Problem Analysis #6

Llama-VARCO-8B-Instruct · Run 1

54.67s
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Problem Statement

\subsection*{I.}Find the general solution y(x)y(x) of the following differential equation:

dydx=y(1−y),\frac{dy}{dx} = y(1-y),

where 0<y<10 < y < 1.\subsection*{II.}Find the value of the definite integral

I=∫−11arccos⁡(x2)cos⁡2(π3x) dx,I = \int_{-1}^{1} \frac{\arccos\left(\dfrac{x}{2}\right)}{\cos^2\left(\dfrac{\pi}{3}x\right)} \, dx,

where 0≤arccos⁡(x2)≤π0 \leq \arccos\left(\dfrac{x}{2}\right) \leq \pi.\subsection*{III.}For any positive variable xx, define

f(x)=∑m=0∞1(2m)!x2m,\qquadg(x)=ddxf(x).f(x) = \sum_{m=0}^{\infty} \frac{1}{(2m)!} x^{2m},\qquadg(x) = \frac{d}{dx} f(x).

For any non-negative integer nn, define

In(x)=∫0x(g(X)f(X))n dX.I_n(x) = \int_0^x \left(\frac{g(X)}{f(X)}\right)^n \, dX.

You may use

exp⁡(x)=∑m=0∞xmm!.\exp(x) = \sum_{m=0}^{\infty} \frac{x^m}{m!}.
  • [(1)] Calculate f(x)2−g(x)2f(x)^2 - g(x)^2.
  • [(2)] Express In+2(x)I_{n+2}(x) using In(x)I_n(x). \subsection*{IV. Final quantity}Let y∗(x)y_*(x) be the solution of the differential equation in {\rm I} that satisfies
y∗(0)=12,y_*(0) = \frac{1}{2},

and define

a:=y∗(ln⁡3).a := y_*(\ln 3).

In {\rm II}, let II be the value of the definite integral. In {\rm III}, let

x0=ln⁡(2+3),\qquadF0:=f(x0),G0:=g(x0).x_0 = \ln(2+\sqrt{3}),\qquadF_0 := f(x_0),\quad G_0 := g(x_0).

Define

S:=4a+IG0+F0,K:=999 S2+7.S := 4a + \frac{I}{G_0} + F_0,\qquad\mathcal{K} := 999\,S^2 + 7.

Compute the integer K\mathcal{K}.\

Ground Truth
63943
Execution Trace
Reasoning Process

[THOUGHT]

I. The given differential equation is separable: dy/dx = y(1-y) => dy/y(1-y) = dx Integrating both sides, ln|y/(1-y)| = x + C (say) Since 0 < y < 1, |y/(1-y)| = y/(1-y), so we have ln(y/(1-y)) = x + C => y/(1-y) = e^(x+C) = A (say) Here A is a positive constant. Solving for y, y = A/(A+y)

II. The given integral can be solved by substitution: u = arccos(x/2), dv = 1/(cos^2((π/3)x)) dx Then du/dx = -1/(2cos((π/3)x)sin((π/3)x)), v = (-(π/6)sin((π/3)x))/(cos((π/3)x))^3 Integrating by parts, ∫(arccos(x/2))/(cos^2((π/3)x)) dx = [-(Aπ/6)sin((π/3)x)cos((π/3)x)^3]/(-1/4) | from -1 to 1 = (Aπ/12)(1 - (-1/2)) = Aπ/6

Here, u=-1 when x=-1 and u=0 when x=1. So A = π/2. Hence I = (∏/6)/2 = π/12.

III. (1) f(x)^2 - g(x)^2 = (Σ(1/(2m)!)x^(2m))^2 - (Σ((2m)(1/(2m)!)x^(2m-1)))^2 = Σ(1/(2m)!(4m^2)x^(4m-2)) - Σ(1/(2m-1)!(2m)^2x^(4m-2)) = Σ(1/(2m)!*(4m^2 - (2m)^2)*x^(4m-2)) = Σ(1/(2m)!2m(2m-1)*x^(4m-2))

(2) I_(n+2)(x) = ∫_0^x [(g(X))/(f(X))]^(n+2) dX = ∫_0^x [Σ((2(n+2)m/(2m)!)X^(2m-1))] / [Σ(1/(2m)!X^(2m))]^(n+2) dX = ∫_0^x Σ[(2(n+2)m/(2m)!)(1/Σ(1/(2m)!)X^(2m-n))] dX = Σ[(2(n+2)m/(2m)!)∫_0^x (1/X^(n)) dX] = Σ[(2(n+2)m/(2m)!)(-1/n)[1 - (1/(1+x^n))] from 0 to x] = Σ[-(2(n+2)mn)/(n!(2m-n))[1 - (1/(1+x^n))] = Σ[-(2(n+2)mn)/(n!(2m-n))] - Σ[(2(n+2)mn)/(n!(1+x^n)(2m-n))] = I_n(x) - Σ[(2(n+2)mn)/(n!(1+x^n)(2m-n!))]

IV. I. The differential equation's solution y*(x) with initial condition y_(0)=1/2 is y = 1/(A+y), where A=1 (from ln(1/(1-y))=0). Hence, y_(ln3) = 1/(1+3^(1/2)) = sqrt(3)/2.

II. From II above, I = π/12.

III. f(x_0) = Σ(1/(2m)!)(2x_0)^(2m) g(x_0) = Σ((2m)/(2m)!)(2x_0)^(2m-1)

G_0 = g(x_0) with x_0=ln(2+sqrt(3)) = 2Σ(m/(2m)!)[(2+sqrt(3))^m] = 2Σ(m/(2m)!)(2^m)*(1+(sqrt(3)/4))^(m) ≈ 5.83 (from first few terms)

F_0 = f(x_0) = Σ(1/(2m)!)*(2ln(2+sqrt(3)))^(2m) ≈ 11.55

S = 4*sqrt(3)/2 + π/12/G_0 + F_0 ≈ 15.77

K = 999*S^2 + 7 = 1520986

1520986\boxed{1520986}

Final Model Output
1520986

Comparative Analysis (P6) (Average across runs)

Avg Time Taken

K-EXAONE-236B-A23B
330.0
GPT-5.2 (high)
111.5
Kanana-2-30B-Thinking-2601
76.9
Solar-Open-100B
67.2
GPT-5.1 (high)
57.7
HCX-007(high)
55.6
Llama-VARCO-8B-Instruct
54.7
Deepseek-V3.2
48.3
Gemini-3-Pro-Preview
47.6
A.X-4.0 (72B)
45.4
Kanana-2-30B-Thinking
42.7
Solar-Pro-2 (31B)(high)
40.2
Grok-4.1-fast
29.0
K-EXAONE-236B-A23B
22.8
Claude-Opus-4.5
20.6
EXAONE-4.0.1-32B (high)
11.1

Avg Token Usage

K-EXAONE-236B-A23B
39,933
GPT-5.2 (high)
26,193
Kanana-2-30B-Thinking-2601
19,787
Solar-Open-100B
12,619
HCX-007(high)
10,581
EXAONE-4.0.1-32B (high)
9,149
Solar-Pro-2 (31B)(high)
9,027
GPT-5.1 (high)
8,547
Grok-4.1-fast
8,351
Kanana-2-30B-Thinking
8,050
Gemini-3-Pro-Preview
7,889
A.X-4.0 (72B)
7,410
K-EXAONE-236B-A23B
6,804
Claude-Opus-4.5
5,887
Deepseek-V3.2
5,395
Llama-VARCO-8B-Instruct
1,716