2019 Germany Team Selection Test

VAIMO 1

1

Let Q+ denote the set of all positive rational numbers. Determine all functions f:Q+Q+ satisfyingf(x2f(y)2)=f(x2)f(y)for all x,yQ+

2

Let ABC be a triangle with AB=AC, and let M be the midpoint of BC. Let P be a point such that PB<PC and PA is parallel to BC. Let X and Y be points on the lines PB and PC, respectively, so that B lies on the segment PX, C lies on the segment PY, and PXM=PYM. Prove that the quadrilateral APXY is cyclic.

3

Let n be a given positive integer. Sisyphus performs a sequence of turns on a board consisting of n+1 squares in a row, numbered 0 to n from left to right. Initially, n stones are put into square 0, and the other squares are empty. At every turn, Sisyphus chooses any nonempty square, say with k stones, takes one of these stones and moves it to the right by at most k squares (the stone should say within the board). Sisyphus' aim is to move all n stones to square n. Prove that Sisyphus cannot reach the aim in less than n1+n2+n3++nnturns. (As usual, x stands for the least integer not smaller than x. )

VAIMO 2

1

Determine all pairs (n,k) of distinct positive integers such that there exists a positive integer s for which the number of divisors of sn and of sk are equal.

2

Does there exist a subset M of positive integers such that for all positive rational numbers r<1 there exists exactly one finite subset of M like S such that sum of reciprocals of elements in S equals r.

3

A point T is chosen inside a triangle ABC. Let A1, B1, and C1 be the reflections of T in BC, CA, and AB, respectively. Let Ω be the circumcircle of the triangle A1B1C1. The lines A1T, B1T, and C1T meet Ω again at A2, B2, and C2, respectively. Prove that the lines AA2, BB2, and CC2 are concurrent on Ω. Proposed by Mongolia