JBMO-SL 2019

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

Let $a, b, c$ be positive real numbers such that $a b c=\frac{2}{3}$. Prove that

$$\frac{a b}{a+b}+\frac{b c}{b+c}+\frac{c a}{c+a} \geqslant \frac{a+b+c}{a^{3}+b^{3}+c^{3}}$$

Let $A$ and $B$ be two non-empty subsets of $X=\{1,2, \ldots, 11\}$ 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.

Let $a, b$ be two distinct real numbers and let $c$ be a positive real number such that

$$a^{4}-2019 a=b^{4}-2019 b=c$$

Prove that $-\sqrt{c}<a b<0$.

Let $a, b, c, d$ be positive real numbers such that $a b c d=1$. Prove the inequality

$$\frac{1}{a^{3}+b+c+d}+\frac{1}{a+b^{3}+c+d}+\frac{1}{a+b+c^{3}+d}+\frac{1}{a+b+c+d^{3}} \leqslant \frac{a+b+c+d}{4}$$

Let $a, b, c$ be positive real numbers. Prove the inequality

$$\left(a^{2}+a c+c^{2}\right)\left(\frac{1}{a+b+c}+\frac{1}{a+c}\right)+b^{2}\left(\frac{1}{b+c}+\frac{1}{a+b}\right)>a+b+c$$

Show that for any positive real numbers $a, b, c$ such that $a+b+c=a b+b c+c a$, the following inequality holds

$$3+\sqrt[3]{\frac{a^{3}+1}{2}}+\sqrt[3]{\frac{b^{3}+1}{2}}+\sqrt[3]{\frac{c^{3}+1}{2}} \leqslant 2(a+b+c)$$

Let $S$ be a set of 100 positive integers 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 each of the other 99 numbers of $S$.

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.

In a $5 \times 100$ table we have coloured black $n$ of its cells. Each of the 500 cells has at most two adjacent (by side) cells coloured black. 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?

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 $\{1,2, \ldots, 99\}$ 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 $\{1,2, \ldots, 2019\}$ 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.

Let $A B C$ be a right-angled triangle with $\hat{A}=90^{\circ}$ and $\hat{B}=30^{\circ}$. The perpendicular at the midpoint $M$ of $B C$ meets the bisector $B K$ of the angle $\hat{B}$ at the point $E$. The perpendicular bisector of $E K$ meets $A B$ at $D$. Prove that $K D$ is perpendicular to $D E$.

Let $A B C$ be a triangle and let $\omega$ be its circumcircle. Let $\ell_{B}$ and $\ell_{C}$ be two parallel lines passing through $B$ and $C$ respectively. The lines $\ell_{B}$ and $\ell_{C}$ intersect with $\omega$ for the second time at the points $D$ and $E$ respectively, with $D$ belonging on the arc $A B$, and $E$ on the arc $A C$. Suppose that $D A$ intersects $\ell_{C}$ at $F$, and $E A$ intersects $\ell_{B}$ at $G$. If $O, O_{1}$ and $O_{2}$ are the circumcenters of the triangles $A B C, A D G$ and $A E F$ respectively, and $P$ is the center of the circumcircle of the triangle $O O_{1} O_{2}$, prove that $O P$ is parallel to $\ell_{B}$ and $\ell_{C}$.

Let $A B C$ be a triangle with incenter $I$. The points $D$ and $E$ lie on the segments $C A$ and $B C$ respectively, such that $C D=C E$. Let $F$ be a point on the segment $C D$. Prove that the quadrilateral $A B E F$ is circumscribable if and only if the quadrilateral $D I E F$ is cyclic.

Let $A B C$ be a triangle such that $A B \neq A C$, and let the perpendicular bisector of the side $B C$ intersect lines $A B$ and $A C$ at points $P$ and $Q$, respectively. If $H$ is the orthocenter of the triangle $A B C$, and $M$ and $N$ are the midpoints of the segments $B C$ and $P Q$ respectively, prove that $H M$ and $A N$ meet on the circumcircle of $A B C$.

Let $P$ be a point in the interior of a triangle $A B C$. The lines $A P, B P$ and $C P$ intersect again the circumcircles of the triangles $P B C, P C A$, and $P A B$ at $D, E$ and $F$ respectively. Prove that $P$ is the orthocenter of the triangle $D E F$ if and only if $P$ is the incenter of the triangle $A B C$.

Let $A B C$ be a non-isosceles triangle with incenter $I$. Let $D$ be a point on the segment $B C$ such that the circumcircle of $B I D$ intersects the segment $A B$ at $E \neq B$, and the circumcircle of $C I D$ intersects the segment $A C$ at $F \neq C$. The circumcircle of $D E F$ intersects $A B$ and $A C$ at the second points $M$ and $N$ respectively. Let $P$ be the point of intersection of $I B$ and $D E$, and let $Q$ be the point of intersection of $I C$ and $D F$. Prove that the three lines $E N, F M$ and $P Q$ are parallel.

Let $A B C$ be a right-angled triangle with $\hat{A}=90^{\circ}$. Let $K$ be the midpoint of $B C$, and let $A K L M$ be a parallelogram with centre $C$. Let $T$ be the intersection of the line $A C$ and the perpendicular bisector of $B M$. Let $\omega_{1}$ be the circle with centre $C$ and radius $C A$ and let $\omega_{2}$ be the circle with centre $T$ and radius $T B$. Prove that one of the points of intersection of $\omega_{1}$ and $\omega_{2}$ is on the line $L M$.

Find all prime numbers $p$ for which there are non-negative integers $x, y$ and $z$ such that the number

$$A=x^{p}+y^{p}+z^{p}-x-y-z$$

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

$$\frac{p^{2}+2 q}{q+r}, \frac{q^{2}+9 r}{r+p}, \frac{r^{2}+3 p}{p+q}$$

Find all prime numbers $p$ and nonnegative integers $x \neq y$ such that $x^{4}-y^{4}=$ $p\left(x^{3}-y^{3}\right)$.

Find all integers $x, y$ such that

$$x^{3}(y+1)+y^{3}(x+1)=19$$

Find all positive integers $x, y, z$ such that

$$45^{x}-6^{y}=2019^{z}$$

Find all triples $(a, b, c)$ of nonnegative integers that satisfy

$$a!+5^{b}=7^{c}$$

Find all perfect squares $n$ such that if the positive integer $a \geqslant 15$ is some divisor of $n$ then $a+15$ is a prime power.