Problem Analysis #2

64.61s
17,622 toks
Problem Statement

Evaluate the integral

I=∫−∞∞P(x) eix(x−3i)8 dx,I = \int_{-\infty}^{\infty} \frac{P(x)\, e^{ix}}{(x - 3i)^{8}}\,dx,

where

P(x)=x7−14ix6+(85−60i)x5−(240+210i)x4+(420−840i)x3−(560+1680i)x2+(1344−2688i)x+3840.P(x) = x^{7} - 14ix^{6} + (85 - 60i)x^{5} - (240 + 210i)x^{4} + (420 - 840i)x^{3} - (560 + 1680i)x^{2} + (1344 - 2688i)x + 3840.

If the value of the integral is I=a+biI = a + bi, ii is the imaginary unit, compute

⌊10a⌋+⌊b⌋.\lfloor 10a \rfloor + \lfloor b \rfloor.
Ground Truth
7
Execution Trace
Reasoning Process

<|channel|>commentary to=python code<|message|>import sympy as sp

x = sp.symbols('x') P = x7 - 14sp.Ix6 + (85-60sp.I)x**5 - (240+210sp.I)x**4 + (420-840sp.I)x**3 - (560+1680sp.I)x**2 + (1344-2688sp.I)x + 3840 expr = P * sp.exp(sp.Ix) der7 = sp.diff(expr, x, 7) val_at_3i = der7.subs(x, 3sp.I).simplify() val_at_3i

Tool Output
[PYTHON OUTPUT] 6*(5479 - 1608*I)*exp(-3) [/PYTHON OUTPUT]
Reasoning Process

<|channel|>commentary to=python code<|message|>import sympy as sp val = 6*(5479 - 1608sp.I)sp.exp(-3) Res = val / sp.factorial(7) # 5040 I_val = 2sp.pisp.I * Res sp.simplify(I_val)

Tool Output
[PYTHON OUTPUT] I*pi*(5479 - 1608*I)*exp(-3)/420 [/PYTHON OUTPUT]
Reasoning Process

<|channel|>commentary to=python code<|message|>import math a = math.pi * math.exp(-3) * 1608 / 420 b = math.pi * math.exp(-3) * 5479 / 420 a, b

Tool Output
[PYTHON OUTPUT] (0.5988294920738872, 2.0404146685776294) [/PYTHON OUTPUT]
Reasoning Process

<|channel|>commentary to=python code<|message|>import sympy as sp sp.nsimplify(10sp.pisp.exp(-3)*1608/420)

Tool Output
[PYTHON OUTPUT] 268*pi*exp(-3)/7 [/PYTHON OUTPUT]
Reasoning Process

<|channel|>commentary to=python code<|message|>import sympy as sp sp.mp.dps = 50 a_exact = (1608 * sp.pi * sp.exp(-3)) / 420 b_exact = (5479 * sp.pi * sp.exp(-3)) / 420 a_val = sp.N(a_exact, 50) b_val = sp.N(b_exact, 50) a_val, b_val

Tool Output
[PYTHON OUTPUT] AttributeError: module 'sympy' has no attribute 'mp' [/PYTHON OUTPUT]
Reasoning Process

<|channel|>commentary to=python code<|message|>import mpmath as mp mp.mp.dps = 50 a_val = (134 * mp.pi * mp.e**(-3)) / 35 b_val = (5479 * mp.pi * mp.e**(-3)) / 420 a_val, b_val

Tool Output
[PYTHON OUTPUT] (mpf('0.59882949207388728212581087414109181624123993032463569'), mpf('2.0404146685776296136612672757581107345682547128412228')) [/PYTHON OUTPUT]
Reasoning Process

<|channel|>analysis to=python code<|message|>import math a = (134/35)math.pimath.exp(-3) b = (5479/420)math.pimath.exp(-3) a, b

Tool Output
[PYTHON OUTPUT] (0.5988294920738872, 2.0404146685776294) [/PYTHON OUTPUT]
Reasoning Process

[THOUGHT] We have the integral value (I = a + bi) with (a \approx 0.5988294920738872) (b \approx 2.0404146685776294). We need to compute (\lfloor 10a \rfloor + \lfloor b \rfloor).

Compute (10a = 5.988294920738872).
Floor of that is (\lfloor 10a \rfloor = 5).

Compute floor of (b = 2.0404146685776294) gives (\lfloor b \rfloor = 2).

Sum: (5 + 2 = 7).

Thus the required integer is (\boxed{7}).

Final Model Output
7

Comparative Analysis (P2) (Average across runs)

Avg Time Taken

Qwen3-30B-A3B-2507
7431.3
Deepseek-R1-distill-Qwen-32B (high)
859.1
Gemini-3-Pro-Preview
199.4
Deepseek-V3.2
116.7
GPT-oss-20B (high)
86.7
Gemma-3-27B
83.2
Solar-Pro-2 (31B)(high)
71.2
HCX-007(high)
61.9
A.X-4.0 (72B)
42.6
Llama-VARCO-8B-Instruct
35.1
Grok-4.1-fast
22.6
Claude-Opus-4.5
15.8
GPT-5.1 (high)
13.1
EXAONE-4.0.1-32B (high)
5.4

Avg Token Usage

Qwen3-30B-A3B-2507
24109.7
GPT-oss-20B (high)
19053.3
Gemini-3-Pro-Preview
17645.0
Deepseek-R1-distill-Qwen-32B (high)
11660.7
HCX-007(high)
11315.0
Solar-Pro-2 (31B)(high)
10817.0
Deepseek-V3.2
5996.0
A.X-4.0 (72B)
5881.3
Grok-4.1-fast
5630.3
Llama-VARCO-8B-Instruct
5237.0
Claude-Opus-4.5
4038.7
EXAONE-4.0.1-32B (high)
3954.7
Gemma-3-27B
3894.7
GPT-5.1 (high)
3378.0