MathDB

2

Part of 2019 Taiwan TST Round 2

Problems(2)

How Many Puddings?

Source: 2019 Taiwan TST Round 2

4/1/2020
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 .
combinatorics
Taiwanese Geometry

Source: 2019 Taiwan TST Round 2

4/1/2020
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.
geometrycircumcircle