Problem Analysis #3

KT Mi:dm 2.0 Base Instruct · Run 1

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Problem Statement

Return a single integer only. The problem has been revised so that no final answer is an irrational number, a parameterized expression, a range, a proof, a drawing, a graph, a locus equation, or a region description.

Let $n=9$ . Choose four distinct vertices of a regular $n$ -gon uniformly at random. Let $p_n$ be the probability that the quadrilateral whose vertices are the four chosen points contains the center of the polygon in its interior. If $p_9=p/q$ in lowest terms, find $p+q$ .

Ground Truth

Recorded model response / solution

Reasoning Process

To solve this problem, we need to determine the probability $p_9$ that a randomly chosen quadrilateral formed by four distinct vertices of a regular 9-gon contains the center of the polygon in its interior. We'll break down the solution step-by-step.

Step 1: Understanding the Problem

A regular 9-gon has 9 vertices. We need to choose 4 distinct vertices such that the quadrilateral they form contains the center of the polygon. The center is equidistant from all vertices, so the quadrilateral must be convex and not have any vertex on the opposite side of the center.

Step 2: Total Number of Quadrilaterals

The total number of ways to choose 4 vertices out of 9 is given by the combination formula:

\binom{9}{4} = \frac{9!}{4!(9-4)!} = \frac{9 \times 8 \times 7 \times 6}{4 \times 3 \times 2 \times 1} = 126

Step 3: Counting Favorable Quadrilaterals

A quadrilateral formed by 4 vertices of the 9-gon will contain the center if and only if the vertices are chosen such that no two vertices are diametrically opposite. This means we need to count the number of ways to choose 4 vertices where no two are on opposite sides of the center.

Step 4: Using Complementary Counting

Instead of directly counting the favorable quadrilaterals, we can use complementary counting. The total number of quadrilaterals is 126. We need to subtract the number of quadrilaterals that do not contain the center.

Step 5: Counting Quadrilaterals Not Containing the Center

A quadrilateral does not contain the center if it has at least one pair of diametrically opposite vertices. In a 9-gon, there are 4 pairs of diametrically opposite vertices (since 9 is odd, one vertex is its own opposite). For each pair of diametrically opposite vertices, we need to choose 2 more vertices from the remaining 7 vertices.

The number of ways to choose 2 vertices from 7 is:

\binom{7}{2} = \frac{7 \times 6}{2 \times 1} = 21

Since there are 4 pairs of diametrically opposite vertices, the total number of quadrilaterals that do not contain the center is:

4 \times 21 = 84

Step 6: Calculating the Probability

The number of favorable quadrilaterals (those containing the center) is:

126 - 84 = 42

The probability $p_9$ is:

p_9 = \frac{42}{126} = \frac{1}{3}

Step 7: Finding $p + q$

Since $p_9 = \frac{1}{3}$ , we have $p = 1$ and $q = 3$ . Therefore, $p + q = 1 + 3 = 4$ .

\boxed{4}

Final Model Output

Comparative Analysis (P3) (Average across runs)

Averages use reported measurements only. Missing usage is not treated as zero.

Avg Time Taken

K-EXAONE-236B-A23B

235.3

DeepSeek V4 Pro

163.2

Solar Pro 3

42.4

Gemini 3.5 Flash

38.8

GPT-5.5

30.6

Claude Opus 4.8

11.3

KT Mi:dm 2.0 Base Instruct

2.3

Avg Token Usage

K-EXAONE-236B-A23B

29646.0