In an alphabet of $n$ letters, is $syllable$ is any ordered pair of two (not necessarily distinct) letters. Some syllables are considered $indecent$. A $word$ is any sequence, finite or infinite, of letters, that does not contain indecent syllables. Find the least possible number of indecent syllables for which infinite words do not exist.

Circles $\Omega$ and $\Gamma$ meet at points $A$ and $B$. The line containing their centres intersects $\Omega$ and $\Gamma$ at point $P$ and $Q$, respectively, such that these points lie on same side of the line $AB$ and point $Q$ is closer to $AB$ than point $P$. The circle $\delta$ lies on the same side of the line $AB$ as $P$ and $Q$, touches the segment $AB$ at point $D$ and touches $\Gamma$ at point $T$. The line $PD$ meets $\delta$ and $\Omega$ again at points $K$ and $L$, respectively. Prove that $\angle QTK=\angle DTL$

Positive integer $d$ is not perfect square. For each positive integer $n$, let $s(n)$ denote the number of digits $1$ among the first $n$ digits in the binary representation of $\sqrt{d}$ (including the digits before the point). Prove that there exists an integer $A$ such that $s(n)>\sqrt{2n}-2$ for all integers $n\ge A$

Ten distinct positive real numbers are given and the sum of each pair is written (So 45 sums). Between these sums there are 5 equal numbers. If we calculate product of each pair, find the biggest number $k$ such that there may be $k$ equal numbers between them.

We are given $m\times n$ table tiled with $3\times 1$ stripes and we are given that $6 | mn$. Prove that there exists a tiling of the table with $2\times 1$ dominoes such that each of these stripes contains one whole domino.

Let $G$ be the centroid of triangle $ABC$. Find the biggest $\alpha$ such that there exists a triangle for which there are at least three angles among $\angle GAB, \angle GAC, \angle GBA, \angle GBC, \angle GCA, \angle GCB$ which are $\geq \alpha$.