The numbers $1, 2, 3, \ldots , 10$ are written on the blackboard. In each step, Andrew chooses two numbers $a, b$ which are written on the blackboard such that $a\geqslant 2b$, he erases them, and in their place writes the number $a-2b$. Find all numbers $n$, such that after a sequence of steps as above, at the end only the number $n$ will remain on the blackboard.

Let $ABCD$ be a rhombus of sides $AB = BC = CD= DA = 13$. On the side $AB$ construct the rhombus $BAFE$ outside $ABCD$ and such that the side $AF$ is parallel to the diagonal $BD$ of $ABCD$. If the area of $BAFE$ is equal to $65$, calculate the area of $ABCD$.

Determine the maximal length $L$ of a sequence $a_1,\dots,a_L$ of positive integers satisfying both the following properties: [list=disc] [*]every term in the sequence is less than or equal to $2^{2023}$, and [*]there does not exist a consecutive subsequence $a_i,a_{i+1},\dots,a_j$ (where $1\le i\le j\le L$) with a choice of signs $s_i,s_{i+1},\dots,s_j\in\{1,-1\}$ for which \[s_ia_i+s_{i+1}a_{i+1}+\dots+s_ja_j=0.\] [/list]

Positive integers 1 to 9 are written in each square of a $ 3 \times 3 $ table. Let us define an operation as follows: Take an arbitrary row or column and replace these numbers $ a, b, c$ with either non-negative numbers $ a-x, b-x, c+x $ or $ a+x, b-x, c-x$, where $ x $ is a positive number and can vary in each operation. (1) Does there exist a series of operations such that all 9 numbers turn out to be equal from the following initial arrangement a)? b)? \[ a) \begin{array}{ccc} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{array} )\] \[ b) \begin{array}{ccc} 2 & 8 & 5 \\ 9 & 3 & 4 \\ 6 & 7 & 1 \end{array} )\] (2) Determine the maximum value which all 9 numbers turn out to be equal to after some steps.

Let triangle $ABC$ be an acute triangle with circumcircle $\Gamma$. Let $X$ and $Y$ be the midpoints of minor arcs $AB$ and $AC$ of $\Gamma$, respectively. If line $XY$ is tangent to the incircle of triangle $ABC$ and the radius of $\Gamma$ is $R$, find, with proof, the value of $XY$ in terms of $R$.

Let $m,n\ge 1$ and $a_1 < a_2 < \ldots < a_n$ be integers. Prove that there exists a subset $T$ of $\mathbb{N}$ such that \[|T| \leq 1+ \frac{a_n-a_1}{2n+1}\] and for every $i \in \{1,2,\ldots , m\}$, there exists $t \in T$ and $s \in [-n,n]$, such that $a_i=t+s$.

Find the largest number $n$ that for which there exists $n$ positive integers such that non of them divides another one, but between every three of them, one divides the sum of the other two. [i]Proposed by Morteza Saghafian[/i]

King Arthur one day had to fight the Dragon with Three Heads and Three Tails. His task became easier when he managed to find a magic sword that could, with a single blow, do one (and only one) of the following things: $\bullet$ cut off a head, $\bullet$ cut off two heads, $\bullet$ cut a tail, $\bullet$ cut off two tails. Furthermore, Fairy Morgana revealed to him the dragon's secret: $\bullet$ if a head is cut off, a new one grows, $\bullet$ if two heads are cut off nothing happens, $\bullet$ in place of a tail, two new tails are born, $\bullet$ if two tails are cut off a new head grow, $\bullet$ and the dragon dies if it loses its three heads and three tails. How many hits are needed to kill the dragon?

A uniform disk rotates at a fixed angular velocity on an axis through its center normal to the plane of the disk, and has kinetic energy $E$. If the same disk rotates at the same angular velocity about an axis on the edge of the disk (still normal to the plane of the disk), what is its kinetic energy? (a) $\frac{1}{2}E$ (b) $\frac{3}{2}E$ (c) $2E$ (d) $3E$ (e) $4E$

Find all integer solutions to the equation $$x^3+2y^3+4z^3+8xyz=0$$

Suppose that for every $n$ the number $m(n)$ is chosen such that $m(n)\ln(m(n))=n-\frac 12$. Show that $b_n$ is asymptotic to the following expression where $b_n$ is the $n-$th Bell number, that is the number of ways to partition $\{1,2,\ldots,n\}$: \[ \frac{m(n)^ne^{m(n)-n-\frac 12}}{\sqrt{\ln n}}. \] Two functions $f(n)$ and $g(n)$ are asymptotic to each other if $\lim_{n\rightarrow \infty}\frac{f(n)}{g(n)}=1$.

