Found problems: 26
2018 Benelux, 1
(a) Determine the minimal value of
$\displaystyle\left(x+\dfrac{1}{y}\right)\left(x+\dfrac{1}{y}-2018\right)+\left(y+\dfrac{1}{x}\right)\left(y+\dfrac{1}{x}-2018\right), $
where $x$ and $y$ vary over the positive reals.
(b) Determine the minimal value of
$\displaystyle\left(x+\dfrac{1}{y}\right)\left(x+\dfrac{1}{y}+2018\right)+\left(y+\dfrac{1}{x}\right)\left(y+\dfrac{1}{x}+2018\right), $
where $x$ and $y$ vary over the positive reals.
2025 Benelux, 3
Let $ABC$ be a triangle with incentre $I$ and circumcircle $\Omega$. Let $D, E, F$ be the midpoints of the arcs $\stackrel{\frown}{BC}, \stackrel{\frown}{CA}, \stackrel{\frown}{AB}$ of $\Omega$ not containing $A, B, C$ respectively. Let $D'$ be the point of $\Omega$ diametrically opposite to $D$. Show that $I, D'$ and the midpoint $M$ of $EF$ lie on a line.
2018 Benelux, 4
An integer $n\geq 2$ having exactly $s$ positive divisors $1=d_1<d_2<\cdots<d_s=n$ is said to be [i]good[/i] if there exists an integer $k$, with $2\leq k\leq s$, such that $d_k>1+d_1+\cdots+d_{k-1}$. An integer $n\geq 2$ is said to be [i]bad[/i] if it is not good.
(a) Show that there are infinitely many bad integers.
(b) Prove that, among any seven consecutive integers all greater than $2$, there are always at least four good integers.
(c) Show that there are infinitely many sequences of seven consecutive good integers.
2021 Benelux, 2
Pebbles are placed on the squares of a $2021\times 2021$ board in such a way that each square contains at most one pebble. The pebble set of a square of the board is the collection of all pebbles which are in the same row or column as this square. (A pebble belongs to the pebble set of the square in which it is placed.) What is the least possible number of pebbles on the board if no two squares have the same pebble set?
2020 Benelux, 3
Let $ABC$ be a triangle. The circle $\omega_A$ through $A$ is tangent to line $BC$ at $B$. The circle $\omega_C$ through $C$ is tangent to line $AB$ at $B$. Let $\omega_A$ and $\omega_C$ meet again at $D$. Let $M$ be the midpoint of line segment $[BC]$, and let $E$ be the intersection of lines $MD$ and $AC$. Show that $E$ lies on $\omega_A$.
2023 Benelux, 1
Find all functions $f\colon\mathbb{R}\to\mathbb{R}$ such that
$(x-y)\bigl(f(x)+f(y)\bigr)\leqslant f\bigl(x^2-y^2\bigr)$ for all $x,y\in\mathbb{R}$.
2025 Benelux, 1
Does there exist a function $f:\mathbb{R}\to \mathbb{R}$ such that $$f(x^2+f(y))=f(x)^2-y$$ for all $x, y\in \mathbb{R}$?
2022 Benelux, 2
Let $n$ be a positive integer. There are $n$ ants walking along a line at constant nonzero speeds. Different ants need not walk at the same speed or walk in the same direction. Whenever two or more ants collide, all the ants involved in this collision instantly change directions. (Different ants need not be moving in opposite directions when they collide, since a faster ant may catch up with a slower one that is moving in the same direction.) The ants keep walking indefinitely.
Assuming that the total number of collisions is finite, determine the largest possible number of collisions in terms of $n$.
2022 Benelux, 1
Let $n\geqslant 0$ be an integer, and let $a_0,a_1,\dots,a_n$ be real numbers. Show that there exists $k\in\{0,1,\dots,n\}$ such that
$$a_0+a_1x+a_2x^2+\cdots+a_nx^n\leqslant a_0+a_1+\cdots+a_k$$
for all real numbers $x\in[0,1]$.
2025 Benelux, 4
Let $a_0, a_1, \ldots, a_{10}$ be integers such that, for each $i \in \{0,1,\ldots,2047\}$, there exists a subset $S \subseteq \{0,1,\ldots,10\}$ with
\[
\sum_{j \in S} a_j \equiv i \pmod{2048}.
\]
Show that for each $i \in \{0,1,\ldots,10\}$, there is exactly one $j \in \{0,1,\ldots,10\}$ such that $a_j$ is divisible by $2^i$ but not by $2^{i+1}$.
Note: $\sum_{j \in S} a_j$ is the summation notation, for instance, $\sum_{j \in \{2,5\}} a_j = a_2 + a_5$, while for the empty set $\varnothing$, one defines $\sum_{j \in \varnothing} a_j = 0$.
2023 Benelux, 2
Determine all integers $k\geqslant 1$ with the following property: given $k$ different colours, if each integer is coloured in one of these $k$ colours, then there must exist integers $a_1<a_2<\cdots<a_{2023}$ of the same colour such that the differences $a_2-a_1,a_3-a_2,\dots,a_{2023}-a_{2022}$ are all powers of $2$.
2020 Benelux, 2
Let $N$ be a positive integer. A collection of $4N^2$ unit tiles with two segments drawn on them as shown is assembled into a $2N\times2N$ board. Tiles can be rotated.
