Problem Analysis #8

2665.83s

300,784 toks

Problem Statement

Answer the following questions. Here, for any complex value $z$ , $\bar{z}$ is the complex conjugate of $z$ , $\arg z$ is the argument of $z$ , $|z|$ is the absolute value of $z$ , and $i$ is the imaginary unit.\subsection*{I.}Sketch the region of $z$ on the complex plane that satisfies $z\bar{z} + \sqrt{2}(z + \bar{z}) + 3i(z - \bar{z}) + 2 \leq 0.$ \subsection*{II.}Consider the complex-valued function $f(z) = \frac{z^2 - 2}{(z^2 + 2i)z^2}.$

[(1)] Find all the poles of $f(z)$ as well as the orders and residues at the poles.
[(2)] By applying the residue theorem, find the value of the integral $I_1 = \oint_C f(z)\,dz,$ where the integration path $C$ is the circle in the counterclockwise direction given by $|z + 1| = 2$ . \subsection*{III.}
[(1)] Let $g(z)$ be a complex-valued function satisfying $\lim_{|z|\to\infty} g(z) = 0$ for $0 $$\leq$$ \arg z $$\leq$$ $$\pi$$$. Let$ C_R $be the semicircle, with radius$ R $, in the upper half of the complex plane with center at the origin. Show that $$ \lim_{R\to\infty} \int_{C_R} e^{iaz} g(z)\,dz = 0, $$ where$ a$ is a positive real number.
[(2)] Find the value of the integral $I_2 = \int_0^{\infty} \frac{\sin x}{x}\,dx.$ \subsection*{IV. Final quantity}In I, the region for $z$ is a disk. Let $R$ be its radius and $c$ be its center. In II(2), let $R_{\mathrm{in}}$ be the sum of the residues of $f(z)$ over all poles lying inside the circle $|z + 1| = 2$ . In II(2) and III(2), let $I_1$ and $I_2$ be the values of the corresponding integrals.Define $S := (R^2 + |c|^2)^2 \left(\frac{|I_1|}{I_2}\right)^3 \cdot \frac{1}{|R_{\mathrm{in}}|},\qquad\mathcal{K} := 999\,S^2 + 7.$ Compute the integer $\mathcal{K}$ .\

Ground Truth

40919040007

Execution Trace

Reasoning Process

\boxed{40919040007}

Reasoning Process

[THOUGHT] The problem is multi-part but reduces to finding certain constants and residues. For part I, the inequality simplifies to ((x + \sqrt{2})^2 + (y - 3)^2 \le 9), giving a disk with center (c = -\sqrt{2} + 3i) and radius (R = 3). So (R^2 + |c|^2 = 9 + 11 = 20).
For part II, (f(z) = \frac{z^2 - 2}{z^2(z^2 + 2i)}). The poles inside (|z+1|=2) are at (z=0) (double) and (z=-1+i) (simple). The residue sum is (R_{\in} = -\frac{1}{2}). The integral is (I_1 = 2\pi i R_{in} = -\pi i), so (|I_1| = \pi).
For part III, the standard Dirichlet integral gives (I_2 = \frac{\pi}{2}).
Thus (\frac{|I_1|}{I_2} = 2) and (\frac{1}{|R_{in}|} = 2).
Hence (S = (20)^2 \cdot 2^3 \cdot 2 = 400 \cdot 8 \cdot 2 = 6400).
Finally, (\mathcal{K} = 999 S^2 + 7 = 999 \cdot 6400^2 + 7 = 999 \cdot 40,960,000 + 7 = 40,919,040,007).
We can verify with Python. [/THOUGHT] [PYTHON] K = 999 * 6400**2 + 7 print(K) [/PYTHON] \boxed{40919040007}

Tool Output

[PYTHON OUTPUT] 40919040007 [/PYTHON OUTPUT]

Reasoning Process

[THOUGHT] I will use Python to verify the residues and the integral I1, and then compute S and K.

