Problem Analysis #9

71.86s

9,311 toks

Problem Statement

In the two-dimensional orthogonal $xy$ coordinate system, consider the curve $L$ represented by the following equations with the parameter $t$ ( $0 $$\leq$$ t $$\leq$$ 2$$\pi$$$). Here,$ a$ is a positive real constant. $\begin{align}x(t) &= a(t - \sin t), \\y(t) &= a(1 - \cos t).\end{align}$

[(I-1)] Obtain the length of the curve $L$ when $t$ varies in the range $0 $\leq$ t $\leq$ 2$$\pi$$$.
[(I-2)] For $0 < t < 2$$\pi$$$, obtain the curvature$ \kappa_L(t) $at an arbitrary point of the curve$ L $. In the three-dimensional orthogonal$ xyz $coordinate system, consider the curved surface represented by the following equations with the parameters$ u $and$ v $($ u $and$ v$ are real numbers): $\begin{align}x(u, v) &= \sinh u \cos v, \\y(u, v) &= 2 \sinh u \sin v, \\z(u, v) &= 3 \cosh u.\end{align}$
[(II-1)] Express the curved surface by an equation without the parameters $u$ and $v$ .
[(II-2)] Sketch the $xy$ -plane view at $z = 5$ and the $xz$ -plane view at $y = 0$ , respectively, of the curved surface. In the sketches, indicate the values at the intersections with each of the axes.
[(II-3)] Express a unit normal vector $\mathbf{n}$ of the curved surface by $u$ and $v$ . Here, the $z$ -component of $\mathbf{n}$ should be positive.
[(II-4)] Let $\kappa$ be the Gaussian curvature at the point $u = v = 0$ . Calculate the absolute value $|\kappa|$ . Finally, define the following quantities:
Let $L$ be the length of the cycloid in {\rm (I-1)}, and let $\kappa_L(\pi)$ be the curvature of $L$ at $t = \pi$ from {\rm (I-2)}. Define $\alpha := L \cdot \kappa_L(\pi).$
In {\rm (II-2)}, denote by $A_x > 0$ and $A_y > 0$ the $x$ - and $y$ -intercepts (respectively) of the cross-section of the surface by the plane $z = 5$ , and by $A_z > 0$ the $z$ -intercept of the cross-section by the plane $y = 0$ . Define $\delta := A_x A_y A_z, \qquad \rho := \frac{A_y}{A_x}.$
In {\rm (II-4)}, let $\beta := |\kappa|$ be the absolute value of the Gaussian curvature at $u = v = 0$ . Using these, define $S := \bigl(\alpha\,\beta\,\delta\,\rho\bigr)^3,\qquad\mathcal{K} := 999\,S^2 + 7.$ Compute the integer $\mathcal{K}$ .\

Ground Truth

781975031906311

Execution Trace

Reasoning Process

[THOUGHT] Compute each quantity exactly: cycloid length L and curvature at t=pi to get alpha; eliminate parameters for the quadric and read off intercepts to get delta and rho; compute Gaussian curvature at (0,0,3) by viewing the surface as the graph z=f(x,y)=3*sqrt(1+x^2+y^2/4) and using K=(f_xx f_yy - f_xy^2) at the origin (since f_x=f_y=0 there). Then form S and finally 𝓚=999 S^2+7, verifying all arithmetic with SymPy rationals. [/THOUGHT] [PYTHON] import sympy as sp

Part I

t, a = sp.symbols('t a', positive=True, real=True) x = a*(t-sp.sin(t)) y = a*(1-sp.cos(t)) dx = sp.diff(x,t) dy = sp.diff(y,t) L = sp.integrate(sp.sqrt(dx2+dy2), (t,0,2*sp.pi))

curvature

ddx = sp.diff(dx,t) ddy = sp.diff(dy,t) kappa = sp.simplify(abs(dxddy - dyddx) / sp.sqrt(dx2+dy2)**3) kappa_pi = sp.simplify(sp.limit(kappa, t, sp.pi)) alpha = sp.simplify(L * kappa_pi)

print("L =", sp.simplify(L)) print("kappa(t) =", kappa) print("kappa(pi) =", kappa_pi) print("alpha =", alpha)

Part II intercepts

Ax = sp.Rational(4,3) Ay = sp.Rational(8,3) Az = sp.Integer(3) delta = sp.simplify(AxAyAz) rho = sp.simplify(Ay/Ax) print("delta =", delta) print("rho =", rho)

Gaussian curvature at (0,0,3) for z = 3*sqrt(1 + x^2 + y^2/4)

x_sym, y_sym = sp.symbols('x_sym y_sym', real=True) f = 3*sp.sqrt(1 + x_sym2 + y_sym2/4) fx = sp.diff(f, x_sym) fy = sp.diff(f, y_sym) fxx = sp.diff(fx, x_sym) fyy = sp.diff(fy, y_sym) fxy = sp.diff(fx, y_sym)

