TH-TMO 2010

Let $a, b, c$ be positive reals. Show that $\frac{a^5}{bc^2} + \frac{b^5}{ca^2} + \frac{c^5}{ab^2} \ge a^2 + b^2 + c^2.$

Define the modulo $2553$ distance $d(x, y)$ between two integers $x, y$ to be the smallest nonnegative integer $d$ equivalent to either $x - y$ or $y - x$ modulo $2553$ . Show that, given a set S of integers such that $|S| \ge 70$ , there must be $m, n \in S$ with $d(m, n) \le 36$ .

Let $a, b, c$ be real numbers so that all roots of the equation $2x^5 + 5x^4 + 5x^3 + ax^2 + bx + c = 0$ are real. Find the smallest real root of the equation above.

Find all primes $p$ such that ${100 \choose p} + 7$ is divisible by $p$ .