IZhO 2024

In an alphabet of $n$ letters, a 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 points $P$ and $Q$, respectively, such that these points lie on the 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 a 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 $n \geq A$.

The teacher has given 10 distinct positive numbers to his students. Serge found all their 45 pairwise sums; five of these sums are equal. Pete found all their 45 pairwise products. What maximum number of equal products can be among Pete’s numbers?

A table $m \times n$ is given, where $mn$ is divisible by 6. In this table a stripe is any $1 \times 3$ or $3 \times 1$ rectangle, and a domino is any $1 \times 2$ or $2 \times 1$ rectangle. The table is tiled with stripes. Prove that on top of this tiling the table can be tiled with dominoes so that in each stripe some two cells are covered by one domino and the remaining cell is covered by another domino. (The table is tiled by rectangles if the rectangles cover the entire table and do not overlap with each other.)

The medians of a triangle $ABC$ concur at point $G$. Among the six angles $GAB, GAC, GBA, GBC, GCA, GCB$ there are at least three angles each of which is at least $\alpha$. Determine the largest $\alpha$ for which this is possible.