CZ-SK-MO 2009

Knowing that the numbers $p, 3p+2, 5p+4, 7p+6, 9p+8$ , and $11p+10$ are all primes, prove that $6p+11$ is a composite number.

Rectangle $ABCD$ is inscribed in circle $O$ . Let the projections of a point $P$ on minor arc $CD$ onto $AB,AC,BD$ be $K,L,M$ , respectively. Prove that $\angle LKM=45$ if and only if $ABCD$ is a square.

Find the least value of $x>0$ such that for all positive real numbers $a,b,c,d$ satisfying $abcd=1$ , the inequality $a^x+b^x+c^x+d^x\ge\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}$ is true.

A positive integer $n$ is called good if and only if there exist exactly $4$ positive integers $k_1, k_2, k_3, k_4$ such that $n+k_i|n+k_i^2$ ( $1 \leq k \leq 4$ ). Prove that:

• $58$ is good;
• $2p$ is good if and only if $p$ and $2p+1$ are both primes ( $p>2$ ).

At every vertex $A_k(1\le k\le n)$ of a regular $n$ -gon, $k$ coins are placed. We can do the following operation: in each step, one can choose two arbitrarily coins and move them to their adjacent vertices respectively, one clockwise and one anticlockwise. Find all positive integers $n$ such that after a finite number of operations, we can reach the following configuration: there are $n+1-k$ coins at vertex $A_k$ for all $1\le k\le n$ .