PL-OM 2013

Find all solutions of the following equation in integers $x,y: x^4+ y= x^3+ y^2$

There are given integers $a$ and $b$ such that $a$ is different from $0$ and the number $3+ a +b^2$ is divisible by $6a$ . Prove that $a$ is negative.

Given is a quadrilateral $ABCD$ in which we can inscribe circle. The segments $AB, BC, CD$ and $DA$ are the diameters of the circles $o1, o2, o3$ and $o4$ , respectively. Prove that there exists a circle tangent to all of the circles $o1, o2, o3$ and $o4$ .

Given is a tetrahedron $ABCD$ in which $AB=CD$ and the sum of measures of the angles $BAD$ and $BCD$ equals $180$ degrees. Prove that the measure of the angle $BAD$ is larger than the measure of the angle $ADC$ .

Let k,m and n be three different positive integers. Prove that $$\left( k-\frac{1}{k} \right)\left( m-\frac{1}{m} \right)\left( n-\frac{1}{n} \right) \le kmn-(k+m+n).$$

For each positive integer $n$ determine the maximum number of points in space creating the set $A$ which has the following properties: $1)$ the coordinates of every point from the set $A$ are integers from the range $[0, n]$ $2)$ for each pair of different points $(x_1,x_2,x_3), (y_1,y_2,y_3)$ belonging to the set $A$ it is satisfied at least one of the following inequalities $x_1< y_1, x_2<y_2, x_3<y_3$ and at least one of the following inequalities $x_1>y_1, x_2>y_2,x_3>y_3$ .