IrMO 2016

If the three-digit number $ABC$ is divisible by $27$ , prove that the three-digit numbers $BCA$ and $CAB$ are also divisible by $27$ .

In triangle $ABC$ we have $|AB| \ne |AC|$ . The bisectors of $\angle ABC$ and $\angle ACB$ meet $AC$ and $AB$ at $E$ and $F$ , respectively, and intersect at I. If $|EI| = |FI|$ find the measure of $\angle BAC$ .

Do there exist four polynomials $P_1(x), P_2(x), P_3(x), P_4(x)$ with real coefficients, such that the sum of any three of them always has a real root, but the sum of any two of them has no real root?

Let $ABC$ be a triangle with $|AC| \ne |BC|$ . Let $P$ and $Q$ be the intersection points of the line $AB$ with the internal and external angle bisectors at $C$ , so that $P$ is between $A$ and $B$ . Prove that if $M$ is any point on the circle with diameter $PQ$ , then $\angle AMP = \angle BMP$ .

Let $a_1, a_2, ..., a_m$ be positive integers, none of which is equal to $10$ , such that $a_1 + a_2 + ...+ a_m = 10m$ . Prove that $(a_1a_2a_3 \cdot ...\cdot a_m)^{1/m} \le 3\sqrt{11}$ .

Triangle $ABC$ has sides $a = |BC| > b = |AC|$ . The points $K$ and $H$ on the segment $BC$ satisfy $|CH| = (a + b)/3$ and $|CK| = (a - b)/3$ . If $G$ is the centroid of triangle $ABC$ , prove that $\angle KGH = 90^o$ .

A rectangular array of positive integers has $4$ rows. The sum of the entries in each column is $20$ . Within each row, all entries are distinct. What is the maximum possible number of columns?

Suppose $a, b, c$ are real numbers such that $abc \ne 0$ . Determine $x, y, z$ in terms of $a, b, c$ such that $bz + cy = a, cx + az = b, ay + bx = c$ . Prove also that $\frac{1 - x^2}{a^2} = \frac{1 - y^2}{b^2} = \frac{1 - z^2}{c^2}$ .

Show that the number $a^3$ where $a=\frac{251}{ \frac{1}{\sqrt[3]{252}-5\sqrt[3]{2}}-10\sqrt[3]{63}}+\frac{1}{\frac{251}{\sqrt[3]{252}+5\sqrt[3]{2}}+10\sqrt[3]{63}}$ is an integer and find its value

Let $AE$ be a diameter of the circumcircle of triangle $ABC$ . Join $E$ to the orthocentre, $H$ , of $\triangle ABC$ and extend $EH$ to meet the circle again at $D$ . Prove that the nine point circle of $\triangle ABC$ passes through the midpoint of $HD$ .

Note. The nine point circle of a triangle is a circle that passes through the midpoints of the sides, the feet of the altitudes and the midpoints of the line segments that join the orthocentre to the vertices.