Problem Analysis #27

GPT-5.1 (high) · Run 1

34.95s
4,722 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 symbolically derive S(A) using the divisor structure of 10^2025 and verify simplifications with Python. Then I'll compute the integer value of S(A)−2025·2026·log10(2), factor it, and finally multiply the distinct prime factors. [/THOUGHT] [PYTHON] from sympy import symbols, factor, factorint

Define symbols

L2, L5 = symbols('L2 L5')

T = sum_{a=1}^{2025} a

T = 2025*2026//2

Derived expression for S:

S = 2026TL2 + 2024TL5

S = 2026TL2 + 2024TL5

Expression to consider: S - 20252026L2

expr = S - 20252026L2 print("Symbolic expr:", factor(expr))

Now, using log10(2) + log10(5) = 1, so L2 + L5 = 1.

So expr should simplify to 2024 * T * (L2 + L5) = 2024 * T.

E = 2024 * T print("Integer E:", E)

Factor E and compute product of distinct primes

factors = factorint(E) print("Prime factorization of E:", factors)

product_distinct_primes = 1 for p in factors.keys(): product_distinct_primes *= p

print("Product of distinct primes:", product_distinct_primes) [/PYTHON]

Tool Output
[PYTHON OUTPUT] Symbolic expr: 4151881800*(L2 + L5) Integer E: 4151881800 Prime factorization of E: {2: 3, 3: 4, 5: 2, 11: 1, 23: 1, 1013: 1} Product of distinct primes: 7688670 [/PYTHON OUTPUT]
Reasoning Process

[THOUGHT] Using the valid Python output, S(A)−2025·2026·log10(2) simplifies to the integer 4151881800, whose prime factorization is {2: 3, 3: 4, 5: 2, 11: 1, 23: 1, 1013: 1}. The product of the distinct primes is 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