UA-UMO 2005

Find all pairs of real numbers $(x,y)$ such that:

$\frac{x-2}{y}+\frac{5}{xy}=\frac{4-y}{x}-\frac{|y-2x|}{xy}.$

Let $AD$ be the median of a triangle $ABC.$ Suppose that $\angle ADB=45^{\circ}$ and $\angle ACB=30^{\circ}$ . Compute $\angle BAD.$

A $6 \times 6$ board is filled out with positive integers. Each move consists of selecting a square larger than $1 \times 1$ , consisting of entire cells, and increasing all numbers inside the selected square by $1$ . Is it always possible to perform several moves so as to reach a situation where all numbers on the board are divisible by $3$ ?

For which real numbers $x>1$ is there a triangle with side lengths $x^4+x^3+2x^2+x+1, 2x^3+x^2+2x+1,$ and $x^4-1$ ?

Are there integers $a,b,c,d,x,y,z,t$ such that each of the numbers:

$|ay-bx|,|az-cx|,|at-dx|,|bz-cy|,|bt-dy|,|ct-dz|$

equals either $1$ or $2005$ ?

A convex quadrilateral $ABCD$ with $BC=CD$ and $\angle CBA+\angle DAB>180^{\circ}$ is given. Suppose that $W$ and $Q$ are points on the sides $BC$ and $DC$ respectively (distinct from the vertices) such that $AD=QD$ and $WQ \parallel AD.$ Also suppose that $AQ$ and $BD$ intersect at a point $M$ that is equidistant from the lines $AD$ and $BC$ . Prove that angle $\angle BWD=\angle ADW.$

Draw the locus of points $M(x,y)$ in the Cartesian plane $xOy$ satisfying $(x^2-1)(|y|-1) \ge 0$ .

Find all pairs of positive integers $(m,n)$ such that $\sqrt{m}+\frac{2005}{\sqrt{n}}=2006.$

On the plane are given $n \ge 3$ points, not all on the same line. For any point $M$ on the same plane, $f(M)$ is defined to be the sum of the distances from $M$ to these $n$ points. Suppose that there is a point $M_1$ such that $f(M_1)\le f(M)$ for any point $M$ on the plane. Prove that if a point $M_2$ satisfies $f(M_1)=f(M_2),$ then $M_1 \equiv M_2.$

Can $1$ be written as a sum of $2005$ different terms of the form $\frac{1}{3n-1},$ where $n$ is a positive integer?

For every positive integer $n$ , prove the inequality:

$\frac{3}{1!+2!+3!}+\frac{4}{2!+3!+4!}+...+\frac{n+2}{n!+(n+1)!+(n+2)!}<\frac{1}{2}.$

Under the rules of the "Sea battle" game (no two ships may have a common point), can the following sets of rectangular ships be arranged on a $10 \times 10$ square board:

$(a)$ two ships $1 \times 4$ , $4$ ships $1 \times 3$ , $6$ ships $1 \times 2$ , and $8$ ships $1 \times 1$ ;

$(b)$ two ships $1 \times 4$ , $4$ ships $1 \times 3$ , $6$ ships $1 \times 2$ , $6$ ships $1 \times 1$ , and $1$ ship $2 \times 2$ ;

$(c)$ two ships $1 \times 4$ , $4$ ships $1 \times 3$ , $6$ ships $1 \times 2$ , $4$ ships $1 \times 1$ , and $2$ ships $2 \times 2$ ?

Let $AB$ and $CD$ be two disjoint chords of a circle. A point $E$ , distinct from $A$ and $B$ , is taken on the chord $AB$ . Consider the arc $AB$ not containing $C$ and $D$ . Using a ruler and a compass, construct a point $F$ on this arc such that $\frac{PE}{EQ}=\frac{1}{2}$ , where $P$ and $Q$ are the intersection points of $AB$ with the segments $FC$ and $FD$ , respectively.

