Problem Analysis #2

10.22s
2,904 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] Parameterize the semicircle by (z=Re^{i\theta}) with (R=\frac52), (0\le\theta\le\pi). Then (dz=iRe^{i\theta}d\theta) and (z^2=R^2e^{2i\theta}). Compute (\int_C z^2,dz = \int_0^\pi R^2e^{2i\theta}\cdot iRe^{i\theta},d\theta) and simplify. Verify exactly with Python. [/THOUGHT] [PYTHON] import sympy as sp

R = sp.Rational(5, 2) theta = sp.symbols('theta', real=True) z = Rsp.exp(sp.Itheta) dz = sp.diff(z, theta) integrand = z**2 * dz res = sp.integrate(integrand, (theta, 0, sp.pi)) print(res) print(sp.nsimplify(res)) print("p+q =", sp.together(-res).as_numer_denom()[0] + sp.together(-res).as_numer_denom()[1]) [/PYTHON]

Tool Output
[PYTHON OUTPUT] -125/12 -125/12 p+q = 137 [/PYTHON OUTPUT]
Reasoning Process

[THOUGHT] The computed contour integral is (\int_C z^2,dz=-125/12), so (p=125), (q=12), hence (p+q=137). [ \boxed{137} ]

Final Model Output
137

Comparative Analysis (P2) (Average across runs)

Avg Time Taken

Gemini-3-Pro-Preview
56.5
naver-hyperclovax/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
model_d_r1
17.4
Solar-Open-100B
16.1
Solar-Pro 2
15.4
GPT-5.2 (high)
10.0
EXAONE-4.0-32B
9.0
axk1
6.4

Avg Token Usage

Gemini-3-Pro-Preview
8640.0
Kanana-2-30B-Thinking-2601
7490.0
Kanana-2-30B-Thinking
7071.0
naver-hyperclovax/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
model_d_r1
3374.0
GPT-5.2 (high)
3069.0
axk1
2817.0