students bought some books in a bookstore. It is known that every student bought exactly three kinds of books, and any two of them shared at least one kind of book. Determine, with proof, how many students bought the most popular book at least? (Note: the most popular book means most students bought this kind of book)

In an infinite grid of regular triangles, Niels and Henrik are playing a game they made up.
Every other time, Niels picks a triangle and writes in it, and every other time, Henrik picks a triangle where he writes a . If one of the players gets four in a row in some direction (see figure), he wins the game.
Determine whether one of the players can force a victory.
IMAGE

Consider a point on the diagonal of a given rectangle , such that . The point is the intersection point between and the parallel line to that contains . Prove that the triangle is equilateral if and only if is a square.

Valentin Vornicu

Round 1

1 Real number satisfies . Find the value of .

Let be a convex quadrilateral, such that

and .

Prove that quadrilateral is a kite.

If prove that

Let Prove that

Let be a triangle with orthocentre . The tangent lines from to the circle with diameter touch this circle at and . Prove that and are collinear.

We have a 4x4 board.All 1x1 squares are white.A move is changing colours of all squares of a 1x3 rectangle from black to white and from white to black.It is possible to make all the 1x1 squares black after several moves?

Let be a sequence that consists of an initial block of positive distinct integers that then repeat periodically. This means that are distinct positive integers and for every positive integer . The terms of the sequence are not known and the goal is to find the period . To do this, at each move it possible to reveal the value of a term of the sequence at your choice.
If is one of the first prime numbers, find for which values of there exist a strategy that allows to find revealing at most terms of the sequence.