Problem Analysis #3

GPT-5.1 (high) · Run 1

3.80s
2,404 toks
Problem Statement

For positive integers (a,b,c)(a,b,c), call (a,b,c)(a,b,c) \emph{cyclically divisible} if

a+1b,b+1c,c+1a\frac{a+1}{b},\quad \frac{b+1}{c},\quad \frac{c+1}{a}

\nare all integers.\nLet T\mathcal T be the set of such triples with 1≤a,b,c≤51\le a,b,c\le 5.\nDetermine ∣T∣|\mathcal T|.

Ground Truth
10
Execution Trace
Reasoning Process

[THOUGHT] We must count triples (a,b,c) with 1<=a,b,c<=5 such that (a+1)/b, (b+1)/c, (c+1)/a are all integers. I'll brute-force all 5^3=125 triples and check divisibility conditions using Python. [/THOUGHT] [PYTHON] count = 0 triples = [] for a in range(1,6): for b in range(1,6): for c in range(1,6): if (a+1) % b == 0 and (b+1) % c == 0 and (c+1) % a == 0: count += 1 triples.append((a,b,c)) print("count:", count) print("triples:", triples) [/PYTHON]

Tool Output
[PYTHON OUTPUT] count: 10 triples: [(1, 1, 1), (1, 1, 2), (1, 2, 1), (1, 2, 3), (2, 1, 1), (2, 3, 1), (3, 1, 2), (3, 4, 5), (4, 5, 3), (5, 3, 4)] [/PYTHON OUTPUT]
Reasoning Process

[THOUGHT] Using the brute-force Python enumeration over all triples (a,b,c) with 1≤a,b,c≤5 that satisfy the divisibility conditions, we found exactly 10 such cyclically divisible triples. Thus, |𝒯|=10. 10\boxed{10}

Final Model Output
10

Comparative Analysis (P3) (Average across runs)

Avg Time Taken

Solar-Open-100B
404.8
Gemini-3-Pro-Preview
47.9
Deepseek-V3.2
11.6
Claude-Opus-4.5
8.4
Grok-4.1-fast
8.0
K-EXAONE-236B-A23B
4.9
GPT-5.1 (high)
4.0

Avg Token Usage

Gemini-3-Pro-Preview
7,194
Grok-4.1-fast
3,033
Claude-Opus-4.5
2,757
Solar-Open-100B
2,726
Deepseek-V3.2
2,656
GPT-5.1 (high)
2,398
K-EXAONE-236B-A23B
2,237
    Open · Problem 3 · GPT-5.1 (high) | EntropyMath