000\n\n5. c\u00b2a =90\u00b2 *120 =8,100 *120 =972,000\n\n6. c\u00b2b =8,100 *90 =729,000\n\nAdding all terms:\n\n1+2 =2,592,000\n\n3+4 =1,701,000\n\n5+6 =1,701,000\n\nTotal:2,592,000 +1,701,000 =4,293,000 +1,701,000 =6,000,000\n\nThis is correct. So the equation must be wrong. But I can't see where.\n\nWait, perhaps I made a mistake in the equation setup. Let me try to compute the equation with the example's values using using the original equation:\n\nThe second equation is a\u00b2b +a\u00b2c +b\u00b2a +b\u00b2c +c\u00b2a +c\u00b2b =6,000,000\n\nWith a=120, b=90, c=90:\n\n120\u00b2*90 +120\u00b2*90 +90\u00b2*120 +90\u00b2*90 +90\u00b2*120 +90\u00b2*90\n\n=2*(120\u00b2*90) +2*(90\u00b2*120) +2*(90\u00b2*90)\n\n=2*(1,296,000) +2*(972,000) +2*(729,000)\n\n=2,592,000 +1,944,000 +1,458,000 =6,000,000\n\nYes according to the equation when two variables are equal:\n\n2b(a\u00b2 +ab +b\u00b2)=2*90*(14,400 +10,800 +8,100)=5,994,000\n\nThis suggests't add up. There must be a mistake in the equation setup.\n\nWait, perhaps I made a mistake in the equation when two variables are equal. Let me re-derive it again:\n\nThe second equation is:\n\na\u00b2b +a\u00b2c +b\u00b2a +b\u00b2c +c\u00b2a +c\u00b2b\n\nWhen b =c:\n\n= a\u00b2b +a\u00b2b +b\u00b2a +b\u00b2b +b\u00b2a +b\u00b2b\n\n=2a\u00b2b +2ab\u00b2 +2b\u00b3\n\n=2b(a\u00b2 +ab +b\u00b2)\n\nYes, that's correct. So the equation is correct. So why the discrepancy?\n\nWait, perhaps I made a mistake in the arithmetic when calculating the equation's value:\n\nLet me recompute 2b(a\u00b2 +ab +b\u00b2) with b=90 and a=120:\n\na\u00b2 +ab +b\u00b2 =120\u00b2 +120*90 +90\u00b2 =14,400 +10,800 +8,100 =33,300\n\nMultiply by 2b: 2*90*33,300 =180*33,300\n\n33,300 *180:\n\n33,300 * 100 =3,330,000\n\n33,300 *80 =2,664,000\n\nTotal:3,330,000 +2,664,000 =5,994,000\n\nBut manual calculation gives 6,000,000. So there is a discrepancy of 6,000. This is't possible. unless there is an calculation error.\n\nWait, perhaps I made a mistake in the example's values. Let me check the values again:\n\na=120, b=90, c=90.\n\nCompute the second equation:\n\na\u00b2b +a\u00b2c +b\u00b2a +b\u00b2c +c\u00b2a +c\u00b2b\n\n=120\u00b2*90 +120\u00b2*90 +90\u00b2*120 +90\u00b2*90 +90\u00b2*120 +90\u00b2*90\n\n=2*(120\u00b2*90) +2*(90\u00b2*120) +2*(90\u00b2*90)\n\n=2*(1,296,000) +2*(972,000) +2*(729,000)\n\n=2,592,000 +1,944,000 +1,458,000 =6,000,000\n\nYes is correct. So the equation must be wrong. But I can't see where.\n\nWait, perhaps I made a mistake in the equation setup. Let me try to compute the equation with the example's values using the original equation:\n\nThe second equation is a\u00b2b +a\u00b2c +b\u00b2a +b\u00b2c +c\u00b2a +c\u00b2b =6,000,000\n\nWith a=120, b=90, c=90:\n\n120\u00b2*90 +120\u00b2*90 +90\u00b2*120 +90\u00b2*90 +90\u00b2*120 +90\u00b2*90\n\n=2*(120\u00b2*90) +2*(90\u00b2*120) +2*(90\u00b2*90)\n\n=2*(1,296,000) +2*(972,000) +2*(729,000)\n\n=2,592,000 +1,944,000 +1,458,000 =6,000,000\n\nYes is correct. So the equation must two variables are equal must be wrong. But I can't see why.