Problem Analysis #29

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10,423 toks

Problem Statement

Consider the point $A(-2,0)$ in the coordinate plane. Suppose two points $P$ and $Q$ satisfy the following conditions:

[(i)] The vectors $\overrightarrow{AP}$ and $\overrightarrow{AQ}$ are parallel.
[(ii)] The dot product satisfies $\overrightarrow{AP} \cdot \overrightarrow{AQ} = 9$ . Let the two circles $C_{1} : x^{2} + y^{2} = 1, \qquadC_{2} : (x+1)^{2} + y^{2} = 1$ intersect at two distinct points $R_{1}$ and $R_{2}$ .Consider the region $\mathcal{R}$ consisting of the arc $R_{1}O'R_{2}$ of circle $C_{1}$ ,which contains the point $O'(-1,0)$ , together with the arc $R_{1}OR_{2}$ of circle $C_{2}$ ,which contains the point $O(0,0)$ .When the point $P$ moves along the region $\mathcal{R}$ , let $k$ denote the total lengthof the locus traced by the corresponding point $Q$ .For any real number $s$ , let $\lfloor s \rfloor$ denote the greatest integer less than or equal to $s$ .Determine the value of $\lfloor k \rfloor$ .

Ground Truth

Execution Trace

Reasoning Process

[THOUGHT] To solve the problem, we need to determine the total length of the locus traced by point ( Q ) as ( P ) moves along the region ( \mathcal{R} ). The key steps involve understanding the transformation from ( P ) to ( Q ) and the geometry of the region ( \mathcal{R} ).

Transformation from ( P ) to ( Q ):
- Given that vectors ( \overrightarrow{AP} ) and ( \overrightarrow{AQ} ) are parallel and their dot product is 9, we find that ( Q ) is the inversion of ( P ) with respect to a circle centered at ( A(-2, 0) ) with radius 3, followed by a translation by ((-2, 0)).
Region ( \mathcal{R} ):
- The region ( \mathcal{R} ) consists of two arcs: one from circle ( C_1 ) (unit circle centered at ( (0, 0) )) and another from circle ( C_2 ) (centered at ( (-1, 0) ) with radius 1). These arcs intersect at points ( R_1 ) and ( R_2 ).
Image of ( \mathcal{R} ) under Inversion:
- The image of the arc on ( C_1 ) under inversion is a circular arc of a circle centered at ( (4, 0) ) with radius 3, subtending an angle of ( 240^\circ ). The length of this arc is ( \frac{240}{360} \times 2\pi \times 3 = 4\pi ).
- The image of the arc on ( C_2 ) under inversion is a vertical line segment from ( \left( \frac{5}{2}, \frac{3\sqrt{3}}{2} \right) ) to ( \left( \frac{5}{2}, -\frac{3\sqrt{3}}{2} \right) ). The length of this segment is ( 3\sqrt{3} ).
Total Length ( k ):
- Summing the lengths of the two images gives ( k = 4\pi + 3\sqrt{3} ).
Numerical Calculation:
- Using Python, we calculate ( 4\pi \approx 12.566 ) and ( 3\sqrt{3} \approx 5.196 ). Their sum is approximately ( 17.762 ), so the floor value is 17.

[PYTHON] import math

Calculate the total length k

k = 4 * math.pi + 3 * math.sqrt(3) print(math.floor(k)) [/PYTHON]

Tool Output

[PYTHON OUTPUT] 17 [/PYTHON OUTPUT]

Reasoning Process

[THOUGHT] The Python code calculates ( k = 4\pi + 3\sqrt{3} \approx 17.762 ), and the floor value is 17. This matches the observation, so the final answer is (\boxed{17}). [/THOUGHT] \boxed{17}

Final Model Output

Comparative Analysis (P29) (Average across runs)

Avg Time Taken

EXAONE-4.0.1-32B (high)

298.6

GPT-5.1 (high)

227.4

Gemini-3-Pro-Preview

139.0

Deepseek-V3.2

109.1

Llama-VARCO-8B-Instruct

74.3

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

67.2

Grok-4.1-fast

38.9

Claude-Opus-4.5

38.3

HCX-007(high)

28.4

A.X-4.0 (72B)

21.1

Avg Token Usage

GPT-5.1 (high)

14833.0

Claude-Opus-4.5

11516.0

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

10423.0

Grok-4.1-fast

9673.0

Deepseek-V3.2

8977.0

Gemini-3-Pro-Preview

8560.0

EXAONE-4.0.1-32B (high)

6140.0

Llama-VARCO-8B-Instruct

5349.0

HCX-007(high)

4304.0

A.X-4.0 (72B)

2252.0