Let be a positive integer. Prove that there exist three permutations , , and of such that for every .

With sonone's pixel art generator and the ascii art I posted earlier, I made my turtle logo black and white in asy.

Let . Prove that for any given positive integer , the product
has at most distinct prime divisors.

Proposed by Géza Kós

Let be an acute-angled triangle with circumference . Let the angle bisectors of and intersect again at and . Let be the intersection point of these angle bisectors. Let and be the respective reflections of and in and . Prove that the center of the circle passing through , , lies on the altitude of triangle from .

Proposed by Victor Domínguez and Ariel García

We are given an infinite deck of cards, each with a real number on it. For every real number , there is exactly one card in the deck that has written on it. Now two players draw disjoint sets and of cards each from this deck. We would like to define a rule that declares one of them a winner. This rule should satisfy the following conditions:
1. The winner only depends on the relative order of the cards: if the cards are laid down in increasing order face down and we are told which card belongs to which player, but not what numbers are written on them, we can still decide the winner.
2. If we write the elements of both sets in increasing order as and , and for all , then beats .
3. If three players draw three disjoint sets from the deck, beats and beats then also beats .
How many ways are there to define such a rule? Here, we consider two rules as different if there exist two sets and such that beats according to one rule, but beats according to the other.

Proposed by Ilya Bogdanov, Russia

Let be a triangle such that

where and denote its semiperimeter and its inradius, respectively. Prove that triangle is similar to a triangle whose side lengths are all positive integers with no common divisors and determine these integers.