MathDB

2018 Taiwan TST Round 2

Part of TST Round 2

Subcontests

(2)
1
4
2
3

Taiwan 2018 TST geometry

Let ABCABC be a triangle with circumcircle Ω\Omega, circumcenter OO and orthocenter HH. Let SS lie on Ω\Omega and PP lie on BCBC such that ∠ASP=90∘\angle ASP=90^\circ, line SHSH intersects the circumcircle of △APS\triangle APS at X≠SX\neq S. Suppose OPOP intersects CA,ABCA,AB at Q,RQ,R, respectively, QY,RZQY,RZ are the altitude of △AQR\triangle AQR. Prove that X,Y,ZX,Y,Z are collinear.
Proposed by Shuang-Yen Lee

Functional equation 011

Find all functions f:Z→Z f: \mathbb{Z} \to \mathbb{Z} such that f(x+f(y))f(y+f(x))=(2x+f(y−x))(2y+f(x−y)) f\left(x+f\left(y\right)\right)f\left(y+f\left(x\right)\right)=\left(2x+f\left(y-x\right)\right)\left(2y+f\left(x-y\right)\right) holds for all integers x,y x,y

Of wolves and sheep

There are nn sheep and a wolf in sheep's clothing . Some of the sheep are friends (friendship is mutual). The goal of the wolf is to eat all the sheep. First, the wolf chooses some sheep to make friend's with. In each of the following days, the wolf eats one of its friends. Whenever the wolf eats a sheep AA: (a) If a friend of AA is originally a friend of the wolf, it un-friends the wolf. (b) If a friend of AA is originally not a friend of the wolf, it becomes a friend of the wolf. Repeat the procedure until the wolf has no friend left. Find the largest integer mm in terms of nn satisfying the following: There exists an initial friendsheep structure such that the wolf has mm different ways of choosing initial sheep to become friends, so that the wolf has a way to eat all of the sheep.