Problem Analysis #8

Solar-Pro-2 (31B)(high) · Run 1

41.81s
9,159 toks
Problem Statement

Answer the following questions. Here, for any complex value zz, zˉ\bar{z} is the complex conjugate of zz, arg⁡z\arg z is the argument of zz, ∣z∣|z| is the absolute value of zz, and ii is the imaginary unit.\subsection*{I.}Sketch the region of zz on the complex plane that satisfies

zzˉ+2(z+zˉ)+3i(z−zˉ)+2≤0.z\bar{z} + \sqrt{2}(z + \bar{z}) + 3i(z - \bar{z}) + 2 \leq 0.

\subsection*{II.}Consider the complex-valued function

f(z)=z2−2(z2+2i)z2.f(z) = \frac{z^2 - 2}{(z^2 + 2i)z^2}.
  • [(1)] Find all the poles of f(z)f(z) as well as the orders and residues at the poles.
  • [(2)] By applying the residue theorem, find the value of the integral
I1=∮Cf(z) dz,I_1 = \oint_C f(z)\,dz,

where the integration path CC is the circle in the counterclockwise direction given by ∣z+1∣=2|z + 1| = 2. \subsection*{III.}

  • [(1)] Let g(z)g(z) be a complex-valued function satisfying
lim⁡∣z∣→∞g(z)=0\lim_{|z|\to\infty} g(z) = 0

for 0≤arg⁡z≤π0 \leq \arg z \leq \pi. Let CRC_R be the semicircle, with radius RR, in the upper half of the complex plane with center at the origin. Show that

lim⁡R→∞∫CReiazg(z) dz=0,\lim_{R\to\infty} \int_{C_R} e^{iaz} g(z)\,dz = 0,

where aa is a positive real number.

  • [(2)] Find the value of the integral
I2=∫0∞sin⁡xx dx.I_2 = \int_0^{\infty} \frac{\sin x}{x}\,dx.

\subsection*{IV. Final quantity}In I, the region for zz is a disk. Let RR be its radius and cc be its center. In II(2), let RinR_{\mathrm{in}} be the sum of the residues of f(z)f(z) over all poles lying inside the circle ∣z+1∣=2|z + 1| = 2. In II(2) and III(2), let I1I_1 and I2I_2 be the values of the corresponding integrals.Define

S:=(R2+∣c∣2)2(∣I1∣I2)3⋅1∣Rin∣,K:=999 S2+7.S := (R^2 + |c|^2)^2 \left(\frac{|I_1|}{I_2}\right)^3 \cdot \frac{1}{|R_{\mathrm{in}}|},\qquad\mathcal{K} := 999\,S^2 + 7.

Compute the integer K\mathcal{K}.\

Ground Truth
40919040007
Execution Trace
Reasoning Process

[THOUGHT] To solve the problem, we break it down into parts as follows:

I. Region of z: The inequality zzˉ+2(z+zˉ)+3i(z−zˉ)+2≤0z\bar{z} + \sqrt{2}(z + \bar{z}) + 3i(z - \bar{z}) + 2 \leq 0 is transformed into a circle equation by substituting z=x+iyz = x + iy. This results in the circle (x+2)2+(y−3)2≤9(x + \sqrt{2})^2 + (y - 3)^2 \leq 9, giving a radius R=3R = 3 and center c=−2+3ic = -\sqrt{2} + 3i. The modulus of the center is ∣c∣=11|c| = \sqrt{11}.

II(1). Poles and Residues of f(z)f(z): The function f(z)=z2−2(z2+2i)z2f(z) = \frac{z^2 - 2}{(z^2 + 2i)z^2} has poles at z=0z = 0 (order 2), z=1−iz = 1 - i, and z=−1+iz = -1 + i (both order 1). The residues are:

  • Res(f,0)=0\text{Res}(f, 0) = 0
  • Res(f,1−i)=12\text{Res}(f, 1 - i) = \frac{1}{2}
  • Res(f,−1+i)=−12\text{Res}(f, -1 + i) = -\frac{1}{2}

II(2). Integral I1I_1: The contour ∣z+1∣=2|z + 1| = 2 includes poles at z=0z = 0 and z=−1+iz = -1 + i. The sum of residues inside is 0+(−12)=−120 + (-\frac{1}{2}) = -\frac{1}{2}. Thus, I1=2πi×(−12)=−πiI_1 = 2\pi i \times (-\frac{1}{2}) = -\pi i, giving ∣I1∣=π|I_1| = \pi.

