Students are in classroom with $n$ rows. In each row there are $m$ tables. It's given that $m,n \geq 3$. At each table there is exactly one student. We call neighbours of the student students sitting one place right, left to him, in front of him and behind him. Each student shook hands with his neighbours. In the end there were $252$ handshakes. How many students were in the classroom?

Find all integer triples $(p,m,n)$ that satisfy: $p^m-n^3=27$ where $p$ is a prime number.

Let $\Gamma$ be circumscribed circle of triangle $ABC $ $(AB \neq AC)$. Let $O$ be circumcenter of the triangle $ABC$. Let $M$ be a point where angle bisector of angle $BAC$ intersects $\Gamma$. Let $D$ $(D \neq M)$ be a point where circumscribed circle of the triangle $BOM$ intersects line segment $AM$ and let $E$ $(E \neq M)$ be a point where circumscribed circle of triangle $COM$ intersects line segment $AM$. Prove that $BD+CE=AM$.

Let $a,b,c$ be real numbers which satisfy: $$a+b+c=2$$$$a^2+b^2+c^2=2$$Prove that at least one of numbers $|a-b|, |b-c|, |c-a|$ is greater or equal than $1$.