XK-MO 2022

Ana has $22$ coins. She can take from her friends either $6$ coins or $18$ coins, or she can give $12$ coins to her friends. She can do these operations many times she wants. Find the least number of coins Ana can have.

Show that for any positive real numbers $a$ and $b$ the following inequality hold, $$\frac{a(a+1)}{b+1}+\frac{b(b+1)}{a+1}\geq a+b.$$

Let $ABCD$ be a parallelogram and $l$ the line parallel to $AC$ which passes through $D$ . Let $E$ and $F$ points on $l$ such that $DE=DF=DB$ . Show that $EA,FC$ and $BD$ are concurrent.

Find all prime numbers $p$ and $q$ such that $pq-p-q+3$ is a perfect square.

Ana has a scale that shows which side weight more or if both side are equal. She has $4$ weights which look the same but they weight $1001g, 1002g, 1004g$ and $1005g$ , respectively. Is it possible for Ana to find out the weight of each of them with only $4$ measurements?

Let $ABC$ be an isosceles triangle with $CA=CB$ and $\angle ACB=20^{\circ}$ . Let $D$ be a point on side $CA$ such that $\angle ADB=30^{\circ}$ . Show that $AB=CD$ .

Let $a,b$ and $c$ be positive integers such that $a!+b+c,b!+c+a$ and $c!+a+b$ are prime numbers. Show that $\frac{a+b+c+1}{2}$ is also a prime number.

Let $a,b$ and $c$ be positive real numbers such that $a+b+c+3abc\geq (ab)^2+(bc)^2+(ca)^2+3$ . Show that the following inequality hold, $$\frac{a^3+b^3+c^3}{3}\geq\frac{abc+2021}{2022}.$$

$22$ light bulbs are given. Each light bulb is connected to exactly one switch, but a switch can be connected to one or more light bulbs. Find the least number of switches we should have such that we can turn on whatever number of light bulbs.

Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that for all real numbers $x$ and $y$ , $$f(f(x-y)-yf(x))=xf(y).$$

Find all positive integers $n$ such that $10^n+3^n+2$ is a palindrome number.

Assume that in the $\triangle ABC$ there exists a point $D$ on $BC$ and a line $l$ passing through $A$ such that $l$ is tangent to $(ADC)$ and $l$ bisects $BD.$ Prove that $a\sqrt{2}\geq b+c.$

Find all real numbers $a,b$ and $c$ such that $a+bc=b+ca=c+ab$ .

Let be given $n$ positive integer. Lets write with $a_n$ the number of positive integer pairs $(x,y)$ such that $x+y$ is even and $1\leq x\leq y\leq n$ . Lets write with $b_n$ the number of positive integer pairs $(x,y)$ such that $x+y\leq n+1$ and $1\leq x\leq y\leq n$ . Prove that $a_n=b_n$ .

Let $\bigtriangleup ABC$ be a triangle and $D$ be a point in line $BC$ such that $AD$ bisects $\angle BAC$ . Furthermore, let $F$ and $G$ be points on the circumcircle of $\bigtriangleup ABC$ and $E\neq D$ point in line $BC$ such that $AF=AE=AD=AG$ . If $X$ and $Y$ are the feet of perpendiculars from $D$ to $EF$ and $EG,$ respectively. Prove that $XY\parallel AD$ .

Find all positive integers $k,m$ and $n$ such that $k!+3^m=3^n$