2024 Bulgarian Spring Mathematical Competition

10.1

The reals $x, y$ satisfy $x(x-6)\leq y(4-y)+7$. Find the minimal and maximal values of the expression $x+2y$.

10.2

Let $ABC$ be a triangle and a circle $\omega$ through $C$ and its incenter $I$ meets $CA, CB$ at $P, Q$. The circumcircles $(CPQ)$ and $(ABC)$ meet at $L$. The angle bisector of $\angle ALB$ meets $AB$ at $K$. Show that, as $\omega$ varies, $\angle PKQ$ is constant.

10.4

A graph $G$ is called $\textit{divisibility graph}$ if the vertices can be assigned distinct positive integers such that between two vertices assigned $u, v$ there is an edge iff $\frac{u} {v}$ or $\frac{v} {u}$ is a positive integer. Show that for any positive integer $n$ and $0 \leq e \leq \frac{n(n-1)}{2}$, there is a $\textit{divisibility graph}$ with $n$ vertices and $e$ edges. Remark on source of 10.3It appears to be Kvant 2022 Issue 10 M2719, so it will not be posted; the same problem was also used as 9.4.

11.2

Let $ABCD$ be a parallelogram and a circle $k$ passes through $A, C$ and meets rays $AB, AD$ at $E, F$. If $BD, EF$ and the tangent at $C$ concur, show that $AC$ is diameter of $k$.

11.4

Given is a convex $2024$-gon $A_1A_2\ldots A_{2024}$ and $1000$ points inside it, so that no three points are collinear. Some pairs of the points are connected with segments so that the interior of the polygon is divided into triangles. Every point is assigned one number among $\{1, -1, 2, - 2\}$, so that the sum of the numbers written in $A_i$ and $A_{i+1012}$ is zero for all $i=1,2, \ldots, 1012$. Prove that there is a triangle, such that the sum of the numbers in some two of its vertices is zero. Remark on source of 11.3It appears as Estonia TST 2004/5, so it will not be posted.

12.1

Given is a sequence $a_1, a_2, \ldots$, such that $a_1=1$ and $a_{n+1}=\frac{9a_n+4}{a_n+6}$ for any $n \in \mathbb{N}$. Which terms of this sequence are positive integers?

12.2

Given is a triangle $ABC$ and two points $D \in AC, E \in BD$ such that $\angle DAE=\angle AED=\angle ABC$. Show that $BE=2CD$ iff $\angle ACB=90^{\circ}$.

12.3

For a positive integer $n$, denote with $b(n)$ the smallest positive integer $k$, such that there exist integers $a_1, a_2, \ldots, a_k$, satisfying $n=a_1^{33}+a_2^{33}+\ldots+a_k^{33}$. Determine whether the set of positive integers $n$ is finite or infinite, which satisfy: a) $b(n)=12;$ b) $b(n)=12^{12^{12}}.$

12.4

Let $d \geq 3$ be a positive integer. The binary strings of length $d$ are splitted into $2^{d-1}$ pairs, such that the strings in each pair differ in exactly one position. Show that there exists an $\textit{alternating cycle}$ of length at most $2d-2$, i.e. at most $2d-2$ binary strings that can be arranged on a circle so that any pair of adjacent strings differ in exactly one position and exactly half of the pairs of adjacent strings are pairs in the split.