III(2). Integral I2I_2: The integral ∫0∞sin⁡xx dx\int_0^\infty \frac{\sin x}{x} \, dx is known to be π2\frac{\pi}{2}.

Final Calculation: Using the values:

  • R=3R = 3, ∣c∣=11|c| = \sqrt{11}
  • ∣I1∣=π|I_1| = \pi, I2=π2I_2 = \frac{\pi}{2}
  • ∣Rin∣=12|R_{\text{in}}| = \frac{1}{2}

We compute:

S=(R2+∣c∣2)2(∣I1∣I2)3⋅1∣Rin∣=(9+11)2⋅23⋅2=400⋅8⋅2=6400S = (R^2 + |c|^2)^2 \left( \frac{|I_1|}{I_2} \right)^3 \cdot \frac{1}{|R_{\text{in}}|} = (9 + 11)^2 \cdot 2^3 \cdot 2 = 400 \cdot 8 \cdot 2 = 6400 K=999⋅S2+7=999⋅40960000+7=40919040007\mathcal{K} = 999 \cdot S^2 + 7 = 999 \cdot 40960000 + 7 = 40919040007

[PYTHON]

Calculations for the final answer

R = 3 c_modulus = (20.5)2 + 32 R_squared = R2 c_squared = c_modulus sum_R_c = R_squared + c_squared S_squared = sum_R_c2 I1_abs = 3.141592653589793 # pi I2 = 3.141592653589793 / 2 # pi/2 ratio = I1_abs / I2 ratio_cubed = ratio3 R_in_abs = 0.5 S = S_squared * ratio_cubed * (1 / R_in_abs) K = 999 * S**2 + 7 print(int(K)) [/PYTHON]

Tool Output
[PYTHON OUTPUT] 40919040007 [/PYTHON OUTPUT]
Reasoning Process

[THOUGHT] The Python output confirms the calculation of K=40919040007\mathcal{K} = 40919040007. [/THOUGHT] 40919040007\boxed{40919040007}

Final Model Output
40919040007

Comparative Analysis (P8) (Average across runs)

Avg Time Taken

K-EXAONE-236B-A23B
1791.1
Gemini-3-Pro-Preview
276.8
Deepseek-V3.2
202.1
Kanana-2-30B-Thinking
159.7
Solar-Open-100B
95.8
GPT-5.1 (high)
91.6
K-EXAONE-236B-A23B
90.0
Llama-VARCO-8B-Instruct
63.3
Grok-4.1-fast
60.2
Kanana-2-30B-Thinking-2601
60.1
A.X-4.0 (72B)
58.6
Claude-Opus-4.5
42.7
Solar-Pro-2 (31B)(high)
41.8
GPT-5.2 (high)
31.0
HCX-007(high)
20.4
EXAONE-4.0.1-32B (high)
5.5

Avg Token Usage

K-EXAONE-236B-A23B
199,182
Solar-Open-100B
32,831
Kanana-2-30B-Thinking
23,414
Kanana-2-30B-Thinking-2601
18,842
Gemini-3-Pro-Preview
15,034
Claude-Opus-4.5
12,309
Grok-4.1-fast
10,596
K-EXAONE-236B-A23B
9,431
Solar-Pro-2 (31B)(high)
9,159
A.X-4.0 (72B)
9,042
GPT-5.1 (high)
8,139
Deepseek-V3.2
8,047
GPT-5.2 (high)
5,585
HCX-007(high)
5,016
EXAONE-4.0.1-32B (high)
4,747
Llama-VARCO-8B-Instruct
1,832