If $a$ and $b$ are positive numbers, prove the inequality $$a^2 +ab+b^2\ge 3(a+b-1).$$

Find the largest natural number which cannot be turned into a multiple of $11$ by reordering its (decimal) digits.

Points $M$ and $N$ are chosen on the sides $AB$ and BC of a triangle $ABC$. The segments $AN$ and $CM$ meet at $O$ such that $AO =CO$. Is the triangle $ABC$ necessarily isosceles, if (a) $AM = CN$? (b) $BM = BN$?

We have a deck of $n$ playing cards, some of which are turned up and some are turned down. In each step we are allowed to take a set of several cards from the top, turn the set and place it back on the top of the deck. What is the smallest number of steps necessary to make all cards in the deck turned down, independent of the initial configuration?

Show that the equation $x^3 +y^3 = 4(x^2y+xy^2 +1)$ has no integer solutions.

Three right-angled triangles have been placed in a halfplane determined by a line $\ell$, each with one leg lying on $\ell$. Assume that there is a line parallel to $\ell$ cutting the triangles in three congruent segments. Show that, if each of the triangles is rotated so that its other leg lies on $\ell$, then there still exists a line parallel to $\ell$ cutting them in three congruent segments.

On the diagonal $AC$ of the rhombus $ABCD$, a point $E$ is taken, which is different from points $A$ and $C$, and on the lines $AB$ and $BC$ are points $N$ and $M$, respectively, with $AE = NE$ and $CE = ME$. Let $K$ be the intersection point of lines $AM$ and $CN$. Prove that points $K, E$ and $D$ are collinear.

Number $ 0$ is written on the board. Two players alternate writing signs and numbers to the right, where the first player always writes either $ +$ or $ -$ sign, while the second player writes one of the numbers $ 1, 2, ... , 1993$,writing each of these numbers exactly once. The game ends after $ 1993$ moves. Then the second player wins the score equal to the absolute value of the expression obtained thereby on the board. What largest score can he always win?

Point $D$ is chosen on the side $AC$ of an acute-angled triangle $ABC$. The median $AM$ intersects the altitude $CH$ and the segment $BD$ at points $N$ and $K$ respectively. Prove that if $AK = BK$, then $AN = 2KM$.

Solve in positive numbers the system $ x_1+\frac{1}{x_2}=4, x_2+\frac{1}{x_3}=1, x_3+\frac{1}{x_4}=4, ..., x_{99}+\frac{1}{x_{100}}=4, x_{100}+\frac{1}{x_1}=1$

Each citizen in a town knows at least $ 30$% of the remaining citizens. A citizen votes in elections if he/she knows at least one candidate. Prove that it is possible to schedule elections with two candidates for the mayor of the city so that at least $ 50$% of the citizen can vote.

Prove the inequality $$ \sqrt {2 + \sqrt [3]{3 + ... + \sqrt [{1993}]{1993}}} < 2$$ All-Russian MO 1993 Regional (Round 4) 10.6

Points $ M,N$ are taken on sides $ BC,CD$ respectively of parallelogram $ ABCD$. Let $ E=BD\cap AM, F=BD\cap AN$. Diagonal $ BD$ cuts triangle $ AMN$ into two parts. Prove that these two parts have equal area if and only if the point $ K$ given by $ EK||AD, FK||AB$ lies on segment $ MN$.

From a square board $1000\times 1000$ four rectangles $2\times 994$ have been cut off as shown on the picture. Initially, on the marked square there is a centaur - a piece that moves to the adjacent square to the left, up, or diagonally up-right in each move. Two players alternately move the centaur. The one who cannot make a move loses the game. Who has a winning strategy?

Find all natural numbers $n$ for which the sum of digits of $5^n$ equals $2^n$.

Prove that, for every integer $n > 2$, the number $$\left[\left( \sqrt[3]{n}+\sqrt[3]{n+2}\right)^3\right]+1$$is divisible by $8$.

Point $O$ is the foot of the altitude of a quadrilateral pyramid. A sphere with center $O$ is tangent to all lateral faces of the pyramid. Points $A,B,C,D$ are taken on successive lateral edges so that segments $AB$, $BC$, and $CD$ pass through the three corresponding tangency points of the sphere with the faces. Prove that the segment $AD$ passes through the fourth tangency point

Given a regular $ 2n$-gon, show that each of its sides and diagonals can be assigned in such a way that the sum of the obtained vectors equals zero.

The expression $ x^3 + . . . x^2 + . . . x + ... = 0$ is written on the blackboard. Two pupils alternately replace the dots by real numbers. The first pupil attempts to obtain an equation having exactly one real root. Can his opponent spoil his efforts?

Seven tetrahedra are placed on the table. For any three of them there exists a horizontal plane cutting them in triangles of equal areas. Show that there exists a plane cutting all seven tetrahedra in triangles of equal areas.

Let $ABC$ be an equilateral triangle. For an arbitrary line $\ell$. through $B$, the orthogonal projections of $A$ and $C$ on $\ell$ are denoted by $D$ and $E$ respectively. If $D\ne E$, equilateral triangles $DEP$ and $DET$ are constructed on different sides of $\ell$. Find the loci of $P$ and $T$.

There are $ 1993$ towns in a country, and at least $ 93$ roads going out of each town. It's known that every town can be reached from any other town. Prove that this can always be done with no more than $ 62$ transfers.