Let $0 \le \alpha, \beta, \gamma \le \frac{\pi}{2}$ satisfy the conditions

$\sin \alpha+\sin \beta+\sin \gamma=1, \sin \alpha \cos 2\alpha+\sin \beta\cos 2\beta+\sin \gamma \cos 2\gamma=-1.$

Find all possible values of $\sin^2 \alpha+\sin^2 \beta+\sin^2 \gamma$ .

Is it possible to divide a $10 \times 10$ board into:

$(a)$ $4$ figures of type $1$ and $21$ figures of type $2$ ? $(b)$ $4$ figures of type $1$ , $19$ figures of type $2$ , and $2$ figures of type $3$ ?

Attachments:

An integer $n>101$ is divisible by $101$ . Suppose that every divisor $d$ of $n$ with $1<d<n$ equals the difference of two divisors of $n$ . Prove that $n$ is divisble by $100$ .

Points $P$ and $Q$ do not lie on the diagonal $AC$ of a parallelogram $ABCD$ and satisfy $\angle ABP=\angle ADP, \angle CBQ=\angle CDQ.$ Prove that $\angle PAQ=\angle PCQ.$

Find all functions $f: \mathbb{R}^{+}\rightarrow \mathbb{R}$ that satisfy:

$f(x)f(y)=f(xy)+2005 \left( \frac{1}{x}+\frac{1}{y}+2004 \right)$ for all $x,y>0.$

If $a,b,c$ are positive real numbers, prove the inequality:

$\frac {a^2}{b} + \frac {b^3}{c^2} + \frac {c^4}{a^3} \ge - a + 2b + 2c.$

P.S. Thank you for the observation. I've already corrected it.

A point $M$ is taken on the perpendicular bisector of the side $AC$ of an acute-angled triangle $ABC$ so that $M$ and $B$ are on the same side of $AC$ . If $\angle BAC=\angle MCB$ and $\angle ABC+\angle MBC=180^{\circ}$ , find $\angle BAC.$

On the plane are marked $2005$ points, no three of which are collinear. A line is drawn through any two of the points. Show that the points can be painted in two colors so that for any two points of the same color the number of the drawn lines separating them is even. (Two points are separated by a line if they lie in different open half-planes determined by the line).

Solve in $\mathbb{R}$ the equation $\left| x -\frac{\pi}{6} \right|+\left| x+\frac{\pi}{3} \right|=\arcsin \frac{x^3-x+2}{2}.$

The sum of positive real numbers $a,b,c$ equals $1$ . Prove that:

$\sqrt{\frac{1}{a}-1}\sqrt{\frac{1}{b}-1}+\sqrt{\frac{1}{b}-1}\sqrt{\frac{1}{c}-1}+\sqrt{\frac{1}{c}-1}\sqrt{\frac{1}{a}-1} \ge 6.$

In an acute-angled triangle $ABC$ , $\omega$ is the circumcircle and $O$ its center, $\omega _1$ the circumcircle of triangle $AOC$ , and $OQ$ the diameter of $\omega _1$ . Let $M$ and $N$ be points on the lines $AQ$ and $AC$ respectively such that the quadrilateral $AMBN$ is a parallelogram. Prove that the lines $MN$ and $BQ$ intersect on $\omega _1$ .

Find all monotone (not necessarily strictly) functions $f: \mathbb{R}^{+}_0\rightarrow \mathbb{R}$ such that:

$f(x+y)-f(x)-f(y)=f(xy+1)-f(xy)-f(1) \forall x,y \ge 0$ ;

$f(3)+3f(1)=3f(2)+f(0).$

A polygon on a coordinate grid is built of $2005$ dominoes $1 \times 2$ . What is the smallest number of sides of an even length such a polygon can have?

Prove that for any integers $n \ge 2$ there is a set $A_n$ of $n$ distinct positive integers such that for any two distinct elements $i,j \in A_n, |i-j|$ divides $i^2+j^2.$

In space are marked $2005$ points, no four of which are in the same plane. A plane is drawn through any three points. Show that the points can be painted in two colors so that for any two points of the same color the number of the drawn planes separating them is odd. (Two points are separated by a plane if they lie in different open half-spaces determined by the plane).