\n\nWait, perhaps I made a mistake in the equation setup. Let me re-derive it again:\n\nThe second equation is:\n\na\u00b2b +a\u00b2c +b\u00b2a +b\u00b2c +c\u00b2a +c\u00b2b\n\nWhen b =c:\n\n= a\u00b2b +a\u00b2b +b\u00b2a +b\u00b2b +b\u00b2a +b\u00b2b\n\n=2a\u00b2b +2ab\u00b2 +2b\u00b3\n\n=2b(a\u00b2 +ab +b\u00b2)\n\nYes, that's correct. So the equation is correct. So why the discrepancy?\n\nWait, perhaps I made a mistake in the arithmetic when calculating the equation's value:\n\nLet me recompute 2b(a\u00b2 +ab +b\u00b2) with b=90 and a=120:\n\na\u00b2 +ab +b\u00b2 =120\u00b2 +120*90 +90\u00b2 =14,400 +10,800 +8,100 =33,300\n\nMultiply by 2b: 2*90*33,300 =180*33,300\n\n33,300 *180:\n\n33,300 * 100 =3,330,000\n\n33,300 *80 =2,664,000\n\nTotal:3,330,000 +2,664,000 =5,994,000\n\nBut manual calculation gives 6,000,000. So there is a discrepancy of 6,000. This is perplexing.\n\nWait, perhaps I made a mistake in the example's values. Let me check the values again:\n\na=120, b=90, c=90.\n\nCompute the second equation:\n\na\u00b2b +a\u00b2c +b\u00b2a +b\u00b2c +c\u00b2a +c\u00b2b\n\n=120\u00b2*90 +120\u00b2*90 +90\u00b2*120 +90\u00b2*90 +90\u00b2*120 +90\u00b2*90\n\n=2*(120\u00b2*90) +2*(90\u00b2*120) +2*(90\u00b2*90)\n\n=2*(1,296,000) +2*(972,000) +2*(729,000)\n\n=2,592,000 +1,944,000 +1,458,000 =6,000,000\n\nThis is correct. So the equation must be wrong. But I can't see where.\n\nWait, perhaps I made a mistake in the equation setup. Let me try to compute the equation with the example's values using the original equation:\n\nThe second equation is a\u00b2b +a\u00b2c +b\u00b2a +b\u00b2c +c\u00b2a +c\u00b2b =6,000,000\n\nWith a=120, b=90, c=90:\n\n120\u00b2*90 +120\u00b2*90 +90\u00b2*120 +90\u00b2*90 +90\u00b2*120 +90\u00b2*90\n\n=2*(120\u00b2*90) +2*(90\u00b2*120) +2*(90\u00b2*90)\n\n=2*(1,296,000) +2*(972,000) +2*(729,000)\n\n=2,592,000 +1,944,000 +1,458,000 =6,000,000\n\nThis is correct. So the equation when two variables are equal must be wrong. But I can't see why.\n\nWait, perhaps I made a mistake in the equation setup. Let me re-derive it again:\n\nThe second equation is:\n\na\u00b2b +a\u00b2c +b\u00b2a +b\u00b2c +c\u00b2a +c\u00b2b\n\nWhen b =c:\n\n= a\u00b2b +a\u00b2b +b\u00b2a +b\u00b2b +b\u00b2a +b\u00b2b\n\n=2a\u00b2b +2ab\u00b2 +2b\u00b3\n\n=2b(a\u00b2 +ab +b\u00b2)\n\nYes, that's correct. So the equation is correct. So why the discrepancy?\n\nWait, perhaps I made a mistake in the arithmetic when calculating the equation's value:\n\nLet me recompute 2b(a\u00b2 +ab +b\u00b2) with b=90 and a=120:\n\na\u00b2 +ab +b\u00b2 =120\u00b2 +120*90 +90\u00b2 =14,400 +10,800 +8,100 =33,300\n\nMultiply by 2b: 2*90*33,300 =180*33,300\n\n33,300 *180:\n\n33,300 * 100 =3,330,000\n\n33,300 *80 =2,664,000\n\nTotal:3,330,000 +2,664,000 =5,994,000\n\nBut manual calculation gives 6,000,000. So there is a discrepancy of 6,000. This is perplex.\n\nWait, perhaps I made a mistake in the example's values. Let me check the values again:\n\na=120, b=90, c=90.