Olympiad problems

2021 Middle European Mathematical Olympiad, 8

Prove that there are infinitely many positive integers $n$ such that $n^2$ written in base $4$ contains only digits $1$ and $2$.

2021 Middle European Mathematical Olympiad, 4

Tags: number theory , memo , MEMO 2021 , Meme

Let $n \ge 3$ be an integer. Zagi the squirrel sits at a vertex of a regular $n$-gon. Zagi plans to make a journey of $n-1$ jumps such that in the $i$-th jump, it jumps by $i$ edges clockwise, for $i \in \{1, \ldots,n-1 \}$. Prove that if after $\lceil \tfrac{n}{2} \rceil$ jumps Zagi has visited $\lceil \tfrac{n}{2} \rceil+1$ distinct vertices, then after $n-1$ jumps Zagi will have visited all of the vertices. ([i]Remark.[/i] For a real number $x$, we denote by $\lceil x \rceil$ the smallest integer larger or equal to $x$.)

2021 Middle European Mathematical Olympiad, 2

Tags: polynomial , algebra , floor function , memo , MEMO 2021

Given a positive integer $n$, we say that a polynomial $P$ with real coefficients is $n$-pretty if the equation $P(\lfloor x \rfloor)=\lfloor P(x) \rfloor$ has exactly $n$ real solutions. Show that for each positive integer $n$ [list=a] [*] there exists an n-pretty polynomial; [*] any $n$-pretty polynomial has a degree of at least $\tfrac{2n+1}{3}$. [/list] ([i]Remark.[/i] For a real number $x$, we denote by $\lfloor x \rfloor$ the largest integer smaller than or equal to $x$.)

2021 Middle European Mathematical Olympiad, 1

Tags: Sequence , algebra , memo , MEMO 2021 , constant

Determine all real numbers A such that every sequence of non-zero real numbers $x_1, x_2, \ldots$ satisfying \[ x_{n+1}=A-\frac{1}{x_n} \] for every integer $n \ge 1$, has only finitely many negative terms.

2021 Middle European Mathematical Olympiad, 3

Tags: combinatorics , Strategy , MEMO 2021 , memo , induction

Let $n, b$ and $c$ be positive integers. A group of $n$ pirates wants to fairly split their treasure. The treasure consists of $c \cdot n$ identical coins distributed over $b \cdot n$ bags, of which at least $n-1$ bags are initially empty. Captain Jack inspects the contents of each bag and then performs a sequence of moves. In one move, he can take any number of coins from a single bag and put them into one empty bag. Prove that no matter how the coins are initially distributed, Jack can perform at most $n-1$ moves and then split the bags among the pirates such that each pirate gets $b$ bags and $c$ coins.

2021 Middle European Mathematical Olympiad, 6

Tags: geometry , power of a point , circumcircle , MEMO 2021 , memo

Let $ABC$ be a triangle and let $M$ be the midpoint of the segment $BC$. Let $X$ be a point on the ray $AB$ such that $2 \angle CXA=\angle CMA$. Let $Y$ be a point on the ray $AC$ such that $2 \angle AYB=\angle AMB$. The line $BC$ intersects the circumcircle of the triangle $AXY$ at $P$ and $Q$, such that the points $P, B, C$, and $Q$ lie in this order on the line $BC$. Prove that $PB=QC$. [i]Proposed by Dominik Burek, Poland[/i]

2021 Middle European Mathematical Olympiad, 1

Tags: functional equation , Functional inequality , algebra , memo , MEMO 2021

Determine all functions $f: \mathbb{R} \to \mathbb{R}$ such that the inequality \[ f(x^2)-f(y^2) \le (f(x)+y)(x-f(y)) \] holds for all real numbers $x$ and $y$.

2021 Middle European Mathematical Olympiad, 4

Tags: regular polygon , combinatorics , memo , MEMO 2021

Let $n$ be a positive integer. Prove that in a regular $6n$-gon, we can draw $3n$ diagonals with pairwise distinct ends and partition the drawn diagonals into $n$ triplets so that: [list] [*] the diagonals in each triplet intersect in one interior point of the polygon and [*] all these $n$ intersection points are distinct. [/list]

2021 Middle European Mathematical Olympiad, 5

Tags: geometry , circumcircle , MEMO 2021 , memo , menelaus theorem

Let $AD$ be the diameter of the circumcircle of an acute triangle $ABC$. The lines through $D$ parallel to $AB$ and $AC$ meet lines $AC$ and $AB$ in points $E$ and $F$, respectively. Lines $EF$ and $BC$ meet at $G$. Prove that $AD$ and $DG$ are perpendicular.

2021 Middle European Mathematical Olympiad, 7

Tags: number theory , Diophantine equation , memo , MEMO 2021 , easy number theorem

Find all pairs $(n, p)$ of positive integers such that $p$ is prime and \[ 1 + 2 + \cdots + n = 3 \cdot (1^2 + 2^2 + \cdot + p^2). \]