TUYMAADA 2017

$ABCD$ is a cyclic quadrilateral such that the diagonals $AC$ and $BD$ are perpendicular and their intersection is $P$ . Point $Q$ on the segment $CP$ is such that $CQ=AP$ . Prove that the perimeter of triangle $BDQ$ is at least $2AC$ .

Tuymaada 2017 Q2 Juniors

In a country every 2 cities are connected either by a direct bus route or a direct plane flight. A $clique$ is a set of cities such that every 2 cities in the set are connected by a direct flight. A $cluque$ is a set of cities such that every 2 cities in the set are connected by a direct flight, and every 2 cities in the set are connected to the same number of cities by a bus route. A $claque$ is a set of cities such that every 2 cities in the set are connected by a direct flight, and every 2 numbers of bus routes from a city in the set are different. Prove that the number of cities of any clique is at most the product of the biggest possible number of cities in a cluque and the the biggest possible number of cities in a claque.

Tuymaada 2017 Q3 Juniors

A right triangle has all its sides rational numbers and the area $S$ . Prove that there exists a right triangle, different from the original one, such that all its sides are rational numbers and its area is $S$ .

Tuymaada 2017 Q4 Juniors