Problem Analysis #9

GPT-5.2 (high) · Run 1

43.48s
6,927 toks
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] I will explore the optimization numerically for small n using nonlinear programming (SLSQP) to guess the exact maximal pattern and value, then infer the formula for even n=2026. I’ll compare the numeric maxima to candidate closed forms (like n/8) and inspect the maximizing configuration structure. [/THOUGHT] [PYTHON] import numpy as np from math import isclose from scipy.optimize import minimize

def max_S_numeric(n, trials=50, seed=0): rng = np.random.default_rng(seed) best = -1 bestx=None

# constraints: x_i>=0, triple sums <=1
cons=[]
for i in range(n):
    cons.append({
        'type':'ineq',
        'fun': lambda x, i=i: 1 - (x[i] + x[(i+1)%n] + x[(i+2)%n])
    })
bounds=[(0,1) for _ in range(n)]

def S(x):
    return sum(x[i]*x[(i+2)%n] for i in range(n))

def obj(x):
    return -S(x)

for t in range(trials):
    # random x, then scale down to satisfy triple constraints
    x = rng.random(n)
    # iterative scaling to satisfy constraints
    for _ in range(10):
        viol = np.array([x[i]+x[(i+1)%n]+x[(i+2)%n] for i in range(n)])
        m = viol.max()
        if m <= 1+1e-9:
            break
        x = x/(m+1e-9)
    res = minimize(obj, x, method='SLSQP', bounds=bounds, constraints=cons,
                   options={'maxiter':2000, 'ftol':1e-12, 'disp':False})
    if res.success:
        val = -res.fun
        if val > best + 1e-7:
            best=val
            bestx=res.x
return best, bestx

def analyze(n): best, x = max_S_numeric(n, trials=120, seed=42+n) print("n", n, "best", best) if x is not None: # print rounded pattern xr = np.round(x, 6) print("x rounded:", xr) # check max triple sum and min triples = np.array([x[i]+x[(i+1)%n]+x[(i+2)%n] for i in range(n)]) print("triple max", triples.max(), "min", triples.min()) # compute S S = sum(x[i]*x[(i+2)%n] for i in range(n)) print("S recomputed", S) print("compare n/8", n/8, "diff", S - n/8) print("compare n/9", n/9, "diff", S - n/9) print("---")

for n in range(4,17): analyze(n) [/PYTHON]

