Problem Analysis #18

GPT-5.2 (high) · Run 1

12.70s

2,910 toks

Problem Statement

Let $\mathbb{R}$ denote the set of all real numbers. Let $f: \mathbb{R} \to \mathbb{R}$ be a function such that $f(x) > 0$ for all $x \in \mathbb{R}$ , and $f(x + y) = f(x)f(y)$ for all $x, y \in \mathbb{R}$ .Let the real numbers $a_1, a_2, \ldots, a_{50}$ be in an arithmetic progression. If $f(a_{31}) = 64f(a_{25})$ , and

\sum_{i=1}^{50} f(a_i) = 3(2^{25} + 1),

find the value of

\sum_{i=6}^{30} f(a_i).

Ground Truth

Execution Trace

Reasoning Process

[THOUGHT] Because $a_i$ is an arithmetic progression and $f(x+y)=f(x)f(y)$ , the values $f(a_i)$ form a geometric progression: $f(a_i)=t r^{i-1}$ where $t=f(a_1)$ and $r=f(d)$ . Use $f(a_{31})=64 f(a_{25})$ to find $r$ , use the total sum to find $t$ , then compute $\sum_{i=6}^{30} f(a_i)$ . I will verify all algebra with Python. [/THOUGHT] [PYTHON] from fractions import Fraction

r = 2 # from r^6=64 t = Fraction(3, 225 - 1) S_6_30 = t * sum(2(i-1) for i in range(6, 31)) t, S_6_30 print(int(S_6_30)) [/PYTHON]

Tool Output

[PYTHON OUTPUT] 96 [/PYTHON OUTPUT]

Reasoning Process

[THOUGHT] The computed value of $\sum_{i=6}^{30} f(a_i)$ is 96.

\boxed{96}

Final Model Output

Comparative Analysis (P18) (Average across runs)

Avg Time Taken

K-EXAONE-236B-A23B

302.6

Llama-VARCO-8B-Instruct

103.6

Gemini-3-Pro-Preview

52.0

Solar-Pro-2 (31B)(high)

48.8

Deepseek-V3.2

38.6

K-EXAONE-236B-A23B

36.3

Kanana-2-30B-Thinking

32.6

HCX-007(high)

32.0

A.X-4.0 (72B)

31.1

Solar-Open-100B

29.4

Grok-4.1-fast

27.7

Claude-Opus-4.5

22.0

Kanana-2-30B-Thinking-2601

20.4

GPT-5.1 (high)

17.3

GPT-5.2 (high)

12.7

EXAONE-4.0.1-32B (high)

5.7

Avg Token Usage

K-EXAONE-236B-A23B

32,979

Solar-Pro-2 (31B)(high)

9,405

Grok-4.1-fast

7,425

Kanana-2-30B-Thinking

7,018

Solar-Open-100B

6,748

Gemini-3-Pro-Preview

6,113

Kanana-2-30B-Thinking-2601

6,034

K-EXAONE-236B-A23B

5,804

Deepseek-V3.2

5,286

Claude-Opus-4.5

5,158

HCX-007(high)

4,565

EXAONE-4.0.1-32B (high)

4,213

Llama-VARCO-8B-Instruct

3,600

GPT-5.1 (high)

3,114

GPT-5.2 (high)

2,910

A.X-4.0 (72B)

2,654