Found problems: 21
2021 European Mathematical Cup, 4
Let $n$ be a positive integer. Morgane has coloured the integers $1,2,\ldots,n$. Each of them is coloured in exactly one colour. It turned out that for all positive integers $a$ and $b$ such that $a<b$ and $a+b \leqslant n$, at least two of the integers among $a$, $b$ and $a+b$ are of the same colour. Prove that there exists a colour that has been used for at least $2n/5$ integers. \\ \\
(Vincent Jugé)
2023 European Mathematical Cup, 4
We say that a $2023$-tuple of nonnegative integers $(a_1,\hdots,a_{2023})$ is [i]sweet[/i] if the following conditions hold:
[list]
[*] $a_1+\hdots+a_{2023}=2023$
[*] $\frac{a_1}{2}+\frac{a_2}{2^2}+\hdots+\frac{a_{2023}}{2^{2023}}\le 1$
[/list]
Determine the greatest positive integer $L$ so that \[a_1+2a_2+\hdots+2023a_{2023}\ge L\] holds for every sweet $2023$-tuple $(a_1,\hdots,a_{2023})$
[i]Ivan Novak[/i]
2018 European Mathematical Cup, 1
Let $a, b, c$ be non-zero real numbers such that $a^2+b+c=\frac{1}{a}, b^2+c+a=\frac{1}{b}, c^2+a+b=\frac{1}{c}.$ Prove that at least two of $a, b, c$ are equal.
2021 European Mathematical Cup, 1
Alice drew a regular $2021$-gon in the plane. Bob then labeled each vertex of the $2021$-gon with a real number, in such a way that the labels of consecutive vertices differ by at most $1$. Then, for every pair of non-consecutive vertices whose labels differ by at most $1$, Alice drew a diagonal connecting them. Let $d$ be the number of diagonals Alice drew. Find the least possible value that $d$ can obtain.
2023 European Mathematical Cup, 1
Determine all sets of real numbers $S$ such that:
[list]
[*] $1$ is the smallest element of $S$,
[*] for all $x,y\in S$ such that $x>y$, $\sqrt{x^2-y^2}\in S$
[/list]
[i]Adian Anibal Santos Sepcic[/i]
2016 European Mathematical Cup, 2
Two circles $C_{1}$ and $C_{2}$ intersect at points $A$ and $B$. Let $P$, $Q$ be points on circles $C_{1}$, $C_{2}$ respectively, such that $|AP| = |AQ|$. The segment $PQ$ intersects circles $C_{1}$ and $C_{2}$ in points $M$, $N$ respectively. Let $C$ be the center of the arc $BP$ of $C_{1}$ which does not contain point $A$ and let $D$ be the center of arc $BQ$ of $C_{2}$ which does not contain point $A$ Let $E$ be the intersection of $CM$ and $DN$. Prove that $AE$ is perpendicular to $CD$.
Proposed by Steve Dinh
2023 European Mathematical Cup, 3
Let $n$ be a positive integer. Let $B_n$ be the set of all binary strings of length $n$. For a binary string $s_1\hdots s_n$, we define it's twist in the following way. First, we count how many blocks of consecutive digits it has. Denote this number by $b$. Then, we replace $s_b$ with $1-s_b$. A string $a$ is said to be a [i]descendant[/i] of $b$ if $a$ can be obtained from $b$ through a finite number of twists. A subset of $B_n$ is called [i]divided[/i] if no two of its members have a common descendant. Find the largest possible cardinality of a divided subset of $B_n$.
[i]Remark.[/i] Here is an example of a twist: $101100 \rightarrow 101000$ because $1\mid 0\mid 11\mid 00$ has $4$ blocks of consecutive digits.
[i]Viktor Simjanoski[/i]
2024 European Mathematical Cup, 2
Let $n$ be a positive integer. The numbers $1, 2, \dots, 2n+1$ are arranged in a circle in that order, and some of them are [i] marked[/i].
We define, for each $k$ such that $1\leq k \leq 2n+1$ , the interval $I_k$ to be the closed circular interval starting at $k$ and ending in $k+n$ (taking remainders mod(2n+1)). We call in interval [i]magical[/i] if it contains strictly more than half of all the marked elements.
Prove that the following two statements are equivalent:
1. At least $n+1$ of the intervals $I_1, I_2, \dots, I_{2n+1}$ are magical
2. The number of marked numbers is odd
2021 European Mathematical Cup, 2
Let $ABC$ be an acute-angled triangle such that $|AB|<|AC|$. Let $X$ and $Y$ be points on the minor arc ${BC}$ of the circumcircle of $ABC$ such that $|BX|=|XY|=|YC|$. Suppose that there exists a point $N$ on the segment $\overline{AY}$ such that $|AB|=|AN|=|NC|$. Prove that the line $NC$ passes through the midpoint of the segment $\overline{AX}$. \\ \\ (Ivan Novak)
2024 European Mathematical Cup, 4
Let $\mathcal{F}$ be a family of (distinct) subsets of the set $\{1,2,\dots,n\}$ such that for all $A$, $B\in \mathcal{F}$,we have that $A^C\cup B\in \mathcal{F}$, where $A^C$ is the set of all members of ${1,2,\dots,n}$ that are not in $A$.
