TUYMAADA 2024

Triangular numbers are numbers of the form $1 + 2 + . . . + n$ with positive integer $n$ , that is $1, 3, 6, 10$ , . . . . Find the largest non-triangular positive integer number that cannot be represented as the sum of distinct triangular numbers. Proposed by A. Golovanov

We will call a hedgehog a graph in which one vertex is connected to all the others and there are no other edges; the number of vertices of this graph will be called the size of the hedgehog. A graph $G$ is given on $n$ vertices (where $n > 1$ ). For each edge $e$ , we denote by $s(e)$ the size of the maximum hedgehog in graph $G$ , which contains this edge. Prove the inequality (summation is carried out over all edges of the graph $G$ ): $$\sum_e \frac{1}{s(e)} \leqslant \frac{n}{2}.$$ Proposed by D. Malec, C. Tompkins

Three athletes ran at different constant speeds along a track of length $1$ . They started moving at the same time at one end of the track. Having reached one of the ends of the track, the athlete immediately turned around and continued running in the opposite direction. After a while, all three athletes met at the start and finished training. At what maximum $S$ can we knowingly say that at some point the sum of the pairwise distances between athletes was at least $S$ ? Proposed by A. Golovanov, I. Rubanov

A triangle $ABC$ is given. $N$ and $M$ are the midpoints of $AB$ and $BC$ , respectively. The bisector of angle $B$ meets the segment $MN$ at $E$ . $H$ is the base of the altitude drawn from $B$ in the triangle $ABC$ . The point $T$ on the circumcircle of $ABC$ is such that the circumcircles of $TMN$ and $ABC$ are tangent. Prove that points $T, H, E, B$ are concyclic. Proposed by M. Yumatov

Extension of angle bisector $BL$ of the triangle $ABC$ (where $AB < BC$ ) meets its circumcircle at $N$ . Let $M$ be the midpoint of $BL$ . Isosceles triangle $BDC$ with base $BC$ and angle equal to $ABC$ at $D$ is constructed outside the triangle $ABC$ . Prove that $CM \perp DN$ . Proposed by А. Mardanov

Given are quadratic trinomials $f$ and $g$ with integral coefficients. For each positive integer $n$ there is an integer $k$ such that $$\frac{f(k)}{g(k)}=\frac{n + 1}{n}.$$ Prove that $f$ and $g$ have a common root. Proposed by A. Golovanov

A toy factory produces several kinds of clay toys. The toys are painted in $k$ colours. Diversity of a colour is the number of different toys of that colour. (Thus, if there are $5$ blue cats, $7$ blue mice and nothing else is blue, the diversity of colour blue is $2$ .) The painting protocol requires that each colour is used and the diversities of each two colours are different. The toys in the store could be painted according to the protocol. However, a batch of clay Cheburashkas arrived at the store before painting (there were no Cheburashkas before). The number of Cheburashkas is not less that the number of the toys of any other kind. The total number of all toys, including Cheburashkas, is at least $\frac{(k+1)(k+2)}{2}$ . Prove that now the toys can be painted in $k + 1$ colours according to the protocol. Proposed by F. Petrov

Prove that a positive integer of the form $n^4 +1$ can have more than $1000$ divisors of the form $a^4 +1$ with integral $a$ .

Chip and Dale play on a $100 \times 100$ table. In the beginning, a chess king stands in the upper left corner of the table. At each move the king is moved one square right, down or right-down diagonally. A player cannot move in the direction used by his opponent in the previous move. The players move in turn, Chip begins. The player that cannot move loses. Which player has a winning strategy?

All perfect squares, and all perfect squares multiplied by two, are written in a row in increasing order. let $f(n)$ be the $n$ -th number in this sequence. (For instance, $f(1)=1,f(2)=2,f(3)=4,f(4)=8$ .) Is there an integer $n$ such that all the numbers $$f(n),f(2n),f(3n),\dots,f(10n^2)$$ are perfect squares?

A triangle $ABC$ is given. The segment connecting the points where the excircles touch $AB$ and $AC$ meets the bisector of angle $C$ at $X$ . The segment connecting the points where the excircles touch $BC$ and $AC$ meets the bisector of angle $A$ at $Y$ . Prove that the midpoint of $XY$ is equidistant from $A$ and $C$ .

The triangle $ABC$ is given. On the arc $BC$ of its circumscribed circle, which does not contain point $A$ , the variable point $X$ is selected, and on the rays $XB$ and $XC$ , the variable points $Y$ and $Z$ , respectively, so that $XA = XY = XZ$ . Prove that the line $YZ$ passes through a fixed point. Proposed by A. Kuznetsov

Given are two polynomial $f$ and $g$ of degree $100$ with real coefficients. For each positive integer $n$ there is an integer $k$ such that $$\frac{f(k)}{g(k)}=\frac{n+1}{n}.$$ Prove that $f$ and $g$ have a common non-constant factor.

A graph $G$ has $n$ vertices ( $n>1$ ). For each edge $e$ let $c(e)$ be the number of vertices of the largest complete subgraph containing $e$ . Prove that the inequality (the summation is over all edges of $G$ ): $$\sum_{e} \frac{c(e)}{c(e)-1}\le \frac{n^2}{2}.$$