Let $n$ be a positive integer. There are $n$ islands with $n-1$ bridges connecting them such that one can travel from any island to another. One afternoon, a fire breaks out in one of the islands. Every morning, it spreads to all neighbouring islands. (Two islands are neighbours if they are connected by a bridge.) To control the spread, one bridge is destroyed every night until the fire has nowhere to spread the next day. Let $X$ be the minimum possible number of bridges one has to destroy before the fire stops spreading. Find the maximum possible value of $X$ over all possible configurations of bridges and island where the fire starts at.

When placing each of the digits $2,4,5,6,9$ in exactly one of the boxes of this subtraction problem, what is the smallest difference that is possible? $\text{(A)}\ 58 \qquad \text{(B)}\ 123 \qquad \text{(C)}\ 149 \qquad \text{(D)}\ 171 \qquad \text{(E)}\ 176$ \[\begin{tabular}[t]{cccc} & \boxed{} & \boxed{} & \boxed{} \\ - & & \boxed{} & \boxed{} \\ \hline \end{tabular}\]

Prove that a number of the form $80\dots01$ (there is at least 1 zero) can't be a perfect square.

Suppose that $a,b,c,d\in\mathbb{R}$ and $f(x)=ax^3+bx^2+cx+d$ such that $f(2)+f(5)<7<f(3)+f(4)$. Prove that there exists $u,v\in\mathbb{R}$ such that $u+v=7 , f(u)+f(v)=7$

Let $ f : \mathbf{R} \to \mathbf{R} $ be a continuous function with $ \displaystyle\int_{0}^{1} f(x) f'(x) \, \mathrm{d}x = 0 $ and $ \displaystyle\int_{0}^{1} f(x)^2 f'(x) \, \mathrm{d}x = 18 $. What is $ \displaystyle\int_{0}^{1} f(x)^4 f'(x) \, \mathrm{d} x $?

Rectangle $p*q,$ where $p,q$ are relatively coprime positive integers with $p <q$ is divided into squares $1*1$.Diagonal which goes from lowest left vertice to highest right cuts triangles from some squares.Find sum of perimeters of all such triangles.

Given a real number $x,$ let $\lfloor x \rfloor$ denote the greatest integer less than or equal to $x.$ For a certain integer $k,$ there are exactly $70$ positive integers $n_{1}, n_{2}, \ldots, n_{70}$ such that $k=\lfloor\sqrt[3]{n_{1}}\rfloor = \lfloor\sqrt[3]{n_{2}}\rfloor = \cdots = \lfloor\sqrt[3]{n_{70}}\rfloor$ and $k$ divides $n_{i}$ for all $i$ such that $1 \leq i \leq 70.$ Find the maximum value of $\frac{n_{i}}{k}$ for $1\leq i \leq 70.$

Given an isosceles trapezoid $ABCD$ with $AB = 6$, $CD = 12$, and area $36$, find $BC$.

The positive reals $a, b, c, x, y, z$ satisfy $$5a+4b+3c=5x+4y+3z.$$ Show that $$\frac{a^5}{x^4}+\frac{b^4}{y^3}+\frac{c^3}{z^2} \geq x+y+z.$$ [i]Proposed by Dominik Burek[/i]

Let $n\ge2$ be a natural number. Prove that $$\prod_{k=2}^n\ln k<\frac{\sqrt{n!}}n.$$

Let $ABCD$ be a convex quadrilateral such that $AB=4$, $BC=5$, $CA=6$, and $\triangle{ABC}$ is similar to $\triangle{ACD}$. Let $P$ be a point on the extension of $DA$ past $A$ such that $\angle{BDC}=\angle{ACP}$. Compute $DP^2$. [i]2020 CCA Math Bonanza Lightning Round #4.3[/i]

Let $ ABC$ a triangle. Let $D$ on $AB$ and $E$ on $AC$ such that $DE||BC$. Let line $DE$ intersect circumcircle of $ABC$ at two distinct points $F$ and $G$ so that line segments $BF$ and $CG$ intersect at P. Let circumcircle of $GDP$ and $FEP$ intersect again at $Q$. Prove that $A, P, Q$ are collinear.

A group of 6 students decided to make [i]study groups[/i] and [i]service activity groups[/i] according to the following principle: Each group must have exactly 3 members. For any pair of students, there are same number of study groups and service activity groups that both of the students are members. Supposing there are at least one group and no three students belong to the same study group and service activity group, find the minimum number of groups.

There are eight rooks on a chessboard, no two attacking each other. Prove that some two of the pairwise distances between the rooks are equal. (The distance between two rooks is the distance between the centers of their cell.)