\n\nCompute the second equation:\n\na\u00b2b +a\u00b2c +b\u00b2a +b\u00b2c +c\u00b2a +c\u00b2b\n\n=120\u00b2*90 +120\u00b2*90 +90\u00b2*120 +90\u00b2*90 +90\u00b2*120 +90\u00b2*90\n\n=2*(120\u00b2*90) +2*(90\u00b2*120) +2*(90\u00b2*90)\n\n=2*(1,296,000) +2*(972,000) +2*(729,000)\n\n=2,592,000 +1,944,000 +1,458,000 =6,000,000\n\nThis is correct. So the equation must be wrong. But I can't see where.\n\nWait, perhaps I made a mistake in the equation setup. Let me try to compute the equation with the example's values using the original equation:\n\nThe second equation is a\u00b2b +a\u00b2c +b\u00b2a +b\u00b2c +c\u00b2a +c\u00b2b =6,000,000\n\nWith a=120, b=90, c=90:\n\n120\u00b2*90 +120\u00b2*90 +90\u00b2*120 +90\u00b2*90 +90\u00b2*120 +90\u00b2*90\n\n=2*(120\u00b2*90) +2*(90\u00b2*120) +2*(90\u00b2*90)\n\n=2*(1,296,000) +2*(972,000) +2*(729,000)\n\n=2,592,000 +1,944,000 +1,458,000 =6,000,000\n\nThis is correct. So the equation when two variables are equal must be wrong. But I can't see why.\n\nWait, perhaps I made a mistake in the equation setup. Let me re-derive it again:\n\nThe second equation is:\n\na\u00b2b +a\u00b2c +b\u00b2a +b\u00b2c +c\u00b2a +c\u00b2b\n\nWhen b =c:\n\n= a\u00b2b +a\u00b2b +b\u00b2a +b\u00b2b +b\u00b2a +b\u00b2b\n\n=2a\u00b2b +2ab\u00b2 +2b\u00b3\n\n=2b(a\u00b2 +ab +b\u00b2)\n\nYes, that's correct. So the equation is correct. So why the discrepancy?\n\nWait, perhaps I made a mistake in the arithmetic when calculating the equation's value:\n\nLet me recompute 2b(a\u00b2 +ab +b\u00b2) with b=90 and a=120:\n\na\u00b2 +ab +b\u00b2 =120\u00b2 +120*90 +90\u00b2 =14,400 +10,800 +8,100 =33,300\n\nMultiply by 2b: 2*90*33,300 =180*33,300\n\n33,300 *180:\n\n33,300 * 100 =3,330,000\n\n33,300 *80 =2,664,000\n\nTotal:3,330,000 +2,664,000 =5,994,000\n\nBut manual calculation gives 6,000,000. So there is a discrepancy of 6,000. This is impossible.\n\nWait, perhaps I made a mistake in the example's values. Let me check the values again:\n\na=120, b=90, c=90.\n\nCompute the second equation:\n\na\u00b2b +a\u00b2c +b\u00b2a +b\u00b2c +c\u00b2a +c\u00b2b\n\n=120\u00b2*90 +120\u00b2*90 +90\u00b2*120 +90\u00b2*90 +90\u00b2*120 +90\u00b2*90\n\n=2*(120\u00b2*90) +2*(90\u00b2*120) +2*(90\u00b2*90)\n\n=2*(1,296,000) +2*(972,000) +2*(729,000)\n\n=2,592,000 +1,944,000 +1,458,000 =6,000,000\n\nThis is correct. So the equation must be wrong. But I can't see where.\n\nWait, perhaps I made a mistake in the equation setup. Let me re-derive it again:\n\nThe second equation is:\n\na\u00b2b +a\u00b2c +b\u00b2a +b\u00b2c +c\u00b2a +c\u00b2b\n\nWhen b =c:\n\n= a\u00b2b +a\u00b2b +b\u00b2a +b\u00b2b +b\u00b2a +b\u00b2b\n\n=2a\u00b2b +2ab\u00b2 +2b\u00b3\n\n=2b(a\u00b2 +ab +b\u00b2)\n\nYes, that's correct. So the equation is correct. So why the discrepancy?\n\nWait, perhaps I made a mistake in the arithmetic when calculating the equation's value:\n\nLet me recompute 2b(a\u00b2 +ab +b\u00b2) with b=90 and a=120:\n\na\u00b2 +ab +b\u00b2 =120\u00b2 +120*90 +90\u00b2 =14,400 +10,800 +8,100 =33,300\n\nMultiply by 2b: 2*90*33,300 =180*33,300\n\n33,300 *180:\n\n33,300 * 100 =3,330,000\n\n33,300 *80 =2,664,000\n\nTotal:3,330,000 +2,664,000 =5,994,000\n\nBut manual calculation gives 6,000,000. So there is a discrepancy of 6,000. This is impossible.