[PYTHON] import sympy as sp

Define variable

z = sp.symbols('z')

Define f(z)

f = (z2 - 2) / (z2 * (z**2 + 2*sp.I))

Poles: solve denominator = 0

poles = sp.solve(z2 * (z2 + 2*sp.I), z) print("Poles:", poles)

For each pole, determine if inside |z+1| = 2

Condition: |z + 1| < 2

inside_poles = [] for p in poles: # Evaluate |p + 1| if abs(p + 1) < 2: inside_poles.append(p) print(f"Inside: {p}, |z+1| = {abs(p+1)}") else: print(f"Outside: {p}, |z+1| = {abs(p+1)}")

Compute residues at poles inside

residues_inside = [] for p in inside_poles: # For multiple poles, sympy.residue can handle residue = sp.residue(f, z, p) residues_inside.append(residue) print(f"Residue at {p}: {residue}")

sum_residues = sum(residues_inside) print("Sum of residues inside:", sum_residues) print("Absolute value of sum of residues:", abs(sum_residues))

Compute I1 = 2piI * sum_residues

I1 = 2 * sp.pi * sp.I * sum_residues print("I1 =", I1) print("|I1| =", abs(I1))

I2 = pi/2

I2 = sp.pi/2 print("I2 =", I2)

Compute |I1|/I2

ratio = abs(I1) / I2 print("|I1|/I2 =", ratio)

Compute S = (R^2 + |c|^2)^2 * (|I1|/I2)^3 * (1/|R_in|)

R = 3 c = -sp.sqrt(2) + 3*sp.I c_sq = abs(c)2 R_sq_plus_c_sq = R2 + c_sq print("R^2 + |c|^2 =", R_sq_plus_c_sq)

S = (R_sq_plus_c_sq2) * (ratio3) * (1/abs(sum_residues)) print("S =", S)

Compute K = 999*S^2 + 7

K = 999 * (float(S)**2) + 7 print("K =", K) [/PYTHON]

Tool Output

[PYTHON OUTPUT] TypeError: Cannot convert complex to float [/PYTHON OUTPUT]

Reasoning Process

[THOUGHT] I'll use Python to compute the residues and then S and K. I'll compute step by step.

[PYTHON] import sympy as sp

Define variable

z = sp.symbols('z') I = sp.I

Define f(z)

f = (z2 - 2) / (z2 * (z**2 + 2*I))

Poles: solve denominator = 0

denom = z2 * (z2 + 2*I) poles = sp.solve(denom, z) print("Poles:", poles)

Function to classify poles

def is_inside(z_val, center=-1, radius=2): return abs(z_val - center) < radius

We'll compute residues at each pole and sum those inside

residues_inside = [] for p in poles: # Check if p is inside |z+1| = 2 if is_inside(p): # Compute residue at p # For multiple poles, use sympy.residue # But sympy.residue expects integer order? We'll compute manually. # For p=0, order 2 if sp.simplify(p) == 0: # Compute residue at 0: Res = limit d/dz [ z^2 f(z) ] at z=0 g = (z2 - 2) / (z2 + 2I) # z^2 f(z) = (z^2-2)/(z^2+2i) # Actually careful: z^2 f(z) = (z^2 - 2) / (z^2 + 2I) g = (z2 - 2) / (z2 + 2I) # residue = derivative at 0 residue_val = sp.diff(g, z).subs(z, 0) residue_val = sp.simplify(residue_val) residues_inside.append(residue_val) print(f"Residue at {p} (double pole): {residue_val}") else: # For simple pole, use limit formula # residue = lim_{z->p} (z-p)*f(z) residue_val = sp.limit((z - p)*f, z, p) residues_inside.append(residue_val) print(f"Residue at {p} (simple pole): {residue_val}") else: print(f"{p} is outside") else: print(f"{p} is outside")

If any poles not computed? Let's ensure we cover all inside poles.

Actually, we need to check all poles: 0, 1-I, -1+I.

Let's compute residues at each pole individually and sum those inside.

Recompute more systematically:

poles_list = poles # [0, -1+I, 1-I]? Actually solve gives [0, -1+I, 1-I]? Let's check.

print("\nDetailed computation:") all_residues = {} for p in poles: # Compute residue using sympy.residue for simple poles, but for double at 0 we handle separately. if sp.simplify(p) == 0: # double pole # residue = limit_{z->0} d/dz [ z^2 f(z) ] g = (z2 - 2) / (z2 + 2*I) residue_val = sp.limit(sp.diff(g, z), z, 0) all_residues[p] = sp.simplify(residue_val) print(f"Residue at {p} (double): {all_residues[p]}") else: # simple pole residue_val = sp.limit((z - p)*f, z, p) all_residues[p] = sp.simplify(residue_val) print(f"Residue at {p} (simple): {all_residues[p]}")

Sum residues inside |z+1|<2

R_in = 0 center = -1 radius = 2 for p, res in all_residues.items(): if abs(p - center) < radius: R_in += res print(f"Adding residue at {p} = {res}")

print(f"\nSum of residues inside C (R_in) = {R_in}") print(f"Absolute value |R_in| = {abs(R_in)}")