K_expr = sp.simplify((fxx*fyy - fxy2) / (1 + fx2 + fy**2)**2) K0 = sp.simplify(K_expr.subs({x_sym:0, y_sym:0})) beta = sp.Abs(K0) print("Gaussian curvature K at origin =", K0) print("beta =", beta)

Compute final

S = sp.simplify((alphabetadeltarho)**3) Kcal = sp.simplify(999S**2 + 7) print("S =", S) print("Kcal =", Kcal) print("Kcal as int?", sp.nsimplify(Kcal)) [/PYTHON][THOUGHT] I will run the Python code to compute all required quantities exactly and then form the integer 𝓚. [/THOUGHT] [PYTHON] import sympy as sp

Part I

t, a = sp.symbols('t a', positive=True, real=True) x = a*(t-sp.sin(t)) y = a*(1-sp.cos(t)) dx = sp.diff(x,t) dy = sp.diff(y,t)

L = sp.integrate(sp.sqrt(dx2+dy2), (t,0,2*sp.pi))

ddx = sp.diff(dx,t) ddy = sp.diff(dy,t) kappa = sp.simplify(abs(dxddy - dyddx) / sp.sqrt(dx2+dy2)**3) kappa_pi = sp.simplify(sp.limit(kappa, t, sp.pi)) alpha = sp.simplify(L * kappa_pi)

Part II intercepts

Ax = sp.Rational(4,3) Ay = sp.Rational(8,3) Az = sp.Integer(3) delta = sp.simplify(AxAyAz) rho = sp.simplify(Ay/Ax)

Gaussian curvature at (0,0,3) for z = 3*sqrt(1 + x^2 + y^2/4)

K_expr = sp.simplify((fxx*fyy - fxy2) / (1 + fx2 + fy**2)**2) K0 = sp.simplify(K_expr.subs({x_sym:0, y_sym:0})) beta = sp.Abs(K0)

S = sp.simplify((alphabetadeltarho)**3) Kcal = sp.simplify(999S**2 + 7)

print("L =", sp.simplify(L)) print("kappa(t) =", kappa) print("kappa(pi) =", kappa_pi) print("alpha =", alpha) print("delta =", delta) print("rho =", rho) print("Gaussian curvature K at origin =", K0) print("beta =", beta) print("S =", S) print("Kcal =", Kcal) print("Kcal integer check =", int(Kcal)) [/PYTHON]

Tool Output

[PYTHON OUTPUT] L = sqrt(2)*a*Integral(sqrt(1 - cos(t)), (t, 0, 2*pi)) kappa(t) = Abs(cos(t) - 1)/(a*(2 - 2*cos(t))**(3/2)) kappa(pi) = 1/(4*a) alpha = sqrt(2)*Integral(sqrt(1 - cos(t)), (t, 0, 2*pi))/4 delta = 32/3 rho = 2 Gaussian curvature K at origin = 9/4 beta = 9/4 S = 3456*sqrt(2)*Integral(sqrt(1 - cos(t)), (t, 0, 2*pi))**3 Kcal = 7 + 23863984128*Integral(sqrt(1 - cos(t)), (t, 0, 2*pi))**6 Kcal as int? 7 + 23863984128*Integral(sqrt(1 - cos(t)), (t, 0, 2*pi))**6 [/PYTHON OUTPUT] [PYTHON OUTPUT] TypeError: cannot compute int value accurately [/PYTHON OUTPUT]

Reasoning Process

[THOUGHT] \boxed{781975031906311}

Final Model Output

781975031906311

Comparative Analysis (P9) (Average across runs)

Avg Time Taken

K-EXAONE-236B-A23B

6180.8

A.X-4.0 (72B)

1385.0

Deepseek-V3.2

484.2

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

298.7

Solar-Open-100B

272.5

Kanana-2-30B-Thinking-2601

149.2

GPT-5.1 (high)

139.8

Llama-VARCO-8B-Instruct

134.8

Kanana-2-30B-Thinking

96.7

Gemini-3-Pro-Preview

94.3

Grok-4.1-fast

79.4

GPT-5.2 (high)

71.9

K-EXAONE-236B-A23B

64.3

HCX-007(high)

44.1

Claude-Opus-4.5

37.4

EXAONE-4.0.1-32B (high)

33.3

axk1

17.2

Avg Token Usage

K-EXAONE-236B-A23B

395093.7

Solar-Open-100B

46567.0

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

23128.0

K-EXAONE-236B-A23B

17006.0

GPT-5.1 (high)

16139.0

Kanana-2-30B-Thinking

15848.0

Kanana-2-30B-Thinking-2601

14597.0

Deepseek-V3.2

14111.0

Grok-4.1-fast

13877.0

Gemini-3-Pro-Preview

13470.0

GPT-5.2 (high)

9311.0

Llama-VARCO-8B-Instruct

8865.0

Claude-Opus-4.5

8603.0

A.X-4.0 (72B)

7948.0

EXAONE-4.0.1-32B (high)

6907.0

HCX-007(high)

6048.0

axk1

5427.0