What is your favorite math topic/subject?

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Let be a set of elements. Let be subsets of such that the union of any of them has more than elements. Show that there's a member of that occurs in at least different subsets .

Let M be the intersection point of the medians of ABC. On the perpendiculars dropped from M to sides BC, AC, AB, points A1, B1, C1 are taken, respectively, with A1B1 perpendicular to MC and A1C1 perpendicular to MB. prove that M is the intersections pf the medians in A1B1C1.
Any solutions without vectors? :)

For a permutation of the set , define its colorfulness as the greatest natural number such that:
- For all , , if , then .
What is the maximum possible colorfulness of a permutation of the set ? Determine how many such permutations have maximal colorfulness.

Proposed by Pavle Martinović

Let be a triangle with incenter and circumcenter . the orthocenter of triangle . The incircle of touches side at respectively. Suppose that and intersects at and respectively (differ from ). Prove that lies on a circle.

( denote the circumcircle of )

For an table filled with natural numbers, we say it is a divisor table if:
- the numbers in the -th row are exactly all the divisors of some natural number ,
- the numbers in the -th column are exactly all the divisors of some natural number ,
- for every .

A prime number is given. Determine the smallest natural number , divisible by , such that there exists an divisor table, or prove that such does not exist.

Proposed by Pavle Martinović

Given a positive integer , prove that there exist infinitely many pairs of positive integers () such that
is a perfect square, where denotes the number of positive divisors of .
Proposed by Dong Zhenyu

Given that




Find the value of

Alice and Bob play a game on a 6 by 6 grid. On his or her turn, a player chooses a rational number not yet appearing in the grid and writes it in an empty square of the grid. Alice goes first and then the players alternate. When all squares have numbers written in them, in each row, the square with the greatest number in that row is colored black. Alice wins if she can then draw a line from the top of the grid to the bottom of the grid that stays in black squares, and Bob wins if she can't. (If two squares share a vertex, Alice can draw a line from one to the other that stays in those two squares.) Find, with proof, a winning strategy for one of the players.

Let be a triangle and let be a point in its interior. Lines , , intersect sides , , at , , , respectively. Prove that

if and only if lies on at least one of the medians of triangle . (Here denotes the area of triangle .)

Can anyone provide the PDF version of the book "Balkan Mathematical Olympiads" by Mircea Becheanu and Bogdan Enescu (published by XYZ press in 2014), thanks!