Prove that every $k\in {1,2,\dots,n}$ appears in at least half of the sets in $\mathcal{F}$.
[i]Stijn Cambie, Mohammad Javad Moghaddas Mehr[/i]
2021 European Mathematical Cup, 4
Find all positive integers $d$ for which there exist polynomials $P(x)$ and $Q(x)$ with real coefficients such that degree of $P$ equals $d$ and
$$P(x)^2+1=(x^2+1)Q(x)^2.$$
2024 European Mathematical Cup, 4
Find all functions $ f: \mathbb{R}^{+} \to \mathbb{R}^{+}$ such that $f(x+yf(x)) = xf(1+y)$
for all x, y positive reals.
2020 European Mathematical Cup, 2
A positive integer $k\geqslant 3$ is called[i] fibby[/i] if there exists a positive integer $n$ and positive integers $d_1 < d_2 < \ldots < d_k$ with the following properties: \\ $\bullet$ $d_{j+2}=d_{j+1}+d_j$ for every $j$ satisfying $1\leqslant j \leqslant k-2$, \\ $\bullet$ $d_1, d_2, \ldots, d_k$ are divisors of $n$, \\ $\bullet$ any other divisor of $n$ is either less than $d_1$ or greater than $d_k$.
Find all fibby numbers. \\ \\ [i]Proposed by Ivan Novak.[/i]
2021 European Mathematical Cup, 3
Let $\mathbb{N}$ denote the set of all positive integers. Find all functions $f:\mathbb{N}\to\mathbb{N}$ such that
$$x^2-y^2+2y(f(x)+f(y))$$
is a square of an integer for all positive integers $x$ and $y$.
2018 European Mathematical Cup, 3
For which real numbers $k > 1$ does there exist a bounded set of positive real numbers $S$ with at
least $3$ elements such that
$$k(a - b)\in S$$
for all $a,b\in S $ with $a > b?$
Remark: A set of positive real numbers $S$ is bounded if there exists a positive real number $M$ such that
$x < M$ for all $x \in S.$
2020 European Mathematical Cup, 3
Two types of tiles, depicted on the figure below, are given.
[img]https://wiki-images.artofproblemsolving.com//2/23/Izrezak.PNG[/img]
Find all positive integers $n$ such that an $n\times n$ board consisting of $n^2$ unit squares can be covered without gaps with these two types of tiles (rotations and reflections are allowed) so that no two tiles overlap and no part of any tile covers an area outside the $n\times n$ board. \\
[i]Proposed by Art Waeterschoot[/i]
2021 European Mathematical Cup, 1
We say that a quadruple of nonnegative real numbers $(a,b,c,d)$ is [i]balanced [/i]if $$a+b+c+d=a^2+b^2+c^2+d^2.$$
Find all positive real numbers $x$ such that $$(x-a)(x-b)(x-c)(x-d)\geq 0$$ for every balanced quadruple $(a,b,c,d)$. \\ \\
(Ivan Novak)
2023 European Mathematical Cup, 2
Let $ABC$ be a triangle such that $\angle BAC = 90^{\circ}$. The incircle of triangle $ABC$ is tangent to the sides $\overline{BC}$, $\overline{CA}$, $\overline{AB}$ at $D,E,F$ respectively. Let $M$ be the midpoint of $\overline{EF}$. Let $P$ be the projection of $A$ onto $BC$ and let $K$ be the intersection of $MP$ and $AD$. Prove that the circumcircles of triangles $AFE$ and $PDK$ have equal radius.
[i]Kyprianos-Iason Prodromidis[/i]
2023 European Mathematical Cup, 1
Suppose $a,b,c$ are positive integers such that \[\gcd(a,b)+\gcd(a,c)+\gcd(b,c)=b+c+2023\] Prove that $\gcd(b,c)=2023$.
[i]Remark.[/i] For positive integers $x$ and $y$, $\gcd(x,y)$ denotes their greatest common divisor.
[i]Ivan Novak[/i]
2021 European Mathematical Cup, 3
Let $\ell$ be a positive integer. We say that a positive integer $k$ is [i]nice [/i] if $k!+\ell$ is a square of an integer. Prove that for every positive integer $n \geqslant \ell$, the set $\{1, 2, \ldots,n^2\}$ contains at most $n^2-n +\ell$ nice integers. \\ \\
(Théo Lenoir)
2020 European Mathematical Cup, 4
Let \(a,b,c\) be positive real numbers such that \(ab+bc+ac = a+b+c\). Prove the following inequality:
\[\sqrt{a+\frac{b}{c}} + \sqrt{b+\frac{c}{a}} + \sqrt{c+\frac{a}{b}} \leq \sqrt{2} \cdot \min \left\{ \frac{a}{b}+\frac{b}{c}+\frac{c}{a},\ \frac{b}{a}+\frac{c}{b}+\frac{a}{c} \right\}.\] \\ \\ [i]Proposed by Dorlir Ahmeti.[/i]