INMO 2020

Let $\Gamma_{1}$ and $\Gamma_{2}$ be two circles of unequal radii, with centres $O_{1}$ and $O_{2}$ respectively, in the plane intersecting in two distinct points $A$ and $B$. Assume that the centre of each of the circles $\Gamma_{1}$ and $\Gamma_{2}$ is outside the other. The tangent to $\Gamma_{1}$ at $B$ intersects $\Gamma_{2}$ again in $C$, different from $B$; the tangent to $\Gamma_{2}$ at $B$ intersects $\Gamma_{1}$ again in $D$, different from $B$. The bisectors of $\angle D A B$ and $\angle C A B$ meet $\Gamma_{1}$ and $\Gamma_{2}$ again in $X$ and $Y$, respectively, different from $A$. Let $P$ and $Q$ be the circumcentres of triangles $A C D$ and $X A Y$, respectively. Prove that $P Q$ is the perpendicular bisector of the line segment $O_{1} O_{2}$.

Suppose $P(x)$ is a polynomial with real coefficients satsfying the condition $P(\cos \theta+\sin \theta)=$ $P(\cos \theta-\sin \theta)$, for every real $\theta$. Prove that $P(x)$ can be expressed in the form

$$P(x)=a_{0}+a_{1}\left(1-x^{2}\right)^{2}+a_{2}\left(1-x^{2}\right)^{4}+\cdots+a_{n}\left(1-x^{2}\right)^{2 n}$$

for some real numbers $a_{0}, a_{1}, a_{2}, \ldots, a_{n}$ and nonnegative integer $n$.

Let $X=\{0,1,2,3,4,5,6,7,8,9\}$. Let $S \subseteq X$ be such that any nonnegative integer $n$ can be written as $p+q$ where the nonnegative integers $p, q$ have all their digits in $S$. Find the smallest possible number of elements in $S$.

Let $n \geq 3$ be an integer and let $1<a_{1} \leq a_{2} \leq a_{3} \leq \cdots \leq a_{n}$ be $n$ real numbers such that $a_{1}+a_{2}+a_{3}+\cdots+a_{n}=2 n$. Prove that

$$a_{1} a_{2} \cdots a_{n-1}+a_{1} a_{2} \cdots a_{n-2}+\cdots+a_{1} a_{2}+a_{1}+2 \leq a_{1} a_{2} \cdots a_{n}$$

Infinitely many equidistant parallel lines are drawn in the plane. A positive integer $n \geq 3$ is called frameable if it is possible to draw a regular polygon with $n$ sides all whose vertices lie on these lines and no line contains more than one vertex of the polygon.

(a) Show that $3,4,6$ are frameable.

(b) Show that any integer $n \geq 7$ is not frameable. (c) Determine whether 5 is frameable.

A stromino is a $3 \times 1$ rectangle. Show that a $5 \times 5$ board divided into twenty-five $1 \times 1$ squares cannot be covered by 16 strominos such that each stromino covers exactly three unit squares of the board and every unit square is covered by either one or two strominos. (A stromino can be placed either horizontally or vertically on the board.)