Problem Analysis #27

GPT-5.2 (high) · Run 1

19.87s
4,020 toks
Problem Statement

\text{Given a set } A={a_1,a_2,\ldots,a_k} \text{ consisting of distinct real numbers, define }S(A) \text{ as the sum of all elements of } A:]

S(A)=a1+a2+⋯+ak.S(A)=a_1+a_2+\cdots+a_k. For example, if p is a prime number and A={1,p}, the set of divisors of p, then S(A)=p+1.\text{For example, if } p \text{ is a prime number and } A=\{1,p\}, \text{ the set of divisors of } p,\text{ then } S(A)=p+1. Now, let A be the set obtained by taking log⁡10 of each divisor of 102025,\text{Now, let } A \text{ be the set obtained by taking } \log_{10} \text{ of each divisor of }10^{2025}, and multiplying the value by +1 if the divisor is even, and by −1 if the divisor is odd.\text{and multiplying the value by } +1 \text{ if the divisor is even, and by }-1 \text{ if the divisor is odd.} Compute the product of the prime factors appearing in the prime factorization of S(A)−2025⋅2026⋅log⁡102.\text{Compute the product of the prime factors appearing in the prime factorization of }S(A)-2025\cdot 2026 \cdot \log_{10} 2.
Ground Truth
7688670
Execution Trace
Reasoning Process

[THOUGHT] I will express S(A) as a double sum over divisors 2^i5^j of 10^2025 with a sign depending on parity (i=0 odd gives negative). Then I will simplify S(A)-2025·2026·log10(2) to an integer and use Python (sympy factorint) to factor it and take the product of distinct primes. [/THOUGHT] [PYTHON] import sympy as sp

n=2025 m=n+1 N = nm(n-1)/2 # 202520262024/2 print(N) fac = sp.factorint(N) print(fac) rad = 1 for p in fac: rad *= p print(rad) [/PYTHON][THOUGHT] I will express S(A) as a double sum over divisors 2^i5^j of 10^2025 with a sign depending on parity (i=0 odd gives negative). Then I will simplify S(A)-2025·2026·log10(2) to an integer and use Python (sympy factorint) to factor it and take the product of distinct primes. [/THOUGHT] [PYTHON] import sympy as sp

n=2025 m=n+1 N = nm(n-1)//2 # 202520262024/2 print(N) fac = sp.factorint(N) print(fac) rad = 1 for p in fac: rad *= p print(rad) [/PYTHON]

Tool Output
[PYTHON OUTPUT] ValueError: 4151881800.0 is not an integer [/PYTHON OUTPUT] [PYTHON OUTPUT] 4151881800 {2: 3, 3: 4, 5: 2, 11: 1, 23: 1, 1013: 1} 7688670 [/PYTHON OUTPUT]
Reasoning Process

[THOUGHT] From the computed value S(A)−2025⋅2026⋅log⁡102=4151881800S(A)-2025\cdot2026\cdot\log_{10}2=4151881800 with prime factorization 23⋅34⋅52⋅11⋅23⋅10132^3\cdot3^4\cdot5^2\cdot11\cdot23\cdot1013, the product of the distinct prime factors is 2⋅3⋅5⋅11⋅23⋅1013=76886702\cdot3\cdot5\cdot11\cdot23\cdot1013=7688670. 7688670\boxed{7688670}

Final Model Output
7688670

Comparative Analysis (P27) (Average across runs)

Avg Time Taken

EXAONE-4.0.1-32B (high)
9270.8
K-EXAONE-236B-A23B
1932.3
Solar-Open-100B
1085.4
Gemini-3-Pro-Preview
193.8
K-EXAONE-236B-A23B
172.3
Llama-VARCO-8B-Instruct
60.5
HCX-007(high)
59.8
Solar-Pro-2 (31B)(high)
50.6
Kanana-2-30B-Thinking
48.3
Deepseek-V3.2
41.0
A.X-4.0 (72B)
40.7
GPT-5.1 (high)
34.9
Kanana-2-30B-Thinking-2601
31.4
Grok-4.1-fast
28.2
Claude-Opus-4.5
27.7
GPT-5.2 (high)
19.9

Avg Token Usage

K-EXAONE-236B-A23B
178,782
K-EXAONE-236B-A23B
32,500
Solar-Open-100B
13,414
Gemini-3-Pro-Preview
12,269
Kanana-2-30B-Thinking
10,313
Grok-4.1-fast
9,410
Solar-Pro-2 (31B)(high)
9,131
Kanana-2-30B-Thinking-2601
8,099
Deepseek-V3.2
6,485
HCX-007(high)
6,470
A.X-4.0 (72B)
6,427
Claude-Opus-4.5
5,967
EXAONE-4.0.1-32B (high)
4,839
GPT-5.1 (high)
4,722
GPT-5.2 (high)
4,020
Llama-VARCO-8B-Instruct
2,568