Problem Analysis #18

GPT-5.1 (high) · Run 1

17.32s
3,114 toks
Problem Statement

Let R\mathbb{R} denote the set of all real numbers. Let f:R→Rf: \mathbb{R} \to \mathbb{R} be a function such that f(x)>0f(x) > 0 for all x∈Rx \in \mathbb{R}, and f(x+y)=f(x)f(y)f(x + y) = f(x)f(y) for all x,y∈Rx, y \in \mathbb{R}.Let the real numbers a1,a2,…,a50a_1, a_2, \ldots, a_{50} be in an arithmetic progression. If f(a31)=64f(a25)f(a_{31}) = 64f(a_{25}), and

∑i=150f(ai)=3(225+1),\sum_{i=1}^{50} f(a_i) = 3(2^{25} + 1),

find the value of

∑i=630f(ai).\sum_{i=6}^{30} f(a_i).

\

Ground Truth
96
Execution Trace
Reasoning Process

[THOUGHT] I'll model the sequence f(ai)f(a_i) as a geometric progression using the functional equation and arithmetic progression structure, then use the given conditions to find the common ratio and first term. Finally, I'll compute the required partial sum using Python for exact arithmetic. [/THOUGHT] [PYTHON] from fractions import Fraction

Given: r^6 = 64 and r>0, so r = 2

r = 2

Let f(a_i) = c * r^(i-1); sum i=1..50 equals 3(2^25 + 1)

from math import isclose

S50 = 3 * (225 + 1) c = Fraction(S50, 250 - 1) # exact rational

Sum from i=6 to 30 of f(a_i) = c * sum_{k=5}^{29} r^k = c * 2^5 * (2^25 - 1)

S_6_30 = c * (25) * (225 - 1)

print("c =", c) print("Sum i=6..30 =", S_6_30) print("As integer:", int(S_6_30)) [/PYTHON]

Tool Output
[PYTHON OUTPUT] c = 3/33554431 Sum i=6..30 = 96 As integer: 96 [/PYTHON OUTPUT]
Reasoning Process

[THOUGHT] Using the Python-verified geometric progression parameters, the required partial sum from i=6i=6 to 3030 is 96.

∑i=630f(ai)=96\sum_{i=6}^{30} f(a_i) = 96

[/THOUGHT] 96\boxed{96}

Final Model Output
96

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