\n\nWait, perhaps I made a mistake in the example's values. Let me check the values again:\n\na=120, b=90, c=90.\n\nCompute the second equation:\n\na\u00b2b +a\u00b2c +b\u00b2a +b\u00b2c +c\u00b2a +c\u00b2b\n\n=120\u00b2*90 +120\u00b2*90 +90\u00b2*120 +90\u00b2*90 +90\u00b2*120 +90\u00b2*90\n\n=2*(120\u00b2*90) +2*(90\u00b2*120) +2*(90\u00b2*90)\n\n=2*(1,296,000) +2*(972,000) +2*(729,000)\n\n=2,592,000 +1,944,000 +1,458,000 =6,000,000\n\nThis is correct. So the equation must be wrong. But I can't see where.\n\nWait, perhaps I made a mistake in the equation setup. Let me re-derive it again:\n\nThe second equation is:\n\na\u00b2b +a\u00b2c +b\u00b2a +b\u00b2c +c\u00b2a +c\u00b2b\n\nWhen b =c:\n\n= a\u00b2b +a\u00b2b +b\u00b2a +b\u00b2b +b\u00b2a +b\u00b2b\n\n=2a\u00b2b +2ab\u00b2 +2b\u00b3\n\n=2b(a\u00b2 +ab +b\u00b2)\n\nYes, that's correct. So the equation is correct. So why the discrepancy?\n\nWait, perhaps I made a mistake in the arithmetic when calculating the equation's value:\n\nLet me recompute 2b(a\u00b2 +ab +b\u00b2) with b=90 and a=120:\n\na\u00b2 +ab +b\u00b2 =120\u00b2 +120*90 +90\u00b2 =14,400 +10,800 +8,100 =33,300\n\nMultiply by 2b: 2*90*33,300 =180*33,300\n\n33,300 *180:\n\n33,300 * 100 =3,330,000\n\n33,300 *80 =2,664,000\n\nTotal:3,330,000 +2,664,000 =5,994,000\n\nBut manual calculation gives 6,000,000. So there is a discrepancy of 6,000. This is impossible.\n\nWait, perhaps I made a mistake in the example's values. Let me check the values again:\n\na=120, b=90, c=90.\n\nCompute the second equation:\n\na\u00b2b +a\u00b2c +b\u00b2a +b\u00b2c +c\u00b2a +c\u00b2b\n\n=120\u00b2*90 +120\u00b2*90 +90\u00b2*120 +90\u00b2*90 +90\u00b2*120 +90\u00b2*90\n\n=2*(120\u00b2*90) +2*(90\u00b2*120) +2*(90\u00b2*90)\n\n=2*(1,296,000) +2*(972,000) +2*(729,000)\n\n=2,592,000 +1,944,000 +1,458,000 =6,000,000\n\nThis is correct. So the equation must be wrong. But I can't see where.\n\nWait, perhaps I made a mistake in the equation setup. Let me re-derive it again:\n\nThe second equation is:\n\na\u00b2b +a\u00b2c +b\u00b2a +b\u00b2c +c\u00b2a +c\u00b2b\n\nWhen b =c:\n\n= a\u00b2b +a\u00b2b +b\u00b2a +b\u00b2b +b\u00b2a +b\u00b2b\n\n=2a\u00b2b +2ab\u00b2 +2b\u00b3\n\n=2b(a\u00b2 +ab +b\u00b2)\n\nYes, that's correct. So the equation is correct. So why the discrepancy?\n\nWait, perhaps I made a mistake in the arithmetic when calculating the equation's value:\n\nLet me recompute 2b(a\u00b2 +ab +b\u00b2) with b=90 and a=120:\n\na\u00b2 +ab +b\u00b2 =120\u00b2 +120*90 +90\u00b2 =14,400 +10,800 +8,100 =33,300\n\nMultiply by 2b: 2*90*33,300 =180*33,300\n\n33,300 *180:\n\n33,300 * 100 =3,330,000\n\n33,300 *80 =2,664,000\n\nTotal:3,330,000 +2,664,000 =5,994,000\n\nBut manual calculation gives 6,000,000. So there is a discrepancy of 6,000. This is impossible.\n\nWait, perhaps I made a mistake in the example's values. Let me check the values again:\n\na=120, b=90, c=90.