Let $a, b, c$ be positive integers such that one of the values $$gcd(a,b) \cdot lcm(b,c), \,\,\,\, gcd(b,c)\cdot lcm(c,a), \,\,\,\, gcd(c,a)-\cdot lcm(a,b)$$is equal to the product of the remaining two. Prove that one of the numbers $a, b, c$ is a multiple of another of them.

Let the interior point $P$ of the convex quadrilateral $ABCD$ be such that $$|\angle PAD| = |\angle ADP| = |\angle CBP| = |\angle PCB| = |\angle CPD|.$$Let $O$ be the center of the circumcircle of the triangle $CPD$. Prove that $|OA| = |OB|$.

Find the largest natural number $n$ such that any set of $n$ tetraminoes, each of which is one of the four shapes in the picture, can be placed without overlapping in a $20 \times 20$ table (no tetramino extends beyond the borders of the table), such that each tetramino covers exactly 4 cells of the 20x20 table. An individual tetramino is allowed to turn and flip at will.

There were $10$ boys and $10$ girls at the party. Every boy likes a different 'positive' number of girls. Every girl likes a different positive number of boys. Define the largest non-negative integer $n$ such that it is always possible to form at least $n$ disjoint pairs in which both like the other.

Let $(a_k)^{\infty}_{k=0}$ be a sequence of real numbers such that if $k$ is a non-negative integer, then $$a_{k+1} = 3a_k - \lfloor 2a_k \rfloor - \lfloor a_k \rfloor.$$Definitely all positive integers $n$ such that if $a_0 = 1/n$, then this sequence is constant after a certain term.

Find all right triangles with integer side lengths in which two congruent circles with prime radius can be inscribed such that they are externally tangent, both touch the hypotenuse, and each is tangent to another leg of the right triangle.