JBMO-SL 2013

Find all ordered triples $(x, y, z)$ of real numbers satisfying the following system of equations:

$$\begin{aligned} x^{3} & =\frac{z}{y}-2 \frac{y}{z} \\ y^{3} & =\frac{x}{z}-2 \frac{z}{x} \\ z^{3} & =\frac{y}{x}-2 \frac{x}{y} \end{aligned}$$

Find the largest possible value of the expression $\left|\sqrt{x^{2}+4 x+8}-\sqrt{x^{2}+8 x+17}\right|$ where $x$ is a real number.

Show that

$$\left(a+2 b+\frac{2}{a+1}\right)\left(b+2 a+\frac{2}{b+1}\right) \geq 16$$

for all positive real numbers $a, b$ satisfying $a b \geq 1$.

All possible pairs of $n$ apples are weighed and the results are given to us in an arbitrary order. Can we determine the weights of the apples if a. $n=4$, b. $n=5$, c. $n=6$ ?

Let $A B$ be a diameter of a circle $\omega$ with center $O$ and $O C$ be a radius of $\omega$ which is perpendicular to $A B$. Let $M$ be a point on the line segment $O C$. Let $N$ be the second point of intersection of the line $A M$ with $\omega$, and let $P$ be the point of intersection of the lines tangent to $\omega$ at $N$ and at $B$. Show that the points $M, O, P, N$ are concyclic.

Let $D$ be a point on the side $B C$ of an acute triangle $A B C$ such that $\angle B A D=\angle C A O$ where $O$ is the center of the circumcircle $\omega$ of the triangle $A B C$. Let $E$ be the second point of intersection of $\omega$ and the line $A D$. Let $M, N, P$ be the midpoints of the line segments $B E, O D, A C$, respectively. Show that $M, N, P$ are collinear.

Let $I$ be the incenter and $A B$ the shortest side of a triangle $A B C$. The circle with center $I$ and passing through $C$ intersects the ray $A B$ at the point $P$ and the ray $B A$ at the point $Q$. Let $D$ be the point where the excircle of the triangle $A B C$ belonging to angle $A$ touches the side $B C$, and let $E$ be the symmetric of the point $C$ with respect to $D$. Show that the lines $P E$ and $C Q$ are perpendicular.

A circle passing through the midpoint $M$ of the side $B C$ and the vertex $A$ of a triangle $A B C$ intersects the sides $A B$ and $A C$ for the second time at the points $P$ and $Q$, respectively. Show that if $\angle B A C=60^{\circ}$ then

$$A P+A Q+P Q<A B+A C+\frac{1}{2} B C$$

Find all positive integere on for othich $1^{2}+2^{2}+\cdots+16^{2}+17$ is a perfort square

Find all ordered pairs $(a, b)$ of positive integers for which the numbers $\frac{a^{3} b-1}{a+1}$ and $\frac{b^{3} a+1}{b-1}$ are positive integers.

Find all ordered triples $(x, y, z)$ of positive integers satisfying the equation

$$\frac{1}{x^{2}}+\frac{y}{x z}+\frac{1}{z^{2}}=\frac{1}{2013}$$

Find all ordered triples $(x, y, z)$ of integers satisfying the following system of equations:

$$\begin{aligned} x^{2}-y^{2} & =z \\ 3 x y+(x-y) z & =z^{2} \end{aligned}$$