Russia1995: For positive $ x,y$ prove that\[\frac1{xy}\geq \frac x{x^4 + y^2} + \frac y{x^2 + y^4}.\]

Is it possible to place $1995$ different natural numbers along a circle so that for any two of these numbers, the ratio of the greatest to the least is a prime?

Two circles with radii $R$ and $r$ intersect at $C$ and $D$ and are tangent to a line $\ell$ at $A$ and $B$. Prove that the circumradius of triangle $ABC$ does not depend on the length of segment $AB$.

Every side and diagonal of a regular $12$-gon is colored in one of $12$ given colors. Can this be done in such a way that, for every three colors, there exist three vertices which are connected to each other by segments of these three colors?

Find all prime numbers $p$ for which number $p^2 + 11$ has exactly six different divisors (counting $1$ and itself).

Circles $S_1$ and $S_2$ with centers $O_1$ and $O_2$ respectively intersect at $A$ and $B$. The circle passing through $O_1$, $O_2$, and $A$ intersects $S_1$, $S_2$ and line $AB$ again at $D$, $E$, and $C$, respectively. Show that $CD = CB = CE$.

A regular hexagon of side $5$ is cut into unit equilateral triangles by lines parallel to the sides of the hexagon. We call the vertices of these triangles knots. If more than half of all knots are marked, show that there exist five marked knots that lie on a circle.

Can the numbers $1,2,...,121$ be written in the cells of an $11\times 11$ board in such a way that any two consecutive numbers are in adjacent cells (sharing a side), and all perfect squares are in the same column?

Given function $f(x) = \dfrac{1}{\sqrt[3]{1-x^3}}$, find $\underbrace{f(... f(f(19))...)}_{95}$. .

Natural numbers $m$ and $n$ satisfy $$gcd(m,n)+lcm(m,n) = m+n. $$Prove that one of numbers $m,n$ divides the other.

In an acute-angled triangle $ABC$, the circle $S$ with the altitude $BK$ as the diameter intersects $AB$ at $E$ and $BC$ at $F$. Prove that the tangents to $S$ at $E$ and $F$ meet on the median from $B$.

There are several equal (possibly overlapping) square-shaped napkins on a rectangular table, with sides parallel to the sides of the table. Prove that it is possible to nail some of them to the table in such a way that every napkin is nailed exactly once.

Consider all quadratic functions $f(x) = ax^2 +bx+c$ with $a < b$ and $f(x) \ge 0$ for all $x$. What is the smallest possible value of the expression $\frac{a+b+c}{b-a}$?

Let a quardilateral $ABCD$ with $AB=AD$ and $\widehat B=\widehat D=90$. At $CD$ we take point $E$ and at $BC$ we take point $Z$ such that $AE\bot DZ$. Prove that $AZ\bot BE$

$N^3$ unit cubes are made into beads by drilling a hole through them along a diagonal, put on a string and binded. Thus the cubes can move freely in space as long as the vertices of two neighboring cubes (including the first and last one) are touching. For which $N$ is it possible to build a cube of edge $N$ using these cubes?

The streets of the city of Duzhinsk are simple polygonal lines not intersecting each other in internal points. Each street connects two crossings and is colored in one of three colors: white, red, or blue. At each crossing exactly three streets meet, one of each color. A crossing is called positive if the streets meeting at it are white, blue and red in counterclockwise direction, and negative otherwise. Prove that the difference between the numbers of positive and negative crossings is a multiple of 4.

A planar section of a parallelepiped is a regular hexagon. Show that this parallelepiped is a cube.

there are some identical squares with sides parallel, in a plane. Among any $k+1$ of them, there are two with a point in common. Prove they can be divided into $2k-1$ sets, such that all the squares in one set aint pairwise disjoint.

Angles $\alpha, \beta, \gamma$ satisfy the inequality $\sin \alpha +\sin \beta +\sin \gamma \ge 2$. Prove that $\cos \alpha + \cos \beta +\cos \gamma \le \sqrt5.$

The sequence $ a_n$ satisfies $ a_{m+n}+ a_{m-n}=\frac12(a_{2m}+a_{2n})$ for all $ m\geq n\geq 0$. If $ a_1=1$, find $ a_{1995}$.

Circles $S_1$ and $S_2$ with centers $O_1$ and $O_2$ respectively intersect at $A$ and $B$. Ray $O_1B$ meets $S_2$ again at $F$, and ray $O_2B$ meets $ S_1$ again at $E$. The line through $B$ parallel to $ EF$ intersects $S_1$ and $S_2$ again at $M$ and $N$, respectively. Prove that $MN = AE +AF$.