Real numbers $a$ and $b$ satisfy $a^3+b^3-6ab=-11$. Prove that $-\frac{7}{3}<a+b<-2$. Proposed by Serbia

Let $a, b, c $ be positive real numbers such that $abc = \frac {2} {3}. $ Prove that: $$\frac {ab}{a + b} + \frac {bc} {b + c} + \frac {ca} {c + a} \geqslant \frac {a+b+c} {a^3+b ^ 3 + c ^ 3}.$$

Let $A$ and $B$ be two non-empty subsets of $X = \{1, 2, . . . , 11 \}$ with $A \cup B = X$. Let $P_A$ be the product of all elements of $A$ and let $P_B$ be the product of all elements of $B$. Find the minimum and maximum possible value of $P_A +P_B$ and find all possible equality cases. Proposed by Greece

Let $a$, $b$ be two distinct real numbers and let $c$ be a positive real numbers such that $a^4 - 2019a = b^4 - 2019b = c$. Prove that $- \sqrt{c} < ab < 0$.

Let $a, b, c, d$ be positive real numbers such that $abcd = 1$. Prove the inequality $\frac{1}{a^3 + b + c + d} +\frac{1}{a + b^3 + c + d}+\frac{1}{a + b + c^3 + d} +\frac{1}{a + b + c + d^3} \leq \frac{a+b+c+d}{4}$ Proposed by Romania

Let $a, b, c$ be positive real numbers. Prove the inequality $(a^2+ac+c^2) \left( \frac{1}{a+b+c}+\frac{1}{a+c} \right)+b^2 \left( \frac{1}{b+c}+\frac{1}{a+b} \right)>a+b+c$. Proposed by Tajikistan

Show that for any positive real numbers $a, b, c$ such that $a + b + c = ab + bc + ca$, the following inequality holds $3 + \sqrt[3]{\frac{a^3+1}{2}}+\sqrt[3]{\frac{b^3+1}{2}}+\sqrt[3]{\frac{c^3+1}{2}}\leq 2(a+b+c)$ Proposed by Dorlir Ahmeti, Albania

Let $S$ be a set of $100$ positive integer numbers having the following property: “Among every four numbers of $S$, there is a number which divides each of the other three or there is a number which is equal to the sum of the other three.” Prove that the set $S$ contains a number which divides all other $99$ numbers of $S$. Proposed by Tajikistan

In a certain city there are $n$ straight streets, such that every two streets intersect, and no three streets pass through the same intersection. The City Council wants to organize the city by designating the main and the side street on every intersection. Prove that this can be done in such way that if one goes along one of the streets, from its beginning to its end, the intersections where this street is the main street, and the ones where it is not, will apear in alternating order. Proposed by Serbia

A $5 \times 100$ table is divided into $500$ unit square cells, where $n$ of them are coloured black and the rest are coloured white. Two unit square cells are called adjacent if they share a common side. Each of the unit square cells has at most two adjacent black unit square cells. Find the largest possible value of $n$.

We have a group of $n$ kids. For each pair of kids, at least one has sent a message to the other one. For each kid $A$, among the kids to whom $A$ has sent a message, exactly $25 \% $ have sent a message to $A$. How many possible two-digit values of $n$ are there? Proposed by Bulgaria

An economist and a statistician play a game on a calculator which does only one operation. The calculator displays only positive integers and it is used in the following way: Denote by $n$ an integer that is shown on the calculator. A person types an integer, $m$, chosen from the set $\{ 1, 2, . . . , 99 \}$ of the first $99$ positive integers, and if $m\%$ of the number $n$ is again a positive integer, then the calculator displays $m\%$ of $n$. Otherwise, the calculator shows an error message and this operation is not allowed. The game consists of doing alternatively these operations and the player that cannot do the operation looses. How many numbers from $\{1, 2, . . . , 2019\}$ guarantee the winning strategy for the statistician, who plays second? For example, if the calculator displays $1200$, the economist can type $50$, giving the number $600$ on the calculator, then the statistician can type $25$ giving the number $150$. Now, for instance, the economist cannot type $75$ as $75\%$ of $150$ is not a positive integer, but can choose $40$ and the game continues until one of them cannot type an allowed number Proposed by Serbia

Let $ABC$ be a right-angled triangle with $\angle A = 90^{\circ}$ and $\angle B = 30^{\circ}$. The perpendicular at the midpoint $M$ of $BC$ meets the bisector $BK$ of the angle $B$ at the point $E$. The perpendicular bisector of $EK$ meets $AB$ at $D$. Prove that $KD$ is perpendicular to $DE$. Proposed by Greece

