Problem Analysis #3

45.11s

7,204 toks

Problem Statement

In the three-dimensional orthogonal coordinate system $xyz$ , consider the surface $S$ defined by $\begin{pmatrix}x(\theta, \phi) \\y(\theta, \phi) \\z(\theta, \phi)\end{pmatrix}=\begin{pmatrix}\cos\theta & -\sin\theta & 0 \\\sin\theta & \cos\theta & 0 \\0 & 0 & 1\end{pmatrix}\begin{pmatrix}\cos\phi + 2 \\0 \\\sin\phi\end{pmatrix},$ where $\theta$ and $\phi$ are parameters of the surface $S$ , and $0 \leq \theta < 2\pi,\qquad 0 \leq \phi < 2\pi.$ Let $V$ be the region surrounded by the surface $S$ , and let $W$ be the region satisfying the inequality $x^2 + y^2 \leq 4$ . Answer the following questions for the surface $S$ .

[I.] Find the unit normal vector oriented inward the region $V$ at the point $P = \begin{pmatrix} \dfrac{1}{\sqrt{2}} \\ \dfrac{1}{\sqrt{2}} \\ 0 \end{pmatrix}$ on the surface $S$ .
[II.] Find the area of the portion of the surface $S$ included in the region $W$ .
[III.] Find the volume of the overlapping region $V \cap W$ .
[IV.] Consider the curve $C$ on the surface $S$ defined by setting $\theta = \phi$ in the parametrization of $S$ . Find the curvature of the curve $C$ at the point $Q = \begin{pmatrix} 0 \\ 2 \\ 1 \end{pmatrix}$ on the curve $C$ . Note that, in general, given a three-dimensional curve defined by $c(t) =\begin{pmatrix}x(t) \\y(t) \\z(t)\end{pmatrix}$ with parameter $t$ , the curvature $\kappa(t)$ of the curve at the point $c(t)$ is given by $\kappa(t)= \frac{\left\lVert \dfrac{dc(t)}{dt} \times \dfrac{d^2c(t)}{dt^2}\right\rVert} {\left\lVert \dfrac{dc(t)}{dt}\right\rVert^3}.$ Let $\mathbf{n} = (n_1,n_2,n_3)$ be the inward unit normal vector at $P$ obtained in {\rm I}. Let $A_S$ be the area obtained in {\rm II}, and let $V_{\mathrm{ov}}$ be the overlapping volume $V\cap W$ obtained in {\rm III}. Let $\kappa_Q$ be the curvature at $Q$ obtained in {\rm IV}.Define $\mathcal{K}= 999 \cdot 123 \cdot\left(\frac{V_{\mathrm{ov}}}{2\pi}- \frac{A_S}{4\pi}\right)\cdot\frac{(n_1 + n_2)^2}{\kappa_Q^2}\;+\; 7.$ Compute the integer $\mathcal{K}$ .\

Ground Truth

249757

Execution Trace

Reasoning Process

[THOUGHT] Model S is a torus with major radius 2 and minor radius 1. I will:

Identify parameters for P and compute the inward unit normal by comparing the surface normal to the direction toward the torus centerline.
Compute the area element |r_θ×r_φ| and integrate over the φ-range where x^2+y^2≤4.
Compute V∩W via cylindrical coordinates integrating the solid torus inequality with ρ≤2.
Parametrize curve C by t with θ=φ=t, compute curvature at t=π/2 (point Q), then assemble 𝒦 and simplify to an integer using sympy. [/THOUGHT] [PYTHON] import sympy as sp

Symbols

t = sp.symbols('t', real=True) theta, phi = sp.symbols('theta phi', real=True)

Parametrization of torus S

x = (sp.cos(phi) + 2) * sp.cos(theta) y = (sp.cos(phi) + 2) * sp.sin(theta) z = sp.sin(phi) r = sp.Matrix([x, y, z])

r_theta = sp.diff(r, theta) r_phi = sp.diff(r, phi)

Normal via cross product

N = sp.simplify(r_theta.cross(r_phi)) # not unit

Point P corresponds to theta=pi/4, phi=pi

subsP = {theta: sp.pi/4, phi: sp.pi} P = sp.Matrix([1/sp.sqrt(2), 1/sp.sqrt(2), 0])

N_P = sp.simplify(N.subs(subsP))

Determine inward direction: towards torus centerline circle at same theta, z=0: (2cosθ,2sinθ,0)

centerline_P = sp.Matrix([2sp.cos(sp.pi/4), 2sp.sin(sp.pi/4), 0]) toward_centerline = sp.simplify(centerline_P - P)

Choose sign so that dot(normal, toward_centerline) > 0 for inward

dot_val = sp.simplify(N_P.dot(toward_centerline))

