IrMO 2013

Prove that $$1-\frac{1}{2012}\left(\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{2013}\right)\ge \frac{1}{\sqrt[2012]{2013}}.$$

The altitudes of a triangle $\triangle ABC$ are used to form the sides of a second triangle $\triangle A_1B_1C_1$ . The altitudes of $\triangle A_1B_1C_1$ are then used to form the sides of a third triangle $\triangle A_2B_2C_2$ . Prove that $\triangle A_2B_2C_2$ is similar to $\triangle ABC$ .

Each of the $36$ squares of a $6 \times 6$ table is to be coloured either Red, Yellow or Blue. (a) No row or column is contain more than two squares of the same colour. (b) In any four squares obtained by intersecting two rows with two columns, no colour is to occur exactly three times. In how many dierent ways can the table be coloured if both of these rules are to be respected?

$A, B$ and $C$ are points on the circumference of a circle with centre $O$ . Tangents are drawn to the circumcircles of triangles $OAB$ and $OAC$ at $P$ and $Q$ respectively, where $P$ and $Q$ are diametrically opposite $O$ . The two tangents intersect at $K$ . The line $CA$ meets the circumcircle of $\triangle OAB$ at $A$ and $X$ . Prove that $X$ lies on the line $KO$ .

Consider the collection of different squares which may be formed by sets of four points chosen from the $12$ labelled points in the diagram on the right. For each possible area such a square may have, determine the number of squares which have this area. Make sure to explain why your list is complete.

Find the smallest positive integer $N$ for which the equation $(x^2 -1)(y^2 -1)=N$ is satised by at least two pairs of integers $(x, y)$ with $1 < x \le y$ .

We say that a doubly infinite sequence $. . . , s_{-2}, s_{-1}, s_{0}, s_1, s_2, . . .$ is subaveraging if $s_n = (s_{n-1} + s_{n+1})/4$ for all integers n. (a) Find a subaveraging sequence in which all entries are different from each other. Prove that all entries are indeed distinct. (b) Show that if $(s_n)$ is a subaveraging sequence such that there exist distinct integers m, n such that $s_m = s_n$ , then there are infinitely many pairs of distinct integers i, j with $s_i = s_j$ .

Let $a,b,c$ be real numbers and let $x=a+b+c,y=a^2+b^2+c^2,z=a^3+b^3+c^3$ and $S=2x^3-9xy+9z .$ $(a)$ Prove that $S$ is unchanged when $a,b,c$ are replaced by $a+t,b+t,c+t$ , respectively , for any real number $t$ . $(b)$ Prove that $(3y-x^2)^3\ge S^2 .$