MathDB

25

Part of 2020 LMT Spring

Problems(1)

Spring 2020 Team Round Problem 25

Source:

8/22/2020
Let △ABC\triangle ABC be a triangle such that AB=5,AC=8,AB=5,AC=8, and ∠BAC=60∘\angle BAC=60^{\circ}. Let Γ\Gamma denote the circumcircle of ABCABC, and let II and OO denote the incenter and circumcenter of △ABC\triangle ABC, respectively. Let PP be the intersection of ray IOIO with Γ\Gamma, and let XX be the intersection of ray BIBI with Γ\Gamma. If the area of quadrilateral XICPXICP can be expressed as ab+cde\frac{a\sqrt{b}+c\sqrt{d}}{e}, where aa and dd are squarefree positive integers and gcd⁡(a,c,e)=1\gcd(a,c,e)=1, compute a+b+c+d+ea+b+c+d+e.