Let $ABC$ be a triangle with circumcircle $\omega$. Let $l_B$ and $l_C$ be two lines through the points $B$ and $C$, respectively, such that $l_B \parallel l_C$. The second intersections of $l_B$ and $l_C$ with $\omega$ are $D$ and $E$, respectively. Assume that $D$ and $E$ are on the same side of $BC$ as $A$. Let $DA$ intersect $l_C$ at $F$ and let $EA$ intersect $l_B$ at $G$. If $O$, $O_1$ and $O_2$ are circumcenters of the triangles $ABC$, $ADG$ and $AEF$, respectively, and $P$ is the circumcenter of the triangle $OO_1O_2$, prove that $l_B \parallel OP \parallel l_C$. Proposed by Stefan Lozanovski, Macedonia

Let $ABC$ be a triangle with incenter $I$. The points $D$ and $E$ lie on the segments $CA$ and $BC$ respectively, such that $CD = CE$. Let $F$ be a point on the segment $CD$. Prove that the quadrilateral $ABEF$ is circumscribable if and only if the quadrilateral $DIEF$ is cyclic. Proposed by Dorlir Ahmeti, Albania

Triangle $ABC$ is such that $AB < AC$. The perpendicular bisector of side $BC$ intersects lines $AB$ and $AC$ at points $P$ and $Q$, respectively. Let $H$ be the orthocentre of triangle $ABC$, and let $M$ and $N$ be the midpoints of segments $BC$ and $PQ$, respectively. Prove that lines $HM$ and $AN$ meet on the circumcircle of $ABC$.

Let $P$ be a point in the interior of a triangle $ABC$. The lines $AP, BP$ and $CP$ intersect again the circumcircles of the triangles $PBC, PCA$ and $PAB$ at $D, E$ and $F$ respectively. Prove that $P$ is the orthocenter of the triangle $DEF$ if and only if $P$ is the incenter of the triangle $ABC$. Proposed by Romania

Let $ABC$ be a non-isosceles triangle with incenter $I$. Let $D$ be a point on the segment $BC$ such that the circumcircle of $BID$ intersects the segment $AB$ at $E\neq B$, and the circumcircle of $CID$ intersects the segment $AC$ at $F\neq C$. The circumcircle of $DEF$ intersects $AB$ and $AC$ at the second points $M$ and $N$ respectively. Let $P$ be the point of intersection of $IB$ and $DE$, and let $Q$ be the point of intersection of $IC$ and $DF$. Prove that the three lines $EN, FM$ and $PQ$ are parallel. Proposed by Saudi Arabia

Let $ABC$ be a right-angled triangle with $\angle A = 90^{\circ}$. Let $K$ be the midpoint of $BC$, and let $AKLM$ be a parallelogram with centre $C$. Let $T$ be the intersection of the line $AC$ and the perpendicular bisector of $BM$. Let $\omega_1$ be the circle with centre $C$ and radius $CA$ and let $\omega_2$ be the circle with centre $T$ and radius $TB$. Prove that one of the points of intersection of $\omega_1$ and $\omega_2$ is on the line $LM$. Proposed by Greece

Find all prime numbers $p$ for which there exist positive integers $x$, $y$, and $z$ such that the number $x^p + y^p + z^p - x - y - z$ is a product of exactly three distinct prime numbers.

Find all triples $(p, q, r)$ of prime numbers such that all of the following numbers are integers $\frac{p^2 + 2q}{q+r}, \frac{q^2+9r}{r+p}, \frac{r^2+3p}{p+q}$ Proposed by Tajikistan

Find all prime numbers $p$ and nonnegative integers $x\neq y$ such that $x^4- y^4=p(x^3-y^3)$. Proposed by Bulgaria

Find all integers $x,y$ such that $x^3(y+1)+y^3(x+1)=19$. Proposed by Bulgaria

Find all positive integers $x, y, z$ such that $45^x-6^y=2019^z$ Proposed by Dorlir Ahmeti, Albania

$a,b,c$ are non-negative integers. Solve: $a!+5^b=7^c$ Proposed by Serbia

Find all perfect squares $n$ such that if the positive integer $a\ge 15$ is some divisor $n$ then $a+15$ is a prime power. Proposed by Saudi Arabia