XK-TST 2012

A student had $18$ papers. He seleced some of these papers, then he cut each of them in $18$ pieces.He took these pieces and selected some of them, which he again cut in $18$ pieces each.The student took this procedure untill he got tired .After a time he counted the pieces and got $2012$ pieces .Prove that the student was wrong during the counting.

Find all three digit numbers, for which the sum of squares of each digit is $90$ .

If $a,b,c$ are the sides of a triangle and $m_a , m_b, m_c$ are the medians prove that $$4(m_a^2+m_b^2+m_c^2)=3(a^2+b^2+c^2)$$

Each term in a sequence $1,0,1,0,1,0...$ starting with the seventh is the sum of the last 6 terms mod 10 .Prove that the sequence $...,0,1,0,1,0,1...$ never occurs

Prove that the equation $$\frac{4}{n}=\frac{1}{x}+\frac{1}{y}+\frac{1}{z}$$ has infinitly many natural solutions