XK-MO 2019

Calculate $1^2-2^2+3^2-4^2+...-2018^2+2019^2$ .

Show that when the product of three conscutive numbers we add arithmetic mean of them it is a perfect cube.

Let $ABCD$ be a rectangle with $AB>BC$ . Let points $E,F$ be on side $CD$ such that $CE=ED$ and $BC=CF$ . Show that if $AC$ is prependicular to $BE$ then $AB=BF$ .

Find all sequence of consecutive positive numbers which the sum of them is equal with $2019$ .

There are given in a table numbers $1,2,...,18$ . What is minimal number of numbers we should erase such that the sum of every two remaining numbers is not perfect square of a positive integer.

Find last three digits of the number $\frac{2019!}{2^{1009}}$ .

Show that for any positive real numbers $a,b,c$ the following inequality is true: $$4(a^3+b^3+c^3+3)\geq 3(a+1)(b+1)(c+1)$$ When does equality hold?

The doctor instructed a person to take $48$ pills for next $30$ days. Every day he take at least $1$ pill and at most $6$ pills. Show that exist the numbers of conscutive days such that the total numbers of pills he take is equal with $11$ .

Find all real numbers $x,y,z$ such that satisfied the following equalities at same time: $\sqrt{x^3-y}=z-1 \wedge \sqrt{y^3-z}=x-1\wedge \sqrt{z^3-x}=y-1$

Let $ABCDE$ be a regular pentagon. Let point $F$ be intersection of segments $AC$ and $BD$ . Let point $G$ be in segment $AD$ such that $2AD=3AG$ . Let point $H$ be the midpoint of side $DE$ . Show that the points $F,G,H$ lie on a line.

Let $a,b$ be real numbers grater then $4$ . Show that at least one of the trinomials $x^2+ax+b$ or $x^2+bx+a$ has two different real zeros.

Find all positive integers $n$ such that $6^n+1$ it has all the same digits when it is writen in decimal representation.

Let $ABC$ be a triangle with $\angle CAB=60^{\circ}$ and with incenter $I$ . Let points $D,E$ be on sides $AC,AB$ , respectively, such that $BD$ and $CE$ are angle bisectors of angles $\angle ABC$ and $\angle BCA$ , respectively. Show that $ID=IE$ .

Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that: $$f(xy+f(x))=xf(y)$$ for all $x,y\in\mathbb{R}$ .

There are given points with integer coordinate $(m,n)$ such that $1\leq m,n\leq 4$ . Two players, Ana and Ben, are playing a game: First Ana color one of the coordinates with red one, then she pass the turn to Ben who color one of the remaining coordinates with yellow one, then this process they repeate again one after other. The game win the first player who can create a rectangle with same color of vertices and the length of sides are positive integer numbers, otherwise the game is a tie. Does there exist a strategy for any of the player to win the game?

Does there exist a triangle with length $a,b,c$ such that:a) $\begin{cases} a+b+c=6 a^2+b^2+c^2=13 a^3+b^3+c^3=28 \end{cases}$ b) $\begin{cases} a+b+c=6 a^2+b^2+c^2=13 a^3+b^3+c^3=30 \end{cases}$

Suppose that each point on a plane is colored with one of the colors red or yellow. Show that exist a convex pentagon with three right angles and all vertices are with same color.

Show that for any non-negative real numbers $a,b,c,d$ such that $a^2+b^2+c^2+d^2=1$ the following inequality hold: $$a+b+c+d-1\geq 16abcd$$ When does equality hold?

Let $ABC$ be an acute triagnle with its circumcircle $\omega$ . Let point $D$ be the foot of triangle $ABC$ from point $A$ . Let points $E,F$ be midpoints of sides $AB,AC$ , respectively. Let points $P$ and $Q$ be the second intersections of of circle $\omega$ with circumcircle of triangles $BDE$ and $CDF$ , respectively. Suppose that $A,P,B,Q$ and $C$ be on a circle in this order. Show that the lines $EF,BQ$ and $CP$ are concurrent.

Find all positive integers $x,y$ such that $2^x+19^y$ is a perfect cube.