Tool Output
[PYTHON OUTPUT] n 4 best 0.5000000000000006 x rounded: [0. 0.5 0. 0.5] triple max 1.0000000000000004 min 0.49999999943726514 S recomputed 0.5000000000000006 compare n/8 0.5 diff 5.551115123125783e-16 compare n/9 0.4444444444444444 diff 0.055555555555556135 --- n 5 best 0.5555555555555556 x rounded: [0.333333 0.333333 0.333333 0.333333 0.333333] triple max 1.0000000000000004 min 0.9999999999999993 S recomputed 0.5555555555555556 compare n/8 0.625 diff -0.06944444444444442 compare n/9 0.5555555555555556 diff 0.0 --- n 6 best 0.75 x rounded: [0. 0.5 0. 0.5 0. 0.5] triple max 1.0000000000000004 min 0.49999999999999983 S recomputed 0.75 compare n/8 0.75 diff 0.0 compare n/9 0.6666666666666666 diff 0.08333333333333337 --- n 7 best 0.7777777777777756 x rounded: [0.333333 0.333333 0.333333 0.333333 0.333333 0.333333 0.333333] triple max 0.9999999999999993 min 0.999999999999998 S recomputed 0.7777777777777756 compare n/8 0.875 diff -0.09722222222222443 compare n/9 0.7777777777777778 diff -2.220446049250313e-15 --- n 8 best 0.999999999999997 x rounded: [0. 0.5 0. 0.5 0. 0.5 0. 0.5] triple max 0.9999999999999994 min 0.499999998767529 S recomputed 0.999999999999997 compare n/8 1.0 diff -2.9976021664879227e-15 compare n/9 0.8888888888888888 diff 0.11111111111110816 --- n 9 best 1.0000000000000024 x rounded: [0.333333 0.333333 0.333333 0.333333 0.333333 0.333333 0.333333 0.333333 0.333333] triple max 1.000000000000002 min 1.0000000000000004 S recomputed 1.0000000000000024 compare n/8 1.125 diff -0.12499999999999756 compare n/9 1.0 diff 2.4424906541753444e-15 --- n 10 best 1.249999999999991 x rounded: [0.5 0. 0.5 0. 0.5 0. 0.5 0. 0.5 0. ] triple max 1.0 min 0.5000000000000021 S recomputed 1.249999999999991 compare n/8 1.25 diff -9.103828801926284e-15 compare n/9 1.1111111111111112 diff 0.13888888888887974 --- n 11 best 1.2499999999999951 x rounded: [0. 0.5 0. 0.5 0. 0.5 0. 0.5 0.470827 0.029173 0.5 ] triple max 0.9999999999999987 min 0.4999999891280222 S recomputed 1.2499999999999951 compare n/8 1.375 diff -0.12500000000000488 compare n/9 1.2222222222222223 diff 0.027777777777772794 --- n 12 best 1.4999999999991451 x rounded: [0. 0.5 0. 0.5 0. 0.5 0. 0.5 0. 0.5 0. 0.5] triple max 0.9999999999999852 min 0.49999999753123897 S recomputed 1.4999999999991451 compare n/8 1.5 diff -8.548717289613705e-13 compare n/9 1.3333333333333333 diff 0.16666666666581187 --- n 13 best 1.5000000000003524 x rounded: [0.5 0. 0.5 0.271169 0.228831 0.5 0. 0.5 0. 0.5 0. 0.5 0. ] triple max 1.0000000000001656 min 0.5000000000000031 S recomputed 1.5000000000003524 compare n/8 1.625 diff -0.12499999999964762 compare n/9 1.4444444444444444 diff 0.055555555555907965 --- n 14 best 1.7500000000000204 x rounded: [0.5 0. 0.5 0. 0.5 0. 0.5 0. 0.5 0. 0.5 0. 0.5 0. ] triple max 1.0000000000000127 min 0.5000000000000001 S recomputed 1.7500000000000204 compare n/8 1.75 diff 2.042810365310288e-14 compare n/9 1.5555555555555556 diff 0.19444444444446485 --- n 15 best 1.7500000000000093 x rounded: [0. 0.5 0. 0.5 0. 0.5 0. 0.5 0. 0.5 0. 0.5 0.187864 0.312136 0.5 ] triple max 1.0000000000000042 min 0.49999999050955957 S recomputed 1.7500000000000093 compare n/8 1.875 diff -0.12499999999999067 compare n/9 1.6666666666666667 diff 0.08333333333334259 --- n 16 best 2.000000000000004 x rounded: [0. 0.5 0. 0.5 0. 0.5 0. 0.5 0. 0.5 0. 0.5 0. 0.5 0. 0.5] triple max 1.0000000000000029 min 0.499999998080272 S recomputed 2.000000000000004 compare n/8 2.0 diff 3.9968028886505635e-15 compare n/9 1.7777777777777777 diff 0.22222222222222632 --- [/PYTHON OUTPUT]
Reasoning Process

[THOUGHT] From the computed maxima for many even nn, the optimum consistently equals n/8n/8 (achieved by the alternating configuration 0,12,0,12,…0,\tfrac12,0,\tfrac12,\dots), hence for n=2026n=2026,

M2026=20268=10134⇒p+q=1013+4=1017.M_{2026}=\frac{2026}{8}=\frac{1013}{4}\Rightarrow p+q=1013+4=1017.

1017\boxed{1017}

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