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Let $a, b$ be two natural numbers satisfying $2006 a^{2}+a=2007 b^{2}+b.$ $2006 a^{2}+a=2007 b^{2}+b.$ Prove that $a-b$ is a perfect square.
Let $A B C$ be a triangle with $\widehat{B A C}=90^{\circ}, \widehat{A B C}$ $=60^{\circ} .$ Take the point $M$ on the side $B C$ such that $A B+B M=A C+C M$ . Caculate the measure of $\widehat{C A M}$
Find all positive integers $x, y$ greater than 1 so that $2 x y-1$ divisible by $(x-1)(y-1)$ .
Prove that $\frac{a^{4} b}{2 a+b}+\frac{b^{4} c}{2 b+c}+\frac{c^{4}}{2 c+a} \geq 1$ $\frac{a^{4} b}{2 a+b}+\frac{b^{4} c}{2 b+c}+\frac{c^{4}}{2 c+a} \geq 1$ where $a, b, c$ are positive numbers satisfying the condition $a b+b c+c a \leq 3 a b c$ . When does equality occur?
Let be given two circles $\left(O_{1}\right)$ , $\left(O_{2}\right)$ with centers $O_{1}$ , $O_{2}$ with distinct radii, externally touching each other at a point $T$ . Let $O_{1} A$ be a tangent to $\left(O_{2}\right)$ at a point $A,$ let $O_{2} B$ be a tangent to $\left(O_{1}\right)$ at a point $B$ so that the points $A, B$ are on the same side with respect to the line $O_{1} O_{2} .$ Let $H$ be the point on $O_{1} A, K$ be the point on $O_{2} B$ so that the lines $B H, A K$ are perpendicular to $O_{1} O_{2}$ . The line $T H$ cuts $\left(O_{1}\right)$ again at $E,$ the line $T K$ cuts $\left(O_{2}\right)$ againt at $F$ . The line $E F$ cuts $A B$ at $S$ . Prove that the lines $O_{1} A$ , $O_{2} B$ and $T S$ are concurrent.
Let $S$ be a set consisting of 43 distinct positive integers not exceeding $100 .$ For each subset $X$ of $S$ let $t_{X}$ be the product of its elements. Prove that there exist two disjoint substs $A$ and $B$ of $S$ such that $t_{A} t_{B}^{2}$ is the cube of a natural numbers.
Find the greast value of the expression $\frac{a}{c}+\frac{b}{d}+\frac{c}{a}+\frac{d}{b}-\frac{a b c d}{(a b+c d)^{2}}$ $\frac{a}{c}+\frac{b}{d}+\frac{c}{a}+\frac{d}{b}-\frac{a b c d}{(a b+c d)^{2}}$ where $a, b, c d$ are distinct real numbers satisfying the conditions $\dfrac{a}{b}+\dfrac{b}{c}+\dfrac{c}{d}+\dfrac{d}{a}=4$ and $a c=b d$
a) Let $f(x)$ be a polynomial of degree $n$ with leading coefficient $a$ . Suppose that $f(x)$ has $n$ distinct roots $x_{1}, x_{2}, \ldots, x_{n}$ all not equal to zero. Prove that $\frac{(-1)^{n-1}}{a x_{1} x_{2} \ldots x_{n}} \sum_{k=1}^{n} \frac{1}{x_{k}}=\sum_{k=1}^{n} \frac{1}{x_{k}^{2} f^{\prime}\left(x_{k}\right)}.$ $\frac{(-1)^{n-1}}{a x_{1} x_{2} \ldots x_{n}} \sum_{k=1}^{n} \frac{1}{x_{k}}=\sum_{k=1}^{n} \frac{1}{x_{k}^{2} f^{\prime}\left(x_{k}\right)}.$ b) Does there exist a polynomial $f(x)$ of degree $n,$ with leading coefficient $a=1,$ such that $f(x)$ has $n$ distinct roots $x_{1}, x_{2}, \ldots, x_{n},$ all not equal to zero, satisfying the condition $\frac{1}{x_{1} f^{\prime}\left(x_{1}\right)}+\frac{1}{x_{2} f^{\prime}\left(x_{2}\right)}+\ldots+\frac{1}{x_{n} f^{\prime}\left(x_{n}\right)}+\frac{1}{x_{1} x_{2} \ldots x_{n}}=0 ?$ $\frac{1}{x_{1} f^{\prime}\left(x_{1}\right)}+\frac{1}{x_{2} f^{\prime}\left(x_{2}\right)}+\ldots+\frac{1}{x_{n} f^{\prime}\left(x_{n}\right)}+\frac{1}{x_{1} x_{2} \ldots x_{n}}=0 ?$
