Problem Analysis #13

GPT-5.2 (high) · Run 1

11.48s
3,541 toks
Problem Statement

Let L1L_1 be the line of intersection of the planes given by the equations

2x+3y+z=4andx+2y+z=5.2x + 3y + z = 4 \quad \text{and} \quad x + 2y + z = 5.

Let L2L_2 be the line passing through the point P(2,−1,3)P(2, -1, 3) and parallel to L1L_1. Let Π\Pi denote the plane given by the equation

2x+y−2z=6.2x + y - 2z = 6.

Suppose that the line L2L_2 meets the plane Π\Pi at the point QQ. Let RR be the foot of the perpendicular drawn from PP to the plane Π\Pi.If the length of the line segment PQPQ is ℓ\ell and the area of △PQR\triangle PQR is AA, find the value of 4ℓ2A24\ell^2 A^2.\

Ground Truth
511758
Execution Trace
Reasoning Process

[THOUGHT] Compute direction vector of L1 as cross product of plane normals. Parametrize L2 through P with that direction, intersect with plane Π to get Q. Find R as orthogonal projection of P onto Π. Then ℓ=|PQ| and area A = 1/2 |(Q-P)×(R-P)|. Finally compute 4ℓ^2A^2 (which simplifies to |PQ|^2 * |(Q-P)×(R-P)|^2). [/THOUGHT] [PYTHON] import sympy as sp

Define normals of the planes for L1

n1 = sp.Matrix([2,3,1]) n2 = sp.Matrix([1,2,1])

Direction of intersection line L1

d = n1.cross(n2) d print("d =", d)

Point P

P = sp.Matrix([2,-1,3])

Plane Pi: 2x + y - 2z = 6

nPi = sp.Matrix([2,1,-2])

Parametric line L2: P + t*d

t = sp.Symbol('t', real=True) X = P + t*d

Solve for intersection with plane Pi

eq = sp.Eq(2X[0] + X[1] - 2X[2], 6) t_sol = sp.solve(eq, t)[0] Q = sp.simplify(X.subs(t, t_sol))

print("t_sol =", t_sol) print("Q =", Q)

Foot of perpendicular from P to plane Pi

Plane: nPi . x = 6, so projection: R = P - ((n.P - 6)/||n||^2) * n

n_norm_sq = (nPi.dot(nPi)) lam = (nPi.dot(P) - 6) / n_norm_sq R = sp.simplify(P - lam*nPi)

print("R =", R)

Compute length PQ

PQ = Q - P ell_sq = sp.simplify(PQ.dot(PQ)) ell = sp.sqrt(ell_sq)

Area of triangle PQR

PR = R - P cross = PQ.cross(PR) cross_sq = sp.simplify(cross.dot(cross)) A_sq = sp.simplify(cross_sq/4)

expr = sp.simplify(4ell_sqA_sq) print("ell_sq =", ell_sq) print("cross_sq =", cross_sq) print("A_sq =", A_sq) print("4ell^2A^2 =", expr) print("as integer?", sp.nsimplify(expr)) [/PYTHON]

Tool Output
[PYTHON OUTPUT] d = Matrix([[1], [-1], [1]]) t_sol = -9 Q = Matrix([[-7], [8], [-6]]) R = Matrix([[4], [0], [1]]) ell_sq = 243 cross_sq = 2106 A_sq = 1053/2 4*ell^2*A^2 = 511758 as integer? 511758 [/PYTHON OUTPUT]
Reasoning Process

[THOUGHT] Using the computed values, 4ℓ2A2=5117584\ell^2A^2 = 511758.

511758\boxed{511758}
Final Model Output
511758

Comparative Analysis (P13) (Average across runs)

Avg Time Taken

K-EXAONE-236B-A23B
479.1
Deepseek-V3.2
97.6
Kanana-2-30B-Thinking
71.3
Gemini-3-Pro-Preview
51.8
Llama-VARCO-8B-Instruct
51.0
Solar-Open-100B
48.1
Kanana-2-30B-Thinking-2601
45.8
Solar-Pro-2 (31B)(high)
38.9
A.X-4.0 (72B)
35.8
HCX-007(high)
35.3
K-EXAONE-236B-A23B
31.5
Grok-4.1-fast
29.2
GPT-5.1 (high)
22.2
Claude-Opus-4.5
21.5
GPT-5.2 (high)
11.5
EXAONE-4.0.1-32B (high)
7.0

Avg Token Usage

K-EXAONE-236B-A23B
51,665
Kanana-2-30B-Thinking
11,762
Kanana-2-30B-Thinking-2601
10,530
Solar-Open-100B
10,064
Solar-Pro-2 (31B)(high)
7,713
Gemini-3-Pro-Preview
7,454
GPT-5.1 (high)
7,218
Grok-4.1-fast
6,529
HCX-007(high)
5,083
Claude-Opus-4.5
4,957
Deepseek-V3.2
4,661
EXAONE-4.0.1-32B (high)
4,487
K-EXAONE-236B-A23B
4,111
GPT-5.2 (high)
3,541
A.X-4.0 (72B)
2,921
Llama-VARCO-8B-Instruct
1,020