Problem Analysis #6

111.49s

26,193 toks

Problem Statement

\subsection*{I.}Find the general solution $y(x)$ of the following differential equation: $\frac{dy}{dx} = y(1-y),$ where $0 < y < 1$ .\subsection*{II.}Find the value of the definite integral $I = \int_{-1}^{1} \frac{\arccos\left(\dfrac{x}{2}\right)}{\cos^2\left(\dfrac{\pi}{3}x\right)} \, dx,$ where $0 $$\leq$$ \arccos\left(\dfrac{x}{2}\right) $$\leq$$ $$\pi$$$.\subsection*{III.}For any positive variable$ x $, define$$f(x) = \sum_{m=0}^{\infty} \frac{1}{(2m)!} x^{2m},\qquadg(x) = \frac{d}{dx} f(x).$$For any non-negative integer$ n$, define $I_n(x) = \int_0^x \left(\frac{g(X)}{f(X)}\right)^n \, dX.$ You may use $\exp(x) = \sum_{m=0}^{\infty} \frac{x^m}{m!}.$

[(1)] Calculate $f(x)^2 - g(x)^2$ .
[(2)] Express $I_{n+2}(x)$ using $I_n(x)$ . \subsection*{IV. Final quantity}Let $y_*(x)$ be the solution of the differential equation in {\rm I} that satisfies $y_*(0) = \frac{1}{2},$ and define $a := y_*(\ln 3).$ In {\rm II}, let $I$ be the value of the definite integral. In {\rm III}, let $x_0 = \ln(2+\sqrt{3}),\qquadF_0 := f(x_0),\quad G_0 := g(x_0).$ Define $S := 4a + \frac{I}{G_0} + F_0,\qquad\mathcal{K} := 999\,S^2 + 7.$ Compute the integer $\mathcal{K}$ .\

Ground Truth

63943

Execution Trace

Reasoning Process

[THOUGHT] Compute each component (logistic solution parameter a, integral I, and hyperbolic values F0,G0) exactly with SymPy, then form S and K and confirm K is an integer. [/THOUGHT] [PYTHON] import sympy as sp x=sp.symbols('x', real=True)

Part I: logistic solution with y(0)=1/2

a = sp.Rational(3,4) # derived directly

Part II: integral I

integrand = sp.acos(x/2) / sp.cos(sp.pi*x/3)**2 I = sp.integrate(integrand, (x, -1, 1)) print("I =", I) print("I simplified =", sp.simplify(I))

Part III/IV hyperbolic values

x0 = sp.log(2+sp.sqrt(3)) F0 = sp.cosh(x0) G0 = sp.sinh(x0) print("F0 =", sp.simplify(F0)) print("G0 =", sp.simplify(G0))

S = 4*a + I/G0 + F0 S_simplified = sp.simplify(S) print("S =", S_simplified)

K = sp.simplify(999*S_simplified**2 + 7) print("K =", K) print("K is integer?", K.is_integer)

numeric check

Kv = sp.N(K, 50) Iv = sp.N(I, 50) Sv = sp.N(S_simplified, 50) print("numeric I =", Iv) print("numeric S =", Sv) print("numeric K =", Kv) PYTHON] [/PYTHON][THOUGHT] Compute each component (logistic solution parameter a, integral I, and hyperbolic values F0,G0) exactly with SymPy, then form S and K and confirm K is an integer. [/THOUGHT] [PYTHON] import sympy as sp x=sp.symbols('x', real=True)

Part I: logistic solution with y(0)=1/2 gives a = y(ln 3) = 3/4

a = sp.Rational(3,4)

Part II: integral I

integrand = sp.acos(x/2) / sp.cos(sp.pi*x/3)**2 I = sp.integrate(integrand, (x, -1, 1)) I_s = sp.simplify(I)

Part IV hyperbolic values

x0 = sp.log(2+sp.sqrt(3)) F0 = sp.simplify(sp.cosh(x0)) G0 = sp.simplify(sp.sinh(x0))

S = sp.simplify(4a + I_s/G0 + F0) K = sp.simplify(999S**2 + 7)

print("I =", I) print("I simplified =", I_s) print("F0 =", F0) print("G0 =", G0) print("S =", S) print("K =", K) print("K is integer?", K.is_integer) print("numeric check:") print("I ~", sp.N(I_s, 80)) print("S ~", sp.N(S, 80)) print("K ~", sp.N(K, 80)) PYTHON] [/PYTHON]

Tool Output

[PYTHON OUTPUT] SyntaxError: unmatched ']' (<string>, line 35) [/PYTHON OUTPUT] [PYTHON OUTPUT] SyntaxError: unmatched ']' (<string>, line 31) [/PYTHON OUTPUT]

Reasoning Process

[THOUGHT] Re-run clean Python to compute I via integration by parts result and confirm numerical integration; compute F0,G0,a,S,K exactly and output K. [/THOUGHT] [PYTHON] import sympy as sp import mpmath as mp

x = sp.Symbol('x', real=True)

Part I

a = sp.Rational(3,4)

Part II analytic via integration by parts and oddness

I_exact = 3*sp.sqrt(3)

numeric check of integral

f_num = lambda t: mp.acos(t/2)/(mp.cos(mp.pi*t/3)**2) I_num = mp.quad(f_num, [-1, 1])

Part IV: f=cosh, g=sinh

x0 = sp.log(2+sp.sqrt(3)) F0 = sp.simplify(sp.cosh(x0)) G0 = sp.simplify(sp.sinh(x0))

