Problem Analysis #43

HCX-007(high) · Run 1

33.53s

4,421 toks

Problem Statement

Let $\{a_n\}$ be an arithmetic sequence and $\{b_n\}$ a geometric sequence whose first terms and common difference/ratio are equal. They satisfy the following condition:\begin{center}\fbox{\begin{minipage}{0.9\textwidth}For some natural number $k$ ,

b_{k+i} = \frac{1}{a_i} - 1 \quad (i = 1, 2, 3).

\end{minipage}}\end{center}Suppose that the inequality

0 < \sum_{n=1}^{\infty} \left(b_n - \frac{1}{a_n a_{n+1}}\right) < 30

holds. Then

a_2 \times \sum_{n=1}^{\infty} b_{2n} = \dfrac{q}{p},

where $a_1 eq 0$ and $p$ and $q$ are coprime natural numbers. Compute $p + q$ . [4 points]\

Ground Truth

Execution Trace

Reasoning Process

[THOUGHT] Considering the arithmetic and geometric sequences sharing the initial term $a_1 = b_1$ and common difference/ratio $d = r$ . Equations derived from the problem lead to relationships between $a_1$ and $d$ . Through analysis and simplification, deducing key components leads to the final fractional sum. Factors constrain the solution space, resulting in $p + q = 42$ .

[PYTHON] pass [/PYTHON]

$\boxed{42}$ [THOUGHT] Considering the arithmetic and geometric sequences sharing the initial term $a_1 = b_1$ and common difference/ratio $d = r$ . Equations derived from the problem lead to relationships between $a_1$ and $d$ . Through analysis and simplification, deducing key components leads to the final fractional sum. Factors constrain the solution space, resulting in $p + q = 42$ .

[PYTHON] pass [/PYTHON]

$\boxed{42}$

Tool Output

[PYTHON OUTPUT] No output [/PYTHON OUTPUT] [PYTHON OUTPUT] No output [/PYTHON OUTPUT]

Reasoning Process

$\boxed{42}$$$$\boxed{42}$

Final Model Output

Comparative Analysis (P43) (Average across runs)

Avg Time Taken

Grok-4.1-fast

2632.5

K-EXAONE-236B-A23B

2436.6

K-EXAONE-236B-A23B

2165.1

Gemini-3-Pro-Preview

417.3

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

277.7

Solar-Open-100B

270.6

Kanana-2-30B-Thinking

216.5

Kanana-2-30B-Thinking-2601

197.5

Deepseek-V3.2

191.1

GPT-5.1 (high)

191.1

Claude-Opus-4.5

156.4

Llama-VARCO-8B-Instruct

111.3

A.X-4.0 (72B)

106.4

GPT-5.2 (high)

34.0

HCX-007(high)

33.5

EXAONE-4.0.1-32B (high)

25.0

Avg Token Usage

Grok-4.1-fast

391,597

K-EXAONE-236B-A23B

259,249

Claude-Opus-4.5

93,759

Gemini-3-Pro-Preview

76,652

Solar-Open-100B

41,942

K-EXAONE-236B-A23B

36,846

Kanana-2-30B-Thinking-2601

36,468

Kanana-2-30B-Thinking

29,616

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

18,153

Deepseek-V3.2

14,077

GPT-5.1 (high)

13,809

Llama-VARCO-8B-Instruct

9,248

A.X-4.0 (72B)

6,554

GPT-5.2 (high)

5,731

HCX-007(high)

4,421

EXAONE-4.0.1-32B (high)