Let and be prime numbers. Prove that if is a perfect square, then is also a perfect square.

Prove that, for any triangle with side lengths , the following inequality holds ( denotes the semiperimeter of a triangle)

A square is given, and a quadratic trinomial with a positive leading coefficient is placed in each of its cells. There are coefficents in total, and these coefficents are chosen from the set , and each coefficient is different from each other. Prove that there exists at least one column such that the sum of the six trinomials in that column has a real root.

Determine all sequences of strictly positive integers satisfying the following two conditions:
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[*]There exists an integer such that, for all indices , .
[*]For any prime number and for any index , the number
is a multiple of .
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Find the minimum value of
where are reals.

Let be a triangle and let denote the circumcircle of . The foot of altitude from to is . The foot of altitudes from to and are , respectively. Let intersect at , and let intersect at . Prove that is the incenter of triangle

Find all positive real solutions to the following system:

Pairwise distinct points , which lie on a circle, are marked by distinct reals . Let be good for a on the circle different than , if and only if is the greatest number on at least one of the two arcs . Let the score of be the number of good points on the circle. Determine the greatest such that regardless of the values of , there exists a point with score at least .

The positive called ‘nice’ if and only if and are both perfect squares. How many positive integers such that are composites for all nice numbers ?

Suppose that and are natural numbers such that
is a prime number. Find all possible values of ,,.

Let be a positive integer. Aslı and Zehra are playing a game on an grid. Initially, stones are placed on some of the unit squares of this grid.

On each move (starting with Aslı), Aslı chooses a row or a column that contains at least two squares with different numbers of stones, and Zehra redistributes the stones in that row or column so that after redistribution, the difference in the number of stones between any two squares in that row or column is at most one. Furthermore, this move must change the number of stones in at least one square.

For which values of , regardless of the initial placement of the stones, can Aslı guarantee that every square ends up with the same number of stones?