UA-UMO 1998

Find all pairs of real numbers $(x,y)$ satisfying the equation $$\sin^2x+\sin^2y=\frac{|x|}x+\frac{|y|}y.$$

Point $M$ is arbitrarily taken on side $AC$ of a triangle $ABC$ . Let $O$ be the intersection of the perpendiculars from the midpoints of segments $AM$ and $MC$ to lines $BC$ and $AB$ , respectively. For which position of $M$ is the distance $OM$ minimal?

A finite set of segments on a line has the following property: In any subset of $1998$ segments there are two having a common point. Show that there exist $1997$ points on the line such that each segment contains at least one of these points.

A function $f$ defined on the interval $[1,+\infty)$ satisfies $$\frac{f(2x)}{\sqrt2}\le f(x)\le x\enspace\text{for }x\ge1.$$ Prove that $f(x)<\sqrt{2x}$ for all $x\ge1$ .

Let $m$ be the smallest of the four numbers $1,x^9,y^9,z^7$ , where $x,y,z$ are nonnegative numbers. Prove that $m\le xy^2z^3$ .

Let $AB$ and $CD$ be diameters of a circle with center $O$ . For a point $M$ on a shorter arc $CB$ , lines $MA$ and $MD$ meet the chord $BC$ at points $P$ and $Q$ respectively. Prove that the sum of the areas of the triangles $CPM$ and $MQB$ equals the area of triangle $DPQ$ .

Baron Munchausen claims that he can accomodate an arbitrary set of guests in the rooms of his castle in such a way that the guests in each room either all know each other or all do not know each other. Is his claim true? (Assume that the rooms are large enough to accomodate any number of people, but that there are finitely many rooms.)

Find all pairs of polynomials $f(x),g(x)$ that satisfy $f(xy)=f(x)+g(x)f(y)$ for all $x,y$ .

Solve the equation $\sqrt{1+\{2x\}}=\lfloor x^2\rfloor+2\lfloor x\rfloor+3$ (where $\{a\}=a-\lfloor a\rfloor$ ).

The altitude $CD$ of triangle $ABC$ meets the bisector $BK$ of this triangle at $M$ and the altitude $KL$ of $\triangle BKC$ at $N$ . The circumcircle of triangle $BKN$ meets the side $AB$ at point $P\ne B$ . Prove that the triangle $KPM$ is isosceles.

For any numbers $0<x,y,z\le1$ prove the inequality $$\frac x{1+y+zx}+\frac y{1+z+xy}+\frac z{1+x+yz}\le\frac3{x+y+z}.$$

Consider a function $f:[0,1]\to[0,1]$ . Suppose that there is a real number $0<\lambda<1$ such that $f(\lambda)\notin\{0,\lambda\}$ and the equality $$f(f(x)+y)=f(x)+f(y)$$ holds whenever the function is defined on the arguments.

(a) Give an example of such a function. (b) Prove that for some $x\in[0,1]$ , $$\underbrace{f(f(\cdots f(}_{19}x)\cdots))=\underbrace{f(f(\cdots f(}_{98}x)\cdots)).$$

Find the number of real roots of the equation $$\frac2\pi\arccos x=\sqrt{1-x^2}+1.$$

The numbers $1,1998,1999$ are written on the board. In each step it is allowed to replace one of the numbers by its square decreased by three times the product of the other two numbers. Is it possible to obtain three numbers with the sum $0$ after several steps?

Two spheres are externally tangent at point $P$ . The segments $AB$ and $CD$ touch the spheres with $A$ and $C$ lying on the first sphere and $B$ and $D$ on the second. Let $M$ and $N$ be the projections of the midpoints of segments $AC$ and $BD$ on the line connecting the centers of the spheres. Prove that $PM=PN$ .

The sequence $(x_n)$ is given by $x_1=1$ and $$x_{n+1}=\frac{n^2}{x_n}+\frac{x_n}{n^2}+2\enspace\text{for }n\ge1.$$ (a) Prove that $x_{n+1}\ge x_n$ for all $n\ge4$ . (b) Prove that $\lfloor x_n\rfloor=n$ for all $n\ge4$ .