The quadrilateral $ABCD$, in which $\angle BAC < \angle DCB$ , is inscribed in a circle $c$, with center $O$. If $\angle BOD = \angle ADC = \alpha$. Find out which values of $\alpha$ the inequality $AB <AD + CD$ occurs.

a, b, c are positive real numbers and a+b+c=k. Find the minimum value of $ b^2/(ka+bc)^1/2+c^2/(kb+ac)^1/2+a^2/(kc+ab)^1/2 $

Let $ M (x,y)=x^2+xy-2y $ , x,y are positive integers a) Solve in positive integers $ x^2+xy-2y=64 $ b) Prove that if M (x,y) is a perfect square, then x+y+2 is composite if x>2.

Given is equilateral triangle $ABC$ with side length $n \geq 3$. It is divided into $n^2$ identical small equilateral triangles with side length $1$. On every vertex of the triangles there is a number. In a move we can choose a rhombus and add or subtract $1$ from all $4$ numbers on the vertices of the rhombus. Let point $D$ have coordinates $(3,2)$ where $3$ is the number of the row and $2$ is the position on it from left to right. On the vertices $A,B,C,D$ there are $1$'s and on the other vertices there are $0$'s. Is it possible, after some operations, all the numbers to become equal?

$ a,b,c,d,e,f $ are real numbers. It is true that: $ a+b+c+d+e+f=20 $ $ (a-2)^2+(b-2)^2+...+(f-2)^2=24 $ Find the maximum value of d.

The vertices of the pentagon $ABCDE$ are on a circle, and the points $H_1, H_2, H_3,H_4$ are the orthocenters of the triangles $ABC, ABE, ACD, ADE$ respectively . Prove that the quadrilateral determined by the four orthocenters is square if and only if $BE \parallel CD$ and the distance between them is $\frac{BE + CD}{2}$.

On the board the number 1 is written. If on the board there is n, write $ n^2 $ or $ (n+1)^2 $ or $ (n+2)^2 $. Can we arrive at a number, devisible by 2015?

Given is a table 4x4 and in every square there is 0 or 1. In a move we choose row or column and we change the numbers there. Call the square "zero" if we cannot decrease the number of zeroes in it. Call "degree of the square" the number zeroes in a "zero" square. Find all possible values of the degree.