Determine all real numbers $x$ such that $$\frac{2002\lfloor x\rfloor}{\lfloor-x\rfloor+x}>\frac{\lfloor2x\rfloor}{x-\lfloor1+x\rfloor}.$$

Let $O$ be a point inside a triangle $ABC$ and let the lines $AO,BO$, and $CO$ meet sides $BC,CA$, and $AB$ at points $A_1,B_1$, and $C_1$, respectively. If $AA_1$ is the longest among the segments $AA_1,BB_1,CC_1$, prove that $$OA_1+OB_1+OC_1\le AA_1.$$

Find all pairs $(n,k)$ of positive integers such that $\binom nk=2002$.

Is it possible to cut a rectangle $2001\times2003$ into pieces of the form each consisting of three unit squares?

Real numbers $x,y,z$ satisfy the inequalities $$x^2\le y+z,\qquad y^2\le z+x\qquad z^2\le x+y.$$Find the minimum and maximum possible values of $z$.

Points $A_0,A_1,\ldots,A_{2k}$, in this order, divide a circumference into $2k+1$ equal arcs. Point $A_0$ is connected by chords to all the other points. These $2k$ chords divide the interior of the circle into $2k+1$ parts. These parts are alternately painted red and blue so that there are $k+1$ red and $k$ blue parts. Show that the blue area is larger than the red area.

Let $m$ and $n$ be positive integers. Prove that the number $2n-1$ is divisible by $(2^m-1)^2$ if and only if $n$ is divisible by $m(2^m-1)$.

Each of the $15$ coaches ranked the $50$ selected football players on the places from $1$ to $50$. For each football player, the highest and lowest obtained ranks differ by at most $5$. For each of the players, the sum of the ranks he obtained is computed, and the sums are denoted by $S_1\le S_2\le\ldots\le S_{50}$. Find the largest possible value of $S_1$.

For any positive numbers $a,b,c$ and natural numbers $n,k$ prove the inequality $$\frac{a^{n+k}}{b^n}+\frac{b^{n+k}}{c^n}+\frac{c^{n+k}}{a^n}\ge a^k+b^k+c^k.$$

The (Fibonacci) sequence $f_n$ is defined by $f_1=f_2=1$ and $f_{n+2}=f_{n+1}+f_n$ for $n\ge1$. Prove that the area of the triangle with the sides $\sqrt{f_{2n+1}},\sqrt{f_{2n+2}},$ and $\sqrt{f_{2n+3}}$ is equal to $\frac12$.

Let $ ABCD$ be a rhombus with $ \angle BAD = 60^{\circ}$. Points $ S$ and $ R$ are chosen inside the triangles $ ABD$ and $ DBC$, respectively, such that $ \angle SBR = \angle RDS = 60^{\circ}$. Prove that $ SR^2\geq AS\cdot CR$.

Is there a positive integer $ k$ such that none of the digits $ 3,4,5,6$ appear in the decimal representation of the number $ 2002!\cdot k$?