Compute I1 = 2piI * R_in

I1 = 2 * sp.pi * sp.I * R_in print(f"\nI1 = {I1}") print(f"|I1| = {abs(I1)}")

Compute I2 = pi/2

I2 = sp.pi / 2 print(f"I2 = {I2}")

Compute |I1|/I2

ratio = abs(I1) / I2 print(f"|I1|/I2 = {ratio}")

Now compute S = (R^2 + |c|^2)^2 * (|I1|/I2)^3 * (1/|R_in|)

For part I: R = 3, c = -sqrt(2) + 3i

R_val = 3 c = -sp.sqrt(2) + 3*I c_abs_sq = c**2.conjugate() # Actually |c|^2 = c * conjugate(c) c_abs_sq = sp.simplify(c * sp.conjugate(c)) print(f"\n|c|^2 = {c_abs_sq}")

Alternatively, compute directly: (-√2)^2 + 3^2 = 2+9=11.

So R^2 + |c|^2 = 9+11=20.

R_sq_plus_c_sq = R_val**2 + c_abs_sq print(f"R^2 + |c|^2 = {R_sq_plus_c_sq}")

Now compute S

S = (R_sq_plus_c_sq2) * (ratio3) * (1/abs(R_in)) S = sp.simplify(S) print(f"S = {S}")

Compute K = 999 * S^2 + 7

K = 999 * (S**2) + 7 K = sp.simplify(K) print(f"K = {K}") [/PYTHON]

Tool Output

[PYTHON OUTPUT] IndentationError: unindent does not match any outer indentation level (<string>, line 44) [/PYTHON OUTPUT]

Reasoning Process

[THOUGHT] I'll use Python to compute the residues and then S and K.

[PYTHON] import sympy as sp

Define variable

z = sp.symbols('z') I = sp.I

Define f(z)

f = (z2 - 2) / (z2 * (z**2 + 2*I))

Poles: solve denominator = 0

poles = sp.solve(z2 * (z2 + 2*I), z) print("Poles:", poles)

Compute residues at each pole

residues = {} for p in poles: if sp.simplify(p) == 0: # double pole at 0: residue = d/dz [z^2 f(z)] at z=0 g = (z2 - 2) / (z2 + 2*I) residue_val = sp.limit(sp.diff(g, z), z, 0) residues[p] = sp.simplify(residue_val) else: # simple pole: residue = lim_{z->p} (z-p) f(z) residue_val = sp.limit((z - p) * f, z, p) residues[p] = sp.simplify(residue_val) print(f"Residue at {p}: {residues[p]}")

Now determine which poles are inside |z+1| = 2

center = -1 radius = 2 inside_poles = [] for p in poles: if abs(p - center) < radius: inside_poles.append(p) print(f"{p} is inside") else: print(f"{p} is outside")

Sum residues of inside poles

R_in = sum(residues[p] for p in inside_poles) print(f"\nSum of residues inside C (R_in) = {R_in}") print(f"|R_in| = {abs(R_in)}")

Compute I1 = 2piI * R_in

I1 = 2 * sp.pi * sp.I * R_in print(f"\nI1 = {I1}") print(f"|I1| = {abs(I1)}")

Compute I2 = pi/2

I2 = sp.pi / 2 print(f"I2 = {I2}")

Compute |I1|/I2

ratio = abs(I1) / I2 print(f"|I1|/I2 = {ratio}")

Now compute S = (R^2 + |c|^2)^2 * (|I1|/I2)^3 * (1/|R_in|)

For part I: R = 3, c = -sqrt(2) + 3i

R_val = 3 c = -sp.sqrt(2) + 3*I c_abs_sq = (sp.sqrt(2))2 + 32 # |c|^2 = (sqrt2)^2 + 3^2 = 2+9=1 troubleshooting: use sympy for exact c_abs_sq = sp.simplify(c * sp.conjugate(c)) print(f"\n|c|^2 = {c_abs_sq}") R_sq_plus_c_sq = R_val**2 + c_abs_sq print(f"R^2 + |c|^2 = {R_sq_plus_c_sq}")

Compute S

S = (R_sq_plus_c_sq2) * (ratio3) * (1/abs(R_in)) S = sp.simplify(S) print(f"\nS = {S}")

Compute K = 999*S^2 + 7

K = 999 * (S**2) + 7 K = sp.simplify(K) print(f"K = {K}")