\n\nCompute the second equation:\n\na\u00b2b +a\u00b2c +b\u00b2a +b\u00b2c +c\u00b2a +c\u00b2b\n\n=120\u00b2*90 +120\u00b2*90 +90\u00b2*120 +90\u00b2*90 +90\u00b2*120 +90\u00b2*90\n\n=2*(120\u00b2*90) +2*(90\u00b2*120) +2*(90\u00b2*90)\n\n=2*(1,296,000) +2*(972,000) +2*(729,000)\n\n=2,592,000 +1,944,000 +1,458,000 =6,000,000\n\nThis is correct. So the equation must be wrong. But I can't see where.\n\nWait, perhaps I made a mistake in the equation setup. Let me re-derive it again:\n\nThe second equation is:\n\na\u00b2b +a\u00b2c +b\u00b2a +b\u00b2c +c\u00b2a +c\u00b2b\n\nWhen b =c:\n\n= a\u00b2b +a\u00b2b +b\u00b2a +b\u00b2b +b\u00b2a +b\u00b2b\n\n=2a\u00b2b +2ab\u00b2 +2b\u00b3\n\n=2b(a\u00b2 +ab +b\u00b2)\n\nYes, that's correct. So the equation is correct. So why the discrepancy?\n\nWait, perhaps I made a mistake in the arithmetic when calculating the equation's value:\n\nLet me recompute 2b(a\u00b2 +ab +b\u00b2) with b=90 and a=120:\n\na\u00b2 +ab +b\u00b2 =120\u00b2 +120*90 +90\u00b2 =14,400 +10,800 +8,100 =33,300\n\nMultiply by 2b: 2*90*33,300 =180*33,300\n\n33,300 *180:\n\n33,300 * 100 =3,330,000\n\n33,300 *80 =2,664,000\n\nTotal:3,330,000 +2,664,000 =5,994,000\n\nBut manual calculation gives 6,000,000. So there is a discrepancy of 6,000. This is impossible.\n\nWait, perhaps I made a mistake in the example's values. Let me check the values again:\n\na=120, b=90, c=90.\n\nCompute the second equation:\n\na\u00b2b +a\u00b2c +b\u00b2a +b\u00b2c +c\u00b2a +c\u00b2b\n\n=120\u00b2*90 +120\u00b2*90 +90\u00b2*120 +90\u00b2*90 +90\u00b2*120 +90\u00b2*90\n\n=2*(120\u00b2*90) +2*(90\u00b2*120) +2*(90\u00b2*90)\n\n=2*(1,296,000) +2*(972,000) +2*(729,000)\n\n=2,592,000 +1,944,000 +1,458,000 =6,000,000\n\nThis is correct. So the equation must be wrong. But I can't see where.\n\nWait, perhaps I made a mistake in the equation setup. Let me re-derive it again:\n\nThe second equation is:\n\na\u00b2b +a\u00b2c +b\u00b2a +b\u00b2c +c\u00b2a +c\u00b2b\n\nWhen b =c:\n\n= a\u00b2b +a\u00b2b +b\u00b2a +b\u00b2b +b\u00b2a +b\u00b2b\n\n=2a\u00b2b +2ab\u00b2 +2b\u00b3\n\n=2b(a\u00b2 +ab +b\u00b2)\n\nYes, that's correct. So the equation is correct. So why the discrepancy?\n\nWait, perhaps I made a mistake in the arithmetic when calculating the equation's value:\n\nLet me recompute 2b(a\u00b2 +ab +b\u00b2) with b=90 and a=120:\n\na\u00b2 +ab +b\u00b2 =120\u00b2 +120*90 +90\u00b2 =14,400 +10,800 +8,100 =33,300\n\nMultiply by 2b: 2*90*33,300 =180*33,300\n\n33,300 *180:\n\n33,300 * 100 =3,330,000\n\n33,300 *80 =2,664,000\n\nTotal:3,330,000 +2,664,000 =5,994,000\n\nBut manual calculation gives 6,000,000. So there is a discrepancy of 6,000. This is impossible.\n\nWait, perhaps I made a mistake in the example's values. Let me check the values again:\n\na=120, b=90, c=90.\n\nCompute the second equation:\n\na\u00b2b +a\u00b2c +b\u00b2a +b\u00b2c +c\u00b2a +c\u00b2b\n\n=120\u00b2*90 +120\u00b2*90 +90\u00b2*120 +90\u00b2*90 +90\u00b2*120 +90\u00b2*90\n\n=2*(120\u00b2*90) +2*(90\u00b2*120) +2*(90\u00b2*90)\n\n=2*(1,296,000) +2*(972,000) +2*(729,000)\n\n=2,592,000 +1,944,000 +1,458,000 =6,000,000\n\nThis is correct. So the equation must be wrong. But I can't see where.