JBMO-SL 2015

MLD

Let $x, y, z$ be real numbers, satisfying the relations

$$\left\{\begin{array}{l} x \geq 20 \\ y \geq 40 \\ z \geq 1675 \\ x+y+z=2015 \end{array}\right.$$

Find the greatest value of the product $P=x \cdot y \cdot z$.

ALB

3. If $x^{3}-3 \sqrt{3} x^{2}+9 x-3 \sqrt{3}-64=0$, find the value of $x^{6}-8 x^{5}+13 x^{4}-5 x^{3}+49 x^{2}-137 x+2015$.

MNE

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

$$\frac{a}{b}+\sqrt{\frac{b}{c}}+\sqrt[3]{\frac{c}{a}}>2$$

MKCD

Let $x, y, z$ be positive real numbers that satisfy the equality $x^{2}+y^{2}+z^{2}=3$. Prove that

$$\frac{x^{2}+y z}{x^{2}+y z+1}+\frac{y^{2}+z x}{y^{2}+z x+1}+\frac{z^{2}+x y}{z^{2}+x y+1} \leq 2$$

BUL

A board $n \times n(n \geq 3)$ is divided into $n^{2}$ unit squares. Integers from 0 to $n$ included are written down: one integer in each unit square, in such a way that the sums of integers in each $2 \times 2$ square of the board are different. Find all $n$ for which such boards exist.

ALB

Positive integers are put into the following table

1	3	6	10	15	21	28	36
2	5	9	14	20	27	35	44
4	8	13	19	26	34	43	53
7	12	18	25	33	42
11	17	24	32	41
16	23
$\ldots$
$\ldots$

Find the number of the line and column where the number 2015 stays.

GRE

Let $n \geq 1$ be a positive integer. A square of side length $n$ is divided by lines parallel to each side into $n^{2}$ squares of side length 1 . Find the number of parallelograms which have vertices among the vertices of the $n^{2}$ squares of side length 1 , with both sides smaller or equal to 2 , and which have the area equal to 2 .

MNE

Around the triangle $A B C$ the circle is circumscribed, and at the vertex $C$ tangent $t$ to this circle is drawn. The line $p$ which is parallel to this tangent intersects the lines $B C$ and $A C$ at the points $D$ and $E$, respectively. Prove that the points $A, B, D, E$ belong to the same circle.

MLD

The point $P$ is outside of the circle $\Omega$. Two tangent lines, passing from the point $P$, touch the circle $\Omega$ at the points $A$ and $B$. The median $A M, M \in(B P)$, intersects the circle $\Omega$ at the point $C$ and the line $P C$ intersects again the circle $\Omega$ at the point $D$. Prove that the lines $A D$ and $B P$ are parallel.

GRE

Let $c \equiv c(O, K)$ be a circle with center $O$ and radius $R$ and $A, B$ be two points on it, not belonging to the same diameter. The bisector of the angle $A \hat{B} O$ intersects the circle $c$ at point $C$, the circumcircle of the triangle $A O B$, say ${ }^{c_{1}}$ at point $K$ and the circumcircle of the triangle $A O C$, say ${ }^{c_{2}}$, at point $L$. Prove that the point $K$ is the circumcenter of the triangle $A O C$ and the point $L$ is the incenter of the triangle $A O B$.

CYP

Let $\triangle A B C$ be an acute triangle. The lines $\left(\varepsilon_{1}\right),\left(\varepsilon_{2}\right)$ are perpendicular to $A B$ at the points $A$, $B$, respectively. The perpendicular lines from the midpoint $M$ of $A B$ to the sides of the triangle $A C_{;} B C$ intersect the lines $\left(\varepsilon_{1}\right),\left(\xi_{2}\right)$ at the points $E, F$, respectively. If $I$ is the intersection point of $E F, M C$, prove that

$$\angle A I B=\angle E M F=\angle C A B+\angle C B A$$

ROU

Let $A B C$ be an acute triangle with $A B \neq A C$. The incircle $\omega$ of the triangle touches the sides $B C, C A$ and $A B$ at $D, E$ and $F$, respectively. The perpendicular line erected at $C$ onto $B C$ meets $E F$ at $M$, and similarly, the perpendicular line erected at $B$ onto $B C$ meets $E F$ at $N$. The line $D M$ meets $\omega$ again in $P$, and the line $D N$ meets $\omega$ again at $Q$. Prove that $D P=D Q$.