IrMO 2011

Suppose $abc\neq 0$ . Express in terms of $a,b,$ and $c$ the solutions $x,y,z,u,v,w$ of the equations $$x+y=a, z+u=b, v+w=c, ay=bz, ub=cv, wc=ax.$$

Let $ABC$ be a triangle whose side lengths are, as usual, denoted by $a=|BC|,$ $b=|CA|,$ $c=|AB|.$ Denote by $m_a,m_b,m_c$ , respectively, the lengths of the medians which connect $A,B,C$ , respectively, with the centers of the corresponding opposite sides.

(a) Prove that $2m_a<b+c$ . Deduce that $m_a+m_b+m_c<a+b+c$ . (b) Give an example of (i) a triangle in which $m_a>\sqrt{bc}$ ; (ii) a triangle in which $m_a\le \sqrt{bc}$ .

The integers $a_0, a_1, a_2, a_3,\ldots$ are defined as follows: $a_0 = 1$ , $a_1 = 3$ , and $a_{n+1} = a_n + a_{n-1}$ for all $n \ge 1$ . Find all integers $n \ge 1$ for which $na_{n+1} + a_n$ and $na_n + a_{n-1}$ share a common factor greater than $1$ .

The incircle $\mathcal{C}_1$ of triangle $ABC$ touches the sides $AB$ and $AC$ at the points $D$ and $E$ , respectively. The incircle $\mathcal{C}_2$ of the triangle $ADE$ touches the sides $AB$ and $AC$ at the points $P$ and $Q$ , and intersects the circle $\mathcal{C}_1$ at the points $M$ and $n$ . Prove that

(a) the center of the circle $\mathcal{C}_2$ lies on the circle $\mathcal{C}_1$ .

(b) the four points $M,N,P,Q$ in appropriate order form a rectangle if and only if twice the radius of $\mathcal{C}_1$ is three times the radius of $\mathcal{C}_2$ .

In the mathematical talent show called "The $X^2$ -factor" contestants are scored by a a panel of $8$ judges. Each judge awards a score of $0$ ('fail'), $X$ ('pass'), or $X^2$ ('pass with distinction'). Three of the contestants were Ann, Barbara and David. Ann was awarded the same score as Barbara by exactly $4$ of the judges. David declares that he obtained different scores to Ann from at least $4$ of the judges, and also that he obtained different scores to Barbara from at least $4$ judges.

In how many ways could scores have been allocated to David, assuming he is telling the truth?

Prove that $$\frac{2}{3}+\frac{4}{5}+\dots +\frac{2010}{2011}$$ is not an integer.

Suppose that $x,y$ and $z$ are positive numbers such that $$1=2xyz+xy+yz+zx$$ Prove that (i) $$\frac{3}{4}\le xy+yz+zx<1$$ (ii) $$xyz\le \frac{1}{8}$$ Using (i) or otherwise, deduce that $$x+y+z\ge \frac{3}{2}$$ and derive the case of equality.

Find with proof all solutions in nonnegative integers $a,b,c,d$ of the equation $$11^a5^b-3^c2^d=1$$