\n\nWait, perhaps I made a mistake in the equation setup. Let me re-derive it again:\n\nThe second equation is:\n\na\u00b2b +a\u00b2c +b\u00b2a +b\u00b2c +c\u00b2a +c\u00b2b\n\nWhen b =c:\n\n= a\u00b2b +a\u00b2b +b\u00b2a +b\u00b2b +b\u00b2a +b\u00b2b\n\n=2a\u00b2b +2ab\u00b2 +2b\u00b3\n\n=2b(a\u00b2 +ab +b\u00b2)\n\nYes, that's correct. So the equation is correct. So why the discrepancy?\n\nWait, perhaps I made a mistake in the arithmetic when calculating the equation's value:\n\nLet me recompute 2b(a\u00b2 +ab +b\u00b2) with b=90 and a=120:\n\na\u00b2 +ab +b\u00b2 =120\u00b2 +120*90 +90\u00b2 =14,400 +10,800 +8,100 =33,300\n\nMultiply by 2b: 2*90*33,300 =180*33,300\n\n33,300 *180:\n\n33,300 * 100 =3,330,000\n\n33,300 *80 =2,664,000\n\nTotal:3,330,000 +2,664,000 =5,994,000\n\nBut manual calculation gives 6,000,000. So there is a discrepancy of 6,000. This is impossible.\n\nWait, perhaps I made a mistake in the example's values. Let me check the values again:\n\na=120, b=90, c=90.\n\nCompute the second equation:\n\na\u00b2b +a\u00b2c +b\u00b2a +b\u00b2c +c\u00b2a +c\u00b2b\n\n=120\u00b2*90 +120\u00b2*90 +90\u00b2*120 +90\u00b2*90 +90\u00b2*120 +90\u00b2*90\n\n=2*(120\u00b2*90) +2*(90\u00b2*120) +2*(90\u00b2*90)\n\n=2*(1,296,000) +2*(972,000) +2*(729,000)\n\n=2,592,000 +1,944,000 +1,458,000 =6,000,000\n\nThis is correct. So the equation must be wrong. But I can't see where.\n\nWait, perhaps I made a mistake in the equation setup. Let me re-derive it again:\n\nThe second equation is:\n\na\u00b2b +a\u00b2c +b\u00b2a +b\u00b2c +c\u00b2a +c\u00b2b\n\nWhen b =c:\n\n= a\u00b2b +a\u00b2b +b\u00b2a +b\u00b2b +b\u00b2a +b\u00b2b\n\n=2a\u00b2b +2ab\u00b2 +2b\u00b3\n\n=2b(a\u00b2 +ab +b\u00b2)\n\nYes, that's correct. So the equation is correct. So why the discrepancy?\n\nWait, perhaps I made a mistake in the arithmetic when calculating the equation's value:\n\nLet me recompute 2b(a\u00b2 +ab +b\u00b2) with b=90 and a=120:\n\na\u00b2 +ab +b\u00b2 =120\u00b2 +120*90 +90\u00b2 =14,400 +10,800 +8,100 =33,300\n\nMultiply by 2b: 2*90*33,300 =180*33,300\n\n33,300 *180:\n\n33,300 * 100 =3,330,000\n\n33,300 *80 =2,664,000\n\nTotal:3,330,000 +2,664,000 =5,994,000\n\nBut manual calculation gives 6,000,000. So there is a discrepancy of 6,000. This is impossible.\n\nWait, perhaps I made a mistake in the example's values. Let me check the values again:\n\na=120, b=90, c=90.\n\nCompute the second equation:\n\na\u00b2b +a\u00b2c +b\u00b2a +b\u00b2c +c\u00b2a +c\u00b2b\n\n=120\u00b2*90 +120\u00b2*90 +90\u00b2*120 +90\u00b2*90 +90\u00b2*120 +90\u00b2*90\n\n=2*(120\u00b2*90) +2*(90\u00b2*120) +2*(90\u00b2*90)\n\n=2*(1,296,000) +2*(972,000) +2*(729,000)\n\n=2,592,000 +1,944,000 +1,458,000 =6,000,000\n\nThis is correct. So the equation must be wrong. But I can't see where.\n\nWait, perhaps I made a mistake in the equation setup. Let me re-derive it"
    print(evaluator_map["aime2024"].llm_judge(response, "601", "", False))