JBMO-SL 2008

If for the real numbers $x, y, z, k$ the following conditions are valid, $x \neq y \neq z \neq x$ and $x^{3}+y^{3}+k\left(x^{2}+y^{2}\right)=y^{3}+z^{3}+k\left(y^{2}+z^{2}\right)=z^{3}+x^{3}+k\left(z^{2}+x^{2}\right)=2008$, find the product $x y z$.

Find all real numbers $a, b, c, d$ such that $a+b+c+d=20$ and $a b+a c+a d+b c+b d+c d=$ 150 .

Let the real parameter $p$ be such that the system

$$\left\{\begin{array}{l} p\left(x^{2}-y^{2}\right)=\left(p^{2}-1\right) x y \\ |x-1|+|y|=1 \end{array}\right.$$

has at least three different real solutions. Find $p$ and solve the system for that $p$.

Find all triples $(x, y, z)$ of real numbers that satisfy the system

$$\left\{\begin{array}{l} x+y+z=2008 \\ x^{2}+y^{2}+z^{2}=6024^{2} \\ \frac{1}{x}+\frac{1}{y}+\frac{1}{z}=\frac{1}{2008} \end{array}\right.$$

Find all triples $(x, y, z)$ of real positive numbers, which satisfy the system

$$\left\{\begin{array}{l} \frac{1}{x}+\frac{4}{y}+\frac{9}{z}=3 \\ x+y+z \leq 12 \end{array}\right.$$

If the real numbers $a, b, c, d$ are such that $0<a, b, c, d<1$, show that

$$1+a b+b c+c d+d a+a c+b d>a+b+c+d$$

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

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

Show that

$$(x+y+z)\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\right) \geq 4\left(\frac{x}{x y+1}+\frac{y}{y z+1}+\frac{z}{z x+1}\right)^{2}$$

for any real positive numbers $x, y$ and $z$.

Consider an integer $n \geq 4$ and a sequence of real numbers $x_{1}, x_{2}, x_{3}, \ldots, x_{n}$. An operation consists in eliminating all numbers not having the rank of the form $4 k+3$, thus leaving only the numbers $x_{3}, x_{7}, x_{11}, \ldots$ (for example, the sequence $4,5,9,3,6,6,1,8$ produces the sequence 9,1 . Upon the sequence $1,2,3, \ldots, 1024$ the operation is performed successively for 5 times. Show that at the end only 1 number remains and find this number.

On a $5 \times 5$ board, $n$ white markers are positioned, each marker in a distinct $1 \times 1$ square. A smart child got an assignment to recolor in black as many markers as possible, in the following manner: a white marker is taken from the board; it is colored in black, and then put back on the board on an empty square such that none of the neighboring squares contains a white marker (two squares are called neighboring if they contain a common side). If it is possible for the child to succeed in coloring all the markers black, we say that the initial positioning of the markers was good.

a) Prove that if $n=20$, then a good initial positioning exists.

b) Prove that if $n=21$, then a good initial positioning does not exist.

Kostas and Helene have the following dialogue:

Kostas: I have in my mind three positive real numbers with product 1 and sum equal to the sum of all their pairwise products.

Helene: I think that I know the numbers you have in mind. They are all equal to 1.

Kostas: In fact, the numbers you mentioned satisfy my conditions, but I did not think of these numbers. The numbers you mentioned have the minimal sum between all possible solutions of the problem.

Can you decide if Kostas is right? (Explain your answer).

Integers $1,2, \ldots, 2 n$ are arbitrarily assigned to boxes labeled with numbers $1,2, \ldots, 2 n$. Now, we add the number assigned to the box to the number on the box label. Show that two such sums give the same remainder modulo $2 n$.

Every cell of table $4 \times 4$ is colored into white. It is permitted to place the cross (pictured below) on the table such that its center lies on the table (the whole figure does not need to lie on the table) and change colors of every cell which is covered into opposite (white and black). Find all $n$ such that after $n$ steps it is possible to get the table with every cell colored black.