Also compute numeric value

print(f"\nNumeric K = {float(K)}") [/PYTHON]

Tool Output

[PYTHON OUTPUT] Poles: [0, -1 + I, 1 - I] Residue at 0: 0 Residue at -1 + I: -1/2 Residue at 1 - I: 1/2 0 is inside -1 + I is inside 1 - I is outside Sum of residues inside C (R_in) = -1/2 |R_in| = 1/2 I1 = -I*pi |I1| = pi I2 = pi/2 |I1|/I2 = 2 |c|^2 = 11 R^2 + |c|^2 = 20 S = 6400 K = 40919040007 Numeric K = 40919040007.0 [/PYTHON OUTPUT]

Reasoning Process

[THOUGHT] I'll use Python to compute the residues and then S and K.

[PYTHON] import sympy as sp

Define variable

z = sp.symbols('z') I = sp.I

Define f(z)

f = (z2 - 2) / (z2 * (z**2 + 2*I))

Poles: solve denominator = 0

poles = sp.solve(z2 * (z2 + 2*I), z) print("Poles:", poles)

Compute residues at each pole

Determine which poles are inside |z+1| = 2

center = -1 radius = 2 inside_poles = [] for p in poles: if abs(p - center) < radius: inside_poles.append(p) print(f"{p} is inside") else: print(f"{p} is outside")

Sum residues of inside poles

R_in = sum(residues[p] for p in inside_poles) print(f"\nSum of residues inside C (R_in) = {R_in}") print(f"|R_in| = {abs(R_in)}")

Compute I1 = 2piI * R_in

I1 = 2 * sp.pi * sp.I * R_in print(f"\nI1 = {I1}") print(f"|I1| = {abs(I1)}")

Compute I2 = pi/2

I2 = sp.pi / 2 print(f"I2 = {I2}")

Compute |I1|/I2

ratio = abs(I1) / I2 print(f"|I1|/I2 = {ratio}")

Now compute S = (R^2 + |c|^2)^2 * (|I1|/I2)^3 * (1/|R_in|)

For part I: R = 3, c = -sqrt(2) + 3i

R_val = 3 c = -sp.sqrt(2) + 3*I c_abs_sq = (sp.sqrt(2))2 + 32 # |c|^2 = (sqrt2)^2 + 3^2 = 2+9=11 print(f"\n|c|^2 = {c_abs_sq}") R_sq_plus_c_sq = R_val**2 + c_abs_sq print(f"R^2 + |c|^2 = {R_sq_plus_c_sq}")

Compute S

S = (R_sq_plus_c_sq2) * (ratio3) * (1/abs(R_in)) S = sp.simccept(S) print(f"\nS = {S}")

Compute K = 999*S^2 + 7

K = 999 * (S**2) + 7 K = sp.simplify(K) print(f"K = {K}")

Also compute numeric value

print(f"Numeric K = {float(K)}") [/PYTHON]

Tool Output

[PYTHON OUTPUT] AttributeError: module 'sympy' has no attribute 'simccept' [/PYTHON OUTPUT]

Reasoning Process

[THOUGHT] I have computed the values and verified with Python:

From I: R = 3, c = -√2 + 3i, so |c|^2 = 11, thus R^2 + |c|^2 = 20.
R_in = -1/2
|I1| = π, I2 = π/2, so |I1|/I2 = 2
S = (20^2) * (2^3) * (1/(1/2)) = 400 * 8 * 2 = 6400
K = 999 * (6400^2) + 7 = 999 * 40,960,000 + 7 = 40,919,040,007

Thus, the integer K is 40,919,040,007.

\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

axk1

17.5

EXAONE-4.0.1-32B (high)

5.5

Avg Token Usage

K-EXAONE-236B-A23B

199182.0

Solar-Open-100B

32831.0

Kanana-2-30B-Thinking

23414.0

Kanana-2-30B-Thinking-2601

18841.5

Gemini-3-Pro-Preview

15034.0

Claude-Opus-4.5

12309.0

Grok-4.1-fast

10596.0

K-EXAONE-236B-A23B

9431.0

Solar-Pro-2 (31B)(high)

9159.0

A.X-4.0 (72B)

9042.0

GPT-5.1 (high)

8139.0

Deepseek-V3.2

8047.0

GPT-5.2 (high)

5585.0

axk1

5107.0

HCX-007(high)

5016.0

EXAONE-4.0.1-32B (high)

4747.0

Llama-VARCO-8B-Instruct

1832.0