Is it true that there exists a triangle with sides $x, y, z$ so that $x^3+y^3+z^3=(x+y)(y+z)(z+x)$?

Let $M$ and $N$ be two palindrome numbers, each having $9$ digits and the palindromes don't start with $0$. If $N>M$ and between $N$ and $M$ there aren't any palindromes, find all values of $N-M$.

Let $ABC$ be a triangle inscribed in the circle $K_1$ and $I$ be center of the inscribed in $ABC$ circle. The lines $IB$ and $IC$ intersect circle $K_1$ again in $J$ and $L$. Circle $K_2$, circumscribed to $IBC$, intersects again $CA$ and $AB$ in $E$ and $F$. Show that $EL$ and $FJ$ intersects on the circle $K_2$.

Let $n=>2$ be a natural number. A set $S$ of natural numbers is called $complete$ if, for any integer $0<=x<n$, there is a subset of $S$ with the property that the remainder of the division by $n$ of the sum of the elements in the subset is $x$. The sum of the elements of the empty set is considered to be $0$. Show that if a set $S$ is $complete$, then there is a subset of $S$ which has at most $n-1$ elements and which is still $complete$.

$p, q, r$ are distinct prime numbers which satisfy $$2pqr + 50pq = 7pqr + 55pr = 8pqr + 12qr = A$$for natural number $A$. Find all values of $A$.

Let $a, b, c$ be reals which satisfy $a+b+c+ab+bc+ac+abc=>7$, prove that $$\sqrt{a^2+b^2+2}+\sqrt{b^2+c^2+2}+\sqrt{c^2+a^2+2}=>6$$

The cube $nxnxn$ consists of $n^3$ unit cubes $1x1x1$, and at least one of these unit cubes is black. Show that we can always cut the cube in $2$ parallelepiped pieces so that each piece contains exactly one black 1x1 square .

Let $ABC$ be a acute triangle in which $O$ and $H$ are the center of the circumscribed circle, respectively the orthocenter. Let $M$ be a point on the small arc $BC$ of the circumscribed circle (different from $B$ and $C$) and be $D, E, F$ be the symmetrical of the point $M$ to the lines $OA, OB, OC$. We note with $K$ the intersection of $BF$ and $CE$ and $I$ is the center of the circle inscribed in the triangle $DEF$. a) Show that the segment bisectors of the segments $EF$ and $IK$ intersect on the circle circumscribed to triangle $ABC$. a) Prove that points $H, K, I$ are collinear.

Let $n$ be a natural composite number. For each proper divisor $d$ of $n$ we write the number $d + 1$ on the board. Determine all natural numbers $n$ for which the numbers written on the board are all the proper divisors of a natural number $m$. (The proper divisors of a natural number $a> 1$ are the positive divisors of $a$ different from $1$ and $a$.)

Let $ABCD$ be a square inscribed in circle $K$. Let $P$ be a point on the small arc $CD$ of circle $K$. The line $PB$ intersects $AC$ in $E$. The line $PA$ intersects $DB$ in $F$. The circle circumscribed to triangle $PEF$ intersects for second time $K$ in $Q$. Prove that $PQ$ is parallel to $CD$.

Prove that in every triangle there are two sides with lengths $x$ and $y$ such that $$\frac{\sqrt{5}-1}{2}\leq\frac{x}{y}\leq\frac{\sqrt{5}+1}{2}$$

Let $n> 2$ be a natural number. We consider $n$ candy bags, each containing exactly one candy. Ali and Omar play the following game in which they move alternately (Ali moves the first): At each move, the player who has to make a move chooses two bags containing $x$, respectively $y$ candy, with $(x,y)=1$, and he puts the $x + y$ candies in one bag (he chooses where). The player who can't make a move loses. Which of the two players has a strategy to win this game?