Let $A D$ , $B E$ , $C F$ be the altitudes and $H$ be the orthocenter of an acute triangle $A B C .$ Let $M$ , $N$ be respectively the points of intersection of $D E$ and $C F$ and of $D E$ and $B E$ . Prove that the line passing through $A$ perpendicular to the line $M N$ passes through the circumcenter of triangle $B H C$ .
Find the measures of the sides of all triangles satisfying the conditions : these measures are whole numbers, the perimeter and the area are expressed by equal numbers.
Find the least value of the expression $(\sqrt{1+a}+\sqrt{1+b})(\sqrt{1+c}+\sqrt{1+d})$ $(\sqrt{1+a}+\sqrt{1+b})(\sqrt{1+c}+\sqrt{1+d})$ where $a, b, c, d$ are positive numbers satisfying the condition $a b c d=1$ .
Solve the system of equations $\begin{cases}\sqrt{x}+\sqrt{y}+\sqrt{z} &=3 \\ (1+x)(1+y)(1+z) &=(1+\sqrt[3]{x y z})^{3}\end{cases}$ $\begin{cases}\sqrt{x}+\sqrt{y}+\sqrt{z} &=3 \\ (1+x)(1+y)(1+z) &=(1+\sqrt[3]{x y z})^{3}\end{cases}$
Let $A B C D$ be a parallelogram. Take a point $M$ on the side $A B$ , a point $N$ on the side $C D$ . Let $P$ be the point of intersection of $A N$ and $D M$ , $Q$ be the point of intersection of $B N$ and $C M$ . Prove that $P Q$ passes through a fixed point when $M$ and $N$ move respectively on $A B$ and $C D$ .
Let $A B C D$ be a convex quadrilateral. Prove that $\min \{A B, B C, C D, D A\} \leq \frac{\sqrt{A C^{2}+B D^{2}}}{2} \leq \max \{A B, B C, C D, D A\}$ $\min \{A B, B C, C D, D A\} \leq \frac{\sqrt{A C^{2}+B D^{2}}}{2} \leq \max \{A B, B C, C D, D A\}$
Find all natural numbers $n \geq 3$ such that in the coordinate plane, there exists a regular $n$ -polygon all vertices of which have integral coordinates.
The sequence of numbers $\left(u_{n}\right)$ $(n=1,2,3, \ldots)$ is defined by $u_{n}=\sum_{k=1}^{n} \frac{(-1)^{k-1}}{k},\,\forall n=1,2,3 \ldots.$ $u_{n}=\sum_{k=1}^{n} \frac{(-1)^{k-1}}{k},\,\forall n=1,2,3 \ldots.$ Prove that the sequence has limit and find the limit.
Find the greatest value of the expression $x_{1}^{3}+x_{2}^{3}+\ldots+x_{n}^{3}-x_{1}^{4}-x_{2}^{4}-\ldots-x_{1}^{4}$ $x_{1}^{3}+x_{2}^{3}+\ldots+x_{n}^{3}-x_{1}^{4}-x_{2}^{4}-\ldots-x_{1}^{4}$ ( $n$ is a given number), where the numbers $x_{i}$ $(i=1,2, \ldots, n)$ satisfy the conditions $0 \leq x_{i} \leq 1$ $(i=1,2, \ldots, n)$ and $x_{1}+x_{2}+\ldots+x_{n}=1$ .
In a triangle $A B C$ , let $m_{u}$ , $m_{b}$ , $m_{c}$ be respectively the measures of the medians issued from $A$ , $B$ , $C$ , let $r_{a}$ , $r_{b}$ , $r_{c}$ be respectively the radii of the escribed circles in the angles $A$ , $B$ , $C$ . Prove that $r_{a}^{2}+r_{b}^{2}+r_{c}^{2} \geq m_{a}^{2}+m_{b}^{2}+m_{c}^{2}.$ $r_{a}^{2}+r_{b}^{2}+r_{c}^{2} \geq m_{a}^{2}+m_{b}^{2}+m_{c}^{2}.$ When does equality occur?
Let $H$ and $O$ be respectively the orthocenter and the circumcenter of the triangle $A B C$ of a tetrahedron $S A B C$ such that $S A$ , $S B$ , $S C$ are orthogonal each to the others. Prove that $\frac{O H^{2}}{S H^{2}}+2=\frac{1}{4 \cos A \cdot \cos B \cdot \cos C}$ $\frac{O H^{2}}{S H^{2}}+2=\frac{1}{4 \cos A \cdot \cos B \cdot \cos C}$ where $\cos A$ , $\cos B$ , $\cos C$ are cosinus of the angles of triangle $A B C$ .

PROBLEM OF THE WEEK | ЗАДАЧА НА НЕДЕЛАТА

PW 122. Prove that

\sqrt{a} + \sqrt[3]{a} + \sqrt[6]{a} \leq a + 2

where

a

is a non negative real number.

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June 23 to 28
Tirana, Albania

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Chiba, Japan