Unit normal candidate

N_P_norm = sp.sqrt(sp.simplify(N_P.dot(N_P))) unit = sp.simplify(N_P / N_P_norm) unit_inward = sp.simplify(unit if dot_val > 0 else -unit) # may not compare symbolically; handle numerically dot_num = float(sp.N(dot_val)) unit_inward = unit if dot_num > 0 else -unit unit_inward_simplified = sp.simplify(unit_inward)

print("N_P:", N_P) print("toward_centerline:", toward_centerline) print("dot_val:", sp.N(dot_val)) print("unit_inward:", unit_inward_simplified)

n1, n2, n3 = [sp.simplify(c) for c in unit_inward_simplified]

II) Area portion where x^2+y^2 <=4 -> (cos phi +2)^2 <=4 -> cos phi <=0

area_element = sp.simplify(sp.sqrt((r_theta.cross(r_phi)).dot(r_theta.cross(r_phi)))) print("area_element simplified:", area_element)

A_S = sp.simplify(sp.integrate(area_element, (phi, sp.pi/2, 3sp.pi/2), (theta, 0, 2sp.pi))) print("A_S:", A_S)

III) Overlapping volume V∩W

rho = sp.symbols('rho', positive=True, real=True) z_sym = sp.symbols('z', real=True)

For rho in [1,2], z bounds +/- sqrt(1-(rho-2)^2)

zmax = sp.sqrt(1-(rho-2)*2) Vov = sp.simplify(sp.integrate(4sp.pirhozmax, (rho, 1, 2))) # 2πρ(2zmax) = 4π ρ zmax print("Vov:", Vov)

IV) curvature of curve C where theta=phi=t

c = sp.Matrix([(sp.cos(t)+2)*sp.cos(t), (sp.cos(t)+2)*sp.sin(t), sp.sin(t)]) c1 = sp.diff(c, t) c2 = sp.diff(c1, t) kappa = sp.simplify(sp.sqrt((c1.cross(c2)).dot(c1.cross(c2))) / (sp.sqrt(c1.dot(c1))**3)) kappa_Q = sp.simplify(kappa.subs(t, sp.pi/2)) print("kappa_Q:", kappa_Q, "numeric:", sp.N(kappa_Q))

Assemble K

K_expr = sp.simplify(999123((Vov/(2sp.pi)) - (A_S/(4sp.pi))) * ((n1+n2)2)/(kappa_Q2) + 7) print("K_expr simplified:", K_expr) print("K_expr numeric:", sp.N(K_expr))

Ensure integer

K_int = sp.nsimplify(K_expr) print("K nsimplify:", K_int) print("Is integer:", K_int.is_integer) print(int(K_int)) [/PYTHON]

Tool Output

[PYTHON OUTPUT] N_P: Matrix([[-sqrt(2)/2], [-sqrt(2)/2], [0]]) toward_centerline: Matrix([[sqrt(2)/2], [sqrt(2)/2], [0]]) dot_val: -1.00000000000000 unit_inward: Matrix([[sqrt(2)/2], [sqrt(2)/2], [0]]) area_element simplified: Abs(cos(phi) + 2) A_S: 4*pi*(-1 + pi) Vov: 2*pi*(-2 + 3*pi)/3 kappa_Q: sqrt(205)/25 numeric: 0.572712842531054 K_expr simplified: 249757 K_expr numeric: 249757.000000000 K nsimplify: 249757 Is integer: True 249757 [/PYTHON OUTPUT]

Reasoning Process

[THOUGHT] Using the computed inward normal, area, overlapping volume, and curvature from the Python output, the expression for (\mathcal{K}) simplifies exactly to an integer. [ \boxed{249757} ]

Final Model Output

249757

Comparative Analysis (P3) (Average across runs)

Avg Time Taken

Solar-Open-100B

924.8

K-EXAONE-236B-A23B

658.0

Gemini-3-Pro-Preview

343.1

Kanana-2-30B-Thinking-2601

217.4

GPT-5.1 (high)

185.0

Deepseek-V3.2

168.5

Kanana-2-30B-Thinking

147.2

Grok-4.1-fast

142.7

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

139.6

Llama-VARCO-8B-Instruct

92.2

A.X-4.0 (72B)

86.0

Claude-Opus-4.5

84.8

K-EXAONE-236B-A23B

63.4

GPT-5.2 (high)

45.1

HCX-007(high)

31.8

axk1

21.6

EXAONE-4.0.1-32B (high)

12.4

Avg Token Usage

K-EXAONE-236B-A23B

67198.0

Solar-Open-100B

49812.0

Kanana-2-30B-Thinking-2601

44837.0

Kanana-2-30B-Thinking

26235.0

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

22878.0

Grok-4.1-fast

21249.0

A.X-4.0 (72B)

19754.0

Claude-Opus-4.5

16732.0

Deepseek-V3.2

15050.0

Gemini-3-Pro-Preview

14578.0

K-EXAONE-236B-A23B

11203.0

GPT-5.1 (high)

9991.0

HCX-007(high)

9092.0

EXAONE-4.0.1-32B (high)

7591.0

GPT-5.2 (high)

7204.0

Llama-VARCO-8B-Instruct

6884.0

axk1

5854.0