2020 CIME I Problems/Problem 14

Problem 14

Let $ABC$ be a triangle with sides $AB = 5, BC = 7, CA = 8$ . Denote by $O$ and $I$ the circumcenter and incenter of $\triangle ABC$ , respectively. The incircle of $\triangle ABC$ touches $\overline{BC}$ at $D$ , and line $OD$ intersects the circumcircle of $\triangle AID$ again at $K$ . Then the length of $DK$ can be expressed in the form $\frac{m \sqrt n}{p}$ , where $m, n, p$ are positive integers, $m$ and $p$ are relatively prime, and $n$ is not divisible by the square of any prime. Find $m+n+p$ .

Solution

Analytic geometry gives us $DK=\frac{17\sqrt{57}}{19}.$ The answer is $93$ .

2020 CIME I (Problems • Answer Key • Resources)
Preceded by Problem 13	Followed by Problem 15
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
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2020 CIME I Problems/Problem 14

Problem 14

Solution

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