Problem Analysis #2

GPT-5.2 (high) · Run 1

10.22s

2,904 toks

Problem Statement

Let

f(z) = z^{2},

and consider the upper semicircular arc

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

Let

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

and form the closed contour

\Gamma = C \cup L,

where $C$ is traversed from $z = \tfrac{5}{2}$ to $z = -\tfrac{5}{2}$ , and $L$ is traversed from $x = -\tfrac{5}{2}$ to $x = \tfrac{5}{2}$ .\

Suppose the contour integral

\int_C f(z)\, dz=-\frac{p}{q},

where $p$ and $q$ are positive integers which are relatively prime. Find $p+q$ .

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

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