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GiM at Yale 2021 Mathathon R1: sillyalg/hextriangles/sirenaprimes

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February 28, 2021
Yalecollege

Problem Statement

1. If 5x+3y−z=45x+3y-z=4, x=yx=y, and z=4z=4, find x+y+zx+y+z.
2. How many ways are there to pick three distinct vertices of a regular hexagon such that the triangle with those three points as its vertices shares exactly one side with the hexagon?
3. Sirena picks five distinct positive primes, p1<p2<p3<p4<p5p_1 < p_2 < p_3 < p_4 < p_5, and finds that they sum to 192192. If the product p1p2p3p4p5p_1p_2p_3p_4p_5 is as large as possible, what is p1−p2+p3−p4+p5p_1 - p_2 + p_3 - p_4 + p_5?
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