IrMO 2014

Given an $8\times 8$ chess board, in how many ways can we select $56$ squares on the board while satisfying both of the following requirements: (a) All black squares are selected. (b) Exactly seven squares are selected in each column and in each row.

Prove that for $N>1$ that $(N^{2})^{2014} - (N^{11})^{106}$ is divisible by $N^6 + N^3 +1$ Is this just a proof by induction or is there a more elegant method? I don't think calculating $N = 2$ was expected.

In the triangle ABC, D is the foot of the altitude from A to BC, and M is the midpoint of the line segment BC. The three angles ∠BAD, ∠DAM and ∠MAC are all equal. Find the angles of the triangle ABC.

Three different non-zero real numbers $a,b,c$ satisfy the equations $a+\frac{2}{b}=b+\frac{2}{c}=c+\frac{2}{a}=p$ , where $p$ is a real number. Prove that $abc+2p=0.$

Suppose $a_1,a_2,\ldots,a_n>0$ , where $n>1$ and $\sum_{i=1}^{n}a_i=1$ . For each $i=1,2,\ldots,n$ , let $b_i=\frac{a^2_i}{\sum\limits_{j=1}^{n}a^2_j}$ . Prove that $$\sum_{i=1}^{n}\frac{a_i}{1-a_i}\le \sum_{i=1}^{n}\frac{b_i}{1-b_i} .$$ When does equality occur ?

Each of the four positive integers $N,N +1,N +2,N +3$ has exactly six positive divisors. There are exactly $20$ dierent positive numbers which are exact divisors of at least one of the numbers. One of these is $27$ . Find all possible values of $N$ .(Both $1$ and $m$ are counted as divisors of the number $m$ .)

The square $ABCD$ is inscribed in a circle with center $O$ . Let $E$ be the midpoint of $AD$ . The line $CE$ meets the circle again at $F$ . The lines $FB$ and $AD$ meet at $H$ . Prove $HD = 2AH$

(a) Let $a_0, a_1,a_2$ be real numbers and consider the polynomial $P(x) = a_0 + a_1x + a_2x^2$ . Assume that $P(-1), P(0)$ and $P(1)$ are integers. Prove that $P(n)$ is an integer for all integers $n$ . (b) Let $a_0,a_1, a_2, a_3$ be real numbers and consider the polynomial $Q(x) = a0 + a_1x + a_2x^2 + a_3x^3$ . Assume that there exists an integer $i$ such that $Q(i),Q(i+1),Q(i+2)$ and $Q(i+3)$ are integers. Prove that $Q(n)$ is an integer for all integers $n$ .

Let $n$ be a positive integer and $a_1,...,a_n$ be positive real numbers. Let $g(x)$ denote the product $(x + a_1)\cdot ... \cdot (x + a_n)$ . Let $a_0$ be a real number and let $f(x) = (x - a_0)g(x)= x^{n+1} + b_1x^n + b_2x^{n-1}+...+ b_nx + b_{n+1}$ . Prove that all the coeffcients $b_1,b_2,..., b_{n+1}$ of the polynomial $f(x)$ are negative if and only if $a_0 > a_1 + a_2 +...+ a_n$ .

Over a period of $k$ consecutive days, a total of $2014$ babies were born in a certain city, with at least one baby being born each day. Show that: (a) If $1014 < k \le 2014$ , there must be a period of consecutive days during which exactly $100$ babies were born. (b) By contrast, if $k = 1014$ , such a period might not exist.