Problem Statement
On a circle are placed nonnegative real numbers
$$\nwith indices taken modulo $n$ (so $x_{n+1}=x_1$, $x_{n+2}=x_2$, etc.).\nThey satisfy the local constraint $$\nx_i + x_{i+1} + x_{i+2} \le 1 \qquad\text{for all } i=1,2,\dots,n.\nFor such a configuration define
$$\nLet $M_n$ be the maximal possible value of $S$ over all admissible configurations $(x_1,\dots,x_n)$ satisfying the above condition. \nIt can be shown that for each positive integer $n$, the maximum $M_n$ is a rational number.\nFor the specific case $n=2026$, write $$\nM_{2026} \;=\; \frac{p}{q} $$\nin lowest terms, where $p$ and $q$ are coprime positive integers. \nDetermine the integer $p+q$.Ground Truth
1017
Execution Trace
Reasoning Process
[THOUGHT] We are given a problem about maximizing the sum S = sum_{i=1}^{n} x_i * x_{i+2} under the constraints that for all i, x_i + x_{i+1} + x_{i+2} ≤ 1, with indices modulo n. The goal is to find M_n for n=2026, express it as p/q in lowest terms, and then compute p+q.
From analysis of small cases and patterns:
- For even n, the maximum M_n is n/8.
- For n=2026 (even), M_n = 2026/8 = 1013/4.
- The fraction 1013/4 is in lowest terms since 1013 is odd and 4 is a power of 2, so gcd(1013, 4) = 1.
- Therefore, p = 1013, q = 4, and p+q = 1017.
The final answer is 1017. [/THOUGHT]
Final Model Output
1017
Comparative Analysis (P9) (Average across runs)
Avg Time Taken
K-EXAONE-236B-A23B
1477.6
Solar-Open-100B
279.6
Gemini-3-Pro-Preview
178.1
Kanana-2-30B-Thinking-2601
106.0
Kanana-2-30B-Thinking
104.4
GPT-5.2 (high)
60.1
GLM-4.5-Air
51.7
HCX-007
40.2
Solar-Pro 2
32.8
EXAONE-4.0-32B
5.9
Avg Token Usage
K-EXAONE-236B-A23B
31533.3
Solar-Open-100B
29943.0
Gemini-3-Pro-Preview
28144.3
Kanana-2-30B-Thinking-2601
20830.7
Kanana-2-30B-Thinking
17433.0
GLM-4.5-Air
9787.3
Solar-Pro 2
8105.0
GPT-5.2 (high)
6624.7
HCX-007
5037.0
EXAONE-4.0-32B
4449.3