Problem Analysis #13

Llama-VARCO-8B-Instruct · Run 1

50.98s
1,020 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]

  1. Find the direction vector of line L1 by subtracting the coefficients of the two planes' equations. Direction vector of L1 = (2-1, 3-2, 0-1) = (1, 1, -1)

  2. Line L2 is parallel to L1 and passes through P(2, -1, 3), so its direction vector is the same as L1's. Parametric equation of L2: (x, y, z) = (2, -1, 3) + t(1, 1, -1)

  3. Find a point Q on L2 that also lies in plane Π by substituting L2's parametric equations into Π's equation. 2*(2+t) + (-1+t) - 2*(3-t) = 6 Solving for t gives: t = 1/2

  4. Substitute t=1/2 back into L2's parametric equation to get point Q: (x, y, z) = (2, -1, 3) + 1/2(1, 1, -1) Point Q = (7/2, 1/2, 1/2)

  5. Find vector PQ: (7/2-2, 1/2+1, 1/2-3) = (3/2, 3/2, -5/2)

  6. Vector PR is P to R, which is parallel to Π's normal vector (2, 1, -2). Normalizing Π's normal vector gives: n = (2/sqrt(10), 1/sqrt(10), -2/sqrt(10))

    So, unit vector in the direction of PR = (2/sqrt(10), 1/sqrt(10), -2/sqrt(10)) Vector PR = (2/sqrt(10)) * |PQ| = (3/2)*sqrt(10)/2

  7. Area A of triangle PQR is half the cross product of PQ and PR: Cross product: (3/2, 3/2, -5/2) x ((3/2)sqrt(10)/2, (1/2)sqrt(10)/2, -(3/2)sqrt(10)/2) = (15sqrt(10)/4 + 15/4(-2), 9sqrt(10)/4 - 9/4*(-5), 9sqrt(10)/4 + 3/4(-15)) = (21.75, 45.75, -6)

    Area A = |cross product / 2| = sqrt((21.75)^2 + (45.75)^2 + (-6)^2) / 2 ≈ 16.39

  8. Length of PQ: |PQ| = sqrt((3/2)^2 + (3/2)^2 + (-5/2)^2) = sqrt(28.5) ≈ 5.35

  9. Compute the answer: 4 * (length PQ)^2 * (Area)^2 = 4 * (5.35)^2 * (16.39)^2 ≈ 1,293,544

1293544\boxed{1293544}

Final Model Output
1293544

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