Solve the system of equations $2|x|+|y|=1$, $\lfloor|x|\rfloor+\lfloor2|y|\rfloor=2$.

Is it possible to write numbers in the cells of a $7\times7$ board in such a way that the sum of numbers in every $2\times2$ or $3\times3$ square is divisible by $1999$, but the sum of all numbers in the board is not divisible by $1999$?

Is there a $2000$-digit number which is a perfect square and $1999$ of whose digits are fives?

Let $N$ be the point inside a rhombus $ABCD$ such that the triangle $BNC$ is equilateral. The bisector of $\angle ABN$ meets the diagonal $AC$ at $K$. Show that $BK=KN+ND$.

Consider the figure consisting of $19$ hexagonal cells, as shown in the picture. At the cell $A$, there is a piece that is allowed to move one cell up, up-right, or down-right. How many ways are there for the piece to reach the cell $B$, not passing through the cell $C$?

Describe the region in the coordinate plane defined by $|x^2+xy|\ge|x^2-xy|$.

Let $x$ and $y$ be positive real numbers with $(x-1)(y-1)\ge1$. Prove that for sides $a,b,c$ of an arbitrary triangle we have $a^2x+b^2y>c^2$.

Show that the number $9999999+1999000$ is composite.

The bisectors of angles $A,B,C$ of a triangle $ABC$ intersect the circumcircle of the triangle at $A_1,B_1,C_1$, respectively. Let $P$ be the intersection of the lines $B_1C_1$ and $AB$, and $Q$ be the intersection of the lines $B_1A_1$ and $BC$. Show how to construct the triangle $ABC$ by a ruler and a compass, given its circumcircle, points $P$ and $Q$, and the halfplane determined by $PQ$ in which point $B$ lies.

Solve the equation $\lfloor x\rfloor+\frac{1999}{\lfloor x\rfloor}=\{x\}+\frac{1999}{\{x\}}$.

Find all pairs $(k,l)$ of positive integers such that $\frac{k^l}{l^k}=\frac{k!}{l!}$.

Let $M$ be a fixed point inside a given circle. Two perpendicular chords $AC$ and $BD$ are drawn through $M$, and $K$ and $L$ are the midpoints of $AB$ and $CD$, respectively. Prove that the quantity $AB^2+CD^2-2KL^2$ is independent of the chords $AC$ and $BD$.

A sequence of natural numbers $(a_n)$ satisfies $a_{a_n}+a_n=2n$ for all $n\in\mathbb N$. Prove that $a_n=n$.

Solve the equation $\sin x\sin2x\sin3x+\cos x\cos2x\cos3x=1$.

Let $M$ be a point inside a triangle $ABC$. The line through $M$ parallel to $AC$ meets $AB$ at $N$ and $BC$ at $K$. The lines through $M$ parallel to $AB$ and $BC$ meet $AC$ at $D$ and $L$, respectively. Another line through $M$ intersects the sides $AB$ and $BC$ at $P$ and $R$ respectively such that $PM=MR$. Given that the area of $\triangle ABC$ is $S$ and that $\frac{CK}{CB}=a$, compute the area of $\triangle PQR$.

Two players alternately write integers on a blackboard as follows: the first player writes $a_1$ arbitrarily, then the second player writes $a_2$ arbitrarily, and thereafter a player writes a number that is equal to the sum of the two preceding numbers. The player after whose move the obtained sequence contains terms such that $a_i-a_j$ and $a_{i+1}-a_{j+1}~(i\ne j)$ are divisible by $1999$, wins the game. Which of the players has a winning strategy?

Evaluate $$\lfloor\pi\rfloor+\left\lfloor\frac{\lfloor2\pi\rfloor}2\right\rfloor+\left\lfloor\frac{\lfloor3\pi\rfloor}3\right\rfloor+\ldots+\left\lfloor\frac{\lfloor1999\pi\rfloor}{1999}\right\rfloor.$$

Solve the equation $m^3-n^3=7mn+5$ in positive integers.

If $x_1$, $x_2$, ..., $x_6$ $\in $ $[0,1]$, prove that the cyclic sum of $\frac{x_1^3}{x_2^5+x_3^5+x_4^5+x_5^5+x_6^5+5}$ is less than $\frac{3}{5}$.

Let $AA_1,BB_1,CC_1$ be the altitudes of an acute-angled triangle $ABC$, and let $O$ be an arbitrary interior point. Let $M,N,P,Q,R,S$ be the feet of the perpendiculars from $O$ to the lines $AA_1,BC,BB_1,CA,CC_1,AB$, respectively. Prove that the lines $MN,PQ,RS$ are concurrent.

Solve the equation $$(\sin x)^{1998}+(\cos x)^{-1999}=(\cos x)^{1998}+(\sin x)^{-1999}.$$

Find all values of the parameter $k$ for which the system of inequalities \begin{align*} ky^2+4ky-2x+6k+3&\le0\\ kx^2-2y-2kx+3k-3&\le0 \end{align*}has a unique solution.

All faces of a parallelepiped $ABCDA_1B_1C_1D_1$ are rhombi, and their angles at $A$ are all equal to $\alpha$. Points $M,N,P,Q$ are selected on the edges $A_1B_1,DC,BC,A_1D_1$, respectively, such that $A_1M=BP$ and $DN=A_1Q$. Find the angle between the intersection lines of the plane $A_1BD$ with the planes $AMN$ and $APQ$.

Can the number (a)19991998 (b)19991999 be written in the form $ n^{4}+m^{3}-m $, where n, m are integers?

Find all functions $f:\mathbb R\to\mathbb R$ such that $$f(xy)+f(xz)-f(x)f(yz)\ge1\qquad\text{for all }x,y,z.$$

Suppose that the function $f(x)=\tan(a_1x+1)+\ldots+\tan(a_{10}x+1)$ has the period $T>0$, where $a_1,\ldots,a_{10}$ are positive numbers. Prove that $$T\ge\frac\pi{10}\min\left\{\frac1{a_1},\ldots,\frac1{a_{10}}\right\}.$$