Two perpendicular chords of a circle, $A M, B N$, which intersect at point $K$, define on the circle four arcs with pairwise different length, with $A B$ being the smallest of them.

We draw the chords $A D, B C$ with $A D \| B C$ and $C, D$ different from $N, M$. If $L$ is the point of intersection of $D N, M C$ and $T$ the point of intersection of $D C, K L$, prove that $\angle K T C=\angle K N L$.

For a fixed triangle $A B C$ we choose a point $M$ on the ray $C A$ (after $A$ ), a point $N$ on the ray $A B$ (after $B$ ) and a point $P$ on the ray $B C$ (after $C$ ) in a way such that $A M-B C=B N-A C=C P-A B$. Prove that the angles of triangle $M N P$ do not depend on the choice of $M, N, P$.

The vertices $A$ and $B$ of an equilateral $\triangle A B C$ lie on a circle $k$ of radius 1 , and the vertex $C$ is inside $k$. The point $D \neq B$ lies on $k, A D=A B$ and the line $D C$ intersects $k$ for the second time in point $E$. Find the length of the segment $C E$.

Let $A B C$ be a triangle, $(B C<A B)$. The line $\ell$ passing trough the vertices $C$ and orthogonal to the angle bisector $B E$ of $\angle B$, meets $B E$ and the median $B D$ of the side $A C$ at points $F$ and $G$, respectively. Prove that segment $D F$ bisect the segment $E G$.

Is it possible to cover a given square with a few congruent right-angled triangles with acute angle equal to $30^{\circ}$ ? (The triangles may not overlap and may not exceed the margins of the square.)

Let $A B C$ be a triangle with $A<90^{\circ}$. Outside of a triangle we consider isosceles triangles $A B E$ and $A C Z$ with bases $A B$ and $A C$, respectively. If the midpoint $D$ of the side $B C$ is such that $D E \perp D Z$ and $E Z=2 \cdot E D$, prove that $\widehat{A E B}=2 \cdot \widehat{A Z C}$.

Let $A B C$ be an isosceles triangle with $A C=B C$. The point $D$ lies on the side $A B$ such that the semicircle with diameter $[B D]$ and center $O$ is tangent to the side $A C$ in the point $P$ and intersects the side $B C$ at the point $Q$. The radius $O P$ intersects the chord $D Q$ at the point $E$ such that $5 \cdot P E=3 \cdot D E$. Find the ratio $\frac{A B}{B C}$.

The side lengths of a parallelogram are $a, b$ and diagonals have lengths $x$ and $y$, Knowing that $a b=\frac{x y}{2}$, show that

$$a=\frac{x}{\sqrt{2}}, b=\frac{y}{\sqrt{2}} \text { or } a=\frac{y}{\sqrt{2}}, b=\frac{x}{\sqrt{2}}$$

Let $O$ be a point inside the parallelogram $A B C D$ such that

$$\angle A O B+\angle C O D=\angle B O C+\angle C O D$$

Prove that there exists a circle $k$ tangent to the circumscribed circles of the triangles $\triangle A O B, \triangle B O C, \triangle C O D$ and $\triangle D O A$.

Let $\Gamma$ be a circle of center $O$, and $\delta$ be a line in the plane of $\Gamma$, not intersecting it. Denote by $A$ the foot of the perpendicular from $O$ onto $\delta$, and let $M$ be a (variable) point on $\Gamma$. Denote by $\gamma$ the circle of diameter $A M$, by $X$ the (other than $M$ ) intersection point of $\gamma$ and $\Gamma$, and by $Y$ the (other than $A$ ) intersection point of $\gamma$ and $\delta$. Prove that the line $X Y$ passes through a fixed point.

Consider $A B C$ an acute-angled triangle with $A B \neq A C$. Denote by $M$ the midpoint of $B C$, by $D, E$ the feet of the altitudes from $B, C$ respectively and let $P$ be the intersection point of the lines $D E$ and $B C$. The perpendicular from $M$ to $A C$ meets the perpendicular from $C$ to $B C$ at point $R$. Prove that lines $P R$ and $A M$ are perpendicular.