IrMO 2017

Determine, with proof, the smallest positive multiple of $99$ all of whose digits are either $1$ or $2$ .

Solve the equations : $$\begin{cases} a + b + c = 0 a^2 + b^2 + c^2 = 1a^3 + b^3 +c^3 = 4abc \end{cases}$$ for $a,b,$ and $c.$

Four circles are drawn with the sides of quadrilateral $ABCD$ as diameters. The two circles passing through $A$ meet again at $A'$ , two circles through $B$ at $B'$ , two circles at $C$ at $C'$ and the two circles at $D$ at $D'$ . Suppose the points $A',B',C'$ and $D'$ are distinct. Prove quadrilateral $A'B'C'D'$ is similar to $ABCD$ .

An equilateral triangle of integer side length $n \geq 1$ is subdivided into small triangles of unit side length, as illustrated in the figure below for $n = 5$ . In this diagram a subtriangle is a triangle of any size which is formed by connecting vertices of the small triangles along the grid lines. It is desired to color each vertex of the small triangles either red or blue in such a way that there is no subtriangle with all of its vertices having the same color. Let $f(n)$ denote the number of distinct colorings satisfying this condition. Determine, with proof, $f(n)$ for every $n \geq 1$

The sequence $a = (a_0, a_1,a_2,...)$ is defined by $a_0 = 0, a_1 =2$ and $$a_{n+2} = 2a_{n+1} + 41a_n$$ Prove that $a_{2016}$ is divisible by $2017.$

Does there exist an even positive integer $n$ for which $n+1$ is divisible by $5$ and the two numbers $2^n + n$ and $2^n -1$ are co-prime?

$5$ teams play in a soccer competition where each team plays one match against each of the other four teams. A winning team gains $5$ points and a losing team $0$ points. For a $0-0$ draw both teams gain $1$ point, and for other draws ( $1-1,2-2,3-3,$ etc.) both teams gain 2 points. At the end of the competition, we write down the total points for each team, and we find that they form 5 consecutive integers. What is the minimum number of goals scored?

A line segment $B_0B_n$ is divided into $n$ equal parts at points $B_1,B_2,...,B_{n-1}$ and $A$ is a point such that $\angle B_0AB_n$ is a right angle. Prove that : $$\sum_{k=0}^{n} |AB_k|^{2} = \sum_{k=0}^{n} |B_0B_k|^2$$

Show that for all non-negative numbers $a,b$ , $$1 + a^{2017} + b^{2017} \geq a^{10}b^{7} + a^{7}b^{2000} + a^{2000}b^{10}$$ When is equality attained?

Given a positive integer $m$ , a sequence of real numbers $a= (a_1,a_2,a_3,...)$ is called $m$ -powerful if it satisfies $$(\sum_{k=1}^{n} a_k )^{m} = \sum_{k=1}^{n} a_k^{m}$$ for all positive integers $n$ . (a) Show that a sequence is $30$ -powerful if and only if at most one of its terms is non-zero. (b) Find a sequence none of whose terms are zero but which is $2017$ -powerful.