Let $S{}$ be the set of nonnegative integers, which cointain only digits $0$ and $1$ in base $4$ numeral system. a) Show that if $x\in S, y\in S, x\neq y,$ then $\frac{x+y}{2}\notin S$. b) Let $T$ be a set of nonnegative integers such that $S\subset T, T\neq S$. Show that there exist $x\in T, y\in T, x\neq y,$ such that $\frac{x+y}{2} \in T$.

Find $\lim_{n\to\infty}\left\{\left(1+n\right)^{\frac{1}{n}}\left(1+\frac{n}{2}\right)^{\frac{2}{n}}\left(1+\frac{n}{3}\right)^{\frac{3}{n}}\cdots\cdots 2\right\}^{\frac{1}{n}}$.

$ \mathcal F$ is a family of 3-subsets of set $ X$. Every two distinct elements of $ X$ are exactly in $ k$ elements of $ \mathcal F$. It is known that there is a partition of $ \mathcal F$ to sets $ X_1,X_2$ such that each element of $ \mathcal F$ has non-empty intersection with both $ X_1,X_2$. Prove that $ |X|\leq4$.

Show that the $n$ digits after the decimal point in $(5 +\sqrt{26})^n$ are all equal.

For $a>1$, evaluate $\int_{\frac{1}{a}}^a \frac{1}{x}(\ln x)\ln\ (x^2+1)dx.$

Board has written numbers: $5$, $7$ and $9$. In every step we do the following: for every pair $(a,b)$, $a>b$ numbers from the board, we also write the number $5a-4b$. Is it possible that after some iterations, $2003$ occurs at the board ?

Let $O$ be the circumcenter of triangle $ABC$ and the line $AO$ intersects segment $BC$ at point $T$ . Assume that lines $m$ and $\ell$ passing through point $T$ are perpendicular to $AB$ and $AC$ respectively. If $E$ is the point of intersection of $m$ and $OB$ and $F$ is the point of intersection of $\ell$ and $OC$, prove that $BE = CF$. (Oleksii Karliuchenko)

The game Battleship is played on a $10\times10$ grid. A [i]fleet[/i] consists of 10 ships: one occupying $4$ cells, two occupying $3$ cells each, three occupying $2$ cells each and four occupying $1$ cell each (see figure). [asy] size(10cm); // Function to draw a square centered at a given position void drawSquare(pair center, real sideLength) { real halfSide = sideLength / 2; draw(shift(center) * box((-halfSide, -halfSide), (halfSide, halfSide))); } // Side length of each square real sideLength = 1; // Coordinates for the squares pair[] positions = { // Top row remains the same (0, 0), (1, 0), (3, 0), (4, 0), (6, 0), (7, 0), (9, 0), (11, 0), (13, 0), (15, 0), // Bottom row moved one square (1 unit) to the right (2, 2), (3, 2), (4, 2), (5, 2), (7, 2), (8, 2), (9, 2), (11, 2), (12, 2), (13, 2) }; // Draw all squares for (pair pos : positions) { drawSquare(pos, sideLength); } [/asy] Ships can be placed either horizontally or vertically, but they must not touch each other, not even at a vertex. Is it possible to place two fleets on the same board according to these rules?

Call a number unremarkable if, when written in base $10$, no two adjacent digits are equal. For example, $123$ is unremarkable, but $122$ is not. Find the sum of all unremarkable $3$-digit numbers. (Note that $012$ and $007$ are not $3$-digit numbers.)

In triangle $\vartriangle ABC$, let $M$ be the midpoint of $\overline{AC}$. Extend $\overline{BM}$ such that it intersects the circumcircle of $\vartriangle ABC$ at a point $X$ not equal to $B$. Let $O$ be the center of the circumcircle of $\vartriangle ABC$. Given that $BM = 4MX$ and $\angle ABC = 45^o$, compute $\sin (\angle BOX)$.

Let $ A $ be the set of real solutions of the equation $ 3^x=x+2, $ and let be the set $ B $ of real solutions of the equation $ \log_3 (x+2) +\log_2 \left( 3^x-x \right) =3^x-1 . $ Prove the validity of the following subpoints: [b]a)[/b] $ A\subset B. $ [b]b)[/b] $ B\not\subset\mathbb{Q} \wedge B\not\subset \mathbb{R}\setminus\mathbb{Q} . $

Suppose that $A$, $B$, $C$, and $D$ are distinct points, no three of which lie on a line, in the Euclidean plane. Show that if the squares of the lengths of the line segments $AB$, $AC$, $AD$, $BC$, $BD$, and $CD$ are rational numbers, then the quotient \[\frac{\mathrm{area}(\triangle ABC)}{\mathrm{area}(\triangle ABD)}\] is a rational number.

Suppose $91$ distinct positive integers greater than $1$ are given such that there are at least $456$ pairs among them which are relatively prime. Show that one can find four integers $a, b, c, d$ among them such that $\gcd(a,b)=\gcd(b,c)=\gcd(c,d)=\gcd(d,a)=1.$

Find the smallest integer $n > 3$ such that, for each partition of $\{3, 4,..., n\}$ in two sets, at least one of these sets contains three (not necessarily distinct) numbers $ a, b, c$ for which $ab = c$. (Alberto Alfarano)

The columns and the row of a $3n \times 3n$ square board are numbered $1,2,\ldots ,3n$. Every square $(x,y)$ with $1 \leq x,y \leq 3n$ is colored asparagus, byzantium or citrine according as the modulo $3$ remainder of $x+y$ is $0,1$ or $2$ respectively. One token colored asparagus, byzantium or citrine is placed on each square, so that there are $3n^2$ tokens of each color. Suppose that one can permute the tokens so that each token is moved to a distance of at most $d$ from its original position, each asparagus token replaces a byzantium token, each byzantium token replaces a citrine token, and each citrine token replaces an asparagus token. Prove that it is possible to permute the tokens so that each token is moved to a distance of at most $d+2$ from its original position, and each square contains a token with the same color as the square.

Cirlce $\Omega$ is inscribed in triangle $ABC$ with $\angle BAC=40$. Point $D$ is inside the angle $BAC$ and is the intersection of exterior bisectors of angles $B$ and $C$ with the common side $BC$. Tangent form $D$ touches $\Omega$ in $E$. FInd $\angle BEC$.