Problem Analysis #2

Kanana-2-30B-Thinking · Run 1

37.60s
7,400 toks
Problem Statement

Let

f(z)=z2,f(z) = z^{2},

and consider the upper semicircular arc

C:z=52eiθ,0≤θ≤π.C : z = \frac{5}{2} e^{i\theta}, \qquad 0 \le \theta \le \pi.

Let

L={ x∈R:−52≤x≤52 },L = \{\, x \in \mathbb{R} : -\tfrac{5}{2} \le x \le \tfrac{5}{2} \,\},

and form the closed contour

Γ=C∪L,\Gamma = C \cup L,

where CC is traversed from z=52z = \tfrac{5}{2} to z=−52z = -\tfrac{5}{2}, and LL is traversed from x=−52x = -\tfrac{5}{2} to x=52x = \tfrac{5}{2}.\

Suppose the contour integral

∫Cf(z) dz=−pq,\int_C f(z)\, dz=-\frac{p}{q},

where pp and qq are positive integers which are relatively prime. Find p+qp+q.

Ground Truth
137
Execution Trace
Reasoning Process

[THOUGHT] I need to compute the contour integral over the upper semicircle C using sympy. The integral is given by ∫_C z^2 dz for z = (5/2) e^(iθ) with θ from 0 to π. I'll parameterize the contour and use sympy to compute the exact value. [/THOUGHT] [PYTHON] import sympy as sp theta = sp.symbols('theta') z = (5/2) * sp.exp(sp.I * theta) dz = sp.diff(z, theta) f_z = z**2 integral = sp.integrate(f_z * dz, (theta, 0, sp.pi)) print(sp.simplify(integral)) [/PYTHON]

Tool Output
[PYTHON OUTPUT] -10.4166666666667 [/PYTHON OUTPUT]
Reasoning Process

[THOUGHT] The Python output of -10.4166666666667 corresponds to the exact fraction -125/12. Since the integral is given as -p/q and p and q are positive integers that are relatively prime, we have p = 125 and q = 12. Therefore, p + q = 137. [/THOUGHT] 137\boxed{137}

Final Model Output
137

Comparative Analysis (P2) (Average across runs)

Avg Time Taken

Gemini-3-Pro-Preview
56.5
HCX-007
42.2
K-EXAONE-236B-A23B
39.8
Kanana-2-30B-Thinking
35.4
GLM-4.5-Air
29.0
Kanana-2-30B-Thinking-2601
28.5
Solar-Open-100B
16.1
Solar-Pro 2
15.4
GPT-5.2 (high)
10.0
EXAONE-4.0-32B
9.0

Avg Token Usage

Gemini-3-Pro-Preview
8640.0
Kanana-2-30B-Thinking-2601
7490.0
Kanana-2-30B-Thinking
7071.0
HCX-007
5399.3
Solar-Pro 2
5336.0
Solar-Open-100B
4666.0
K-EXAONE-236B-A23B
4476.7
GLM-4.5-Air
3903.0
EXAONE-4.0-32B
3641.0
GPT-5.2 (high)
3069.0