Problems
For positive real numbers \(a,b,c\) prove the inequality: \[(a^2b + b^2c + c^2a)(ab^2 + bc^2 + ca^2)\geq 9a^2b^2c^2.\]
On a \(10\times 10\) board, a bacterium sits in one of the cells. In one move, the bacterium shifts to a cell adjacent to the side (i.e. not diagonal) and divides into two bacteria (both remain in the same new cell). Then, again, one of the bacteria sitting on the board shifts to a new adjacent cell, either horizontally or vertically, and divides into two, and so on. Is it possible for there to be an equal number of bacteria in all cells after several such moves?
Let \(p\) and \(q\) be two prime numbers such that \(q = p + 2\). Prove that \(p^q + q^p\) is divisible by \(p + q\).
Find the angles of the triangle \(ABC\) if the center of the inscribed circle \(E\) and the center of the superscribed circle \(D\) are symmetric with respect to the segment \(AC\).
In the arithmetic puzzle different letters denote different digits and the same letters denote the same digit. \[P.Z + T.C + D.R + O.B + E.Y\] It turned out that all five terms are not integers, but the sum itself is an integer. Find the sum of the expression. For each possible answer, write one example with these five terms. Explain why other numbers cannot be obtained.