S = sp.simplify(4a + I_exact/G0 + F0) K = sp.simplify(999S**2 + 7)

print("I_exact =", I_exact) print("I_num =", mp.nsum(lambda k: 0, [0,0]) if False else I_num) # just prints I_num print("F0 =", F0) print("G0 =", G0) print("S =", S) print("K =", K) print("K integer?", bool(sp.Integer(K) == K)) PYTHON]

Tool Output

[PYTHON OUTPUT] SyntaxError: unmatched ']' (<string>, line 31) [/PYTHON OUTPUT]

Reasoning Process

[THOUGHT] The previous SyntaxError indicates the executed string likely had a stray bracket; I'll rerun with a minimal, clean script (no odd constructs) to compute I numerically for verification, compute F0,G0 exactly, then S and K exactly. [/THOUGHT] [PYTHON] import sympy as sp import mpmath as mp

Exact values derived

a = sp.Rational(3,4) I_exact = 3*sp.sqrt(3)

Numerical check for the integral

f_num = lambda t: mp.acos(t/2)/(mp.cos(mp.pi*t/3)**2) I_num = mp.quad(f_num, [-1, 0, 1]) print("I_num =", I_num)

Hyperbolic quantities

x0 = sp.log(2+sp.sqrt(3)) F0 = sp.simplify(sp.cosh(x0)) G0 = sp.simplify(sp.sinh(x0)) print("F0 =", F0) print("G0 =", G0)

S = sp.simplify(4a + I_exact/G0 + F0) K = sp.simplify(999S**2 + 7)

print("S =", S) print("K =", K) print("K as int =", int(K)) PYTHON]

Tool Output

[PYTHON OUTPUT] SyntaxError: unmatched ']' (<string>, line 26) [/PYTHON OUTPUT]

Reasoning Process

[THOUGHT] Test the execution environment with a minimal Python snippet to diagnose why it's reporting an unmatched bracket. [/THOUGHT] [PYTHON] print(2) PYTHON]

Tool Output

[PYTHON OUTPUT] SyntaxError: unmatched ']' (<string>, line 2) [/PYTHON OUTPUT]

Reasoning Process

[THOUGHT] The earlier SyntaxError came from malformed closing tags; rerun with correct [PYTHON] ... [/PYTHON] tags and compute K exactly with SymPy. [/THOUGHT] [PYTHON] import sympy as sp

Part I

a = sp.Rational(3,4)

Part II

I = 3*sp.sqrt(3)

Part IV: f=cosh, g=sinh

x0 = sp.log(2+sp.sqrt(3)) F0 = sp.simplify(sp.cosh(x0)) G0 = sp.simplify(sp.sinh(x0))

S = sp.simplify(4a + I/G0 + F0) K = sp.simplify(999S**2 + 7)

print("a =", a) print("I =", I) print("F0 =", F0) print("G0 =", G0) print("S =", S) print("K =", K) PYTHON] [/PYTHON]

Tool Output

[PYTHON OUTPUT] Ellipsis [/PYTHON OUTPUT] [PYTHON OUTPUT] SyntaxError: unmatched ']' (<string>, line 23) [/PYTHON OUTPUT]

Reasoning Process

[THOUGHT] Run a minimal correct Python block to ensure the tag parsing is correct, then compute K exactly. [/THOUGHT] [PYTHON] import sympy as sp

a = sp.Rational(3,4) I = 3*sp.sqrt(3)

x0 = sp.log(2+sp.sqrt(3)) F0 = sp.cosh(x0) G0 = sp.sinh(x0)

S = sp.simplify(4a + I/G0 + F0) K = sp.simplify(999S**2 + 7)

print(S) print(K) [/PYTHON]

Tool Output

[PYTHON OUTPUT] cosh(log(sqrt(3) + 2)) + 3 + 3*sqrt(3)/sinh(log(sqrt(3) + 2)) 7 + 999*(sinh(log(sqrt(3) + 2))*cosh(log(sqrt(3) + 2)) + 3*sqrt(3) + 3*sinh(log(sqrt(3) + 2)))**2/sinh(log(sqrt(3) + 2))**2 [/PYTHON OUTPUT]

Reasoning Process

[THOUGHT] From the Python output, (S=\cosh(\ln(2+\sqrt3))+3+\frac{3\sqrt3}{\sinh(\ln(2+\sqrt3))}). Since (e^{\ln(2+\sqrt3)}=2+\sqrt3) and (e^{-\ln(2+\sqrt3)}=2-\sqrt3), we get (\cosh(\ln(2+\sqrt3))=2) and (\sinh(\ln(2+\sqrt3))=\sqrt3). Thus (S=2+3+3=8), so (\mathcal K=999\cdot 8^2+7=63943). \boxed{63943}

Final Model Output

63943

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

axk1

12.9

EXAONE-4.0.1-32B (high)

11.1

Avg Token Usage

K-EXAONE-236B-A23B

39933.0

GPT-5.2 (high)

26193.0

Kanana-2-30B-Thinking-2601

19787.0

Solar-Open-100B

12619.0

HCX-007(high)

10581.0

EXAONE-4.0.1-32B (high)

9149.0

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

9027.0

GPT-5.1 (high)

8547.0

Grok-4.1-fast

8351.0

Kanana-2-30B-Thinking

8050.0

Gemini-3-Pro-Preview

7889.0

A.X-4.0 (72B)

7410.0

K-EXAONE-236B-A23B

6804.0

Claude-Opus-4.5

5887.0

Deepseek-V3.2

5395.0

axk1

4182.0

Llama-VARCO-8B-Instruct

1716.0