MathDB

2019 Taiwan TST Round 2

Part of TST Round 2

Subcontests

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How Many Puddings?

There are n≥3 n \ge 3 puddings in a room. If a pudding A A hates a pudding B B , then B B hates A A as well. Suppose the following two conditions holds:
1. Given any four puddings, there are two puddings who like each other.
2. For any positive integer m m , if there are m m puddings who like each other, then there exists 3 3 puddings (from the other n−m n-m puddings) that hate each other.
Find the smallest possible value of n n .

Taiwanese Geometry

Given a triangle △ABC \triangle{ABC} . Denote its incircle and circumcircle by ω,Ω \omega, \Omega , respectively. Assume that ω \omega tangents the sides AB,AC AB, AC at F,E F, E , respectively. Then, let the intersections of line EF EF and Ω \Omega to be P,Q P,Q . Let M M to be the mid-point of BC BC . Take a point R R on the circumcircle of △MPQ \triangle{MPQ} , say Γ \Gamma , such that MR⊥EF MR \perp EF . Prove that the line AR AR , ω \omega and Γ \Gamma intersect at one point.
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