[asy]size(1.5cm);draw((0,0)--(1,0)--(1,1)--(0,1)--cycle);draw((0,0.5)--(0.5,0),red);draw((0.5,1)--(1,0.5),red);[/asy]
The segments on the tiles define paths on the board. Determine the least possible number and the largest possible number of such paths.
[i]For example, there are $9$ paths on the $4\times4$ board shown below.[/i]
[asy]size(4cm);draw((0,0)--(4,0)--(4,4)--(0,4)--cycle);draw((0,1)--(4,1));draw((0,2)--(4,2));draw((0,3)--(4,3));draw((1,0)--(1,4));draw((2,0)--(2,4));draw((3,0)--(3,4));draw((0,3.5)--(0.5,4),red);draw((0,2.5)--(1.5,4),red);draw((3.5,0)--(4,0.5),red);draw((2.5,0)--(4,1.5),red);draw((0.5,0)--(0,0.5),red);draw((2.5,4)--(3,3.5)--(3.5,4),red);draw((4,3.5)--(3.5,3)--(4,2.5),red);draw((0,1.5)--(1,2.5)--(1.5,2)--(0.5,1)--(1.5,0),red);draw((1.5,3)--(2,3.5)--(3.5,2)--(2,0.5)--(1.5,1)--(2.5,2)--cycle,red);[/asy]
2018 Benelux, 3
Let $ABC$ be a triangle with orthocentre $H$, and let $D$, $E$, and $F$ denote the respective midpoints of line segments $AB$, $AC$, and $AH$. The reflections of $B$ and $C$ in $F$ are $P$ and $Q$, respectively.
(a) Show that lines $PE$ and $QD$ intersect on the circumcircle of triangle $ABC$.
(b) Prove that lines $PD$ and $QE$ intersect on line segment $AH$.
2019 Benelux, 4
An integer $m>1$ is [i]rich[/i] if for any positive integer $n$, there exist positive integers $x,y,z$ such that $n=mx^2-y^2-z^2$. An integer $m>1$ is [i]poor[/i] if it is not rich.
[list=a]
[*]Find a poor integer.[/*]
[*]Find a rich integer.[/*]
[/list]
2022 Benelux, 3
Let $ABC$ be a scalene acute triangle. Let $B_1$ be the point on ray $[AC$ such that $|AB_1|=|BB_1|$. Let $C_1$ be the point on ray $[AB$ such that $|AC_1|=|CC_1|$. Let $B_2$ and $C_2$ be the points on line $BC$ such that $|AB_2|=|CB_2|$ and $|BC_2|=|AC_2|$. Prove that $B_1$, $C_1$, $B_2$, $C_2$ are concyclic.
2018 Benelux, 2
In the land of Heptanomisma, four different coins and three different banknotes are used, and their denominations are seven different natural numbers. The denomination of the smallest banknote is greater than the sum of the denominations of the four different coins. A tourist has exactly one coin of each denomination and exactly one banknote of each denomination, but he cannot afford the book on numismatics he wishes to buy. However, the mathematically inclined shopkeeper offers to sell the book to the tourist at a price of his choosing, provided that he can pay this price in more than one way.
([i]The tourist can pay a price in more than one way if there are two different subsets of his coins and notes, the denominations of which both add up to this price.[/i])
(a) Prove that the tourist can purchase the book if the denomination of each banknote is smaller than $49$.
(b) Show that the tourist may have to leave the shop empty-handed if the denomination of the largest banknote is $49$.
2025 Benelux, 2
Let $N\geq 2$ be a natural number. At a mathematical olympiad training camp the same $N$ courses are organised every day. Each student takes exactly one of the $N$ courses each day. At the end of the camp, every student has takes each course exactly once, and any two students took the same course on at least one day, but took different courses on at least one other day. What is, in terms of $N$, the largest possible number of students at the camp?
2023 Benelux, 3
Let $ABC$ be a triangle with incentre $I$ and circumcircle $\omega$. Let $N$ denote the second point of intersection of line $AI$ and $\omega$. The line through $I$ perpendicular to $AI$ intersects line $BC$, segment $[AB]$, and segment $[AC]$ at the points $D$, $E$, and $F$, respectively. The circumcircle of triangle $AEF$ meets $\omega$ again at $P$, and lines $PN$ and $BC$ intersect at $Q$. Prove that lines $IQ$ and $DN$ intersect on $\omega$.
2020 Benelux, 1
Find all positive integers $d$ with the following property: there exists a polynomial $P$ of degree $d$ with integer coefficients such that $\left|P(m)\right|=1$ for at least $d+1$ different integers $m$.
2023 Benelux, 4
A positive integer $n$ is [i]friendly[/i] if the difference of each pair of neighbouring digits of $n$, written in base $10$, is exactly $1$. [i]For example, 6787 is friendly, but 211 and 901 are not.[/i]
Find all odd natural numbers $m$ for which there exists a friendly integer divisible by $64m$.
2021 Benelux, 4
A sequence $a_1, a_2, a_3, \ldots$ of positive integers satisfies $a_1 > 5$ and $a_{n+1} = 5 + 6 + \cdots + a_n$ for all positive integers $n$. Determine all prime numbers $p$ such that, regardless of the value of $a_1$, this sequence must contain a multiple of $p$.
2020 Benelux, 4
A divisor $d$ of a positive integer $n$ is said to be a [i]close[/i] divisor of $n$ if $\sqrt{n}<d<2\sqrt{n}$. Does there exist a positive integer with exactly $2020$ close divisors?
2010 Benelux, 4
Find all quadruples $(a, b, p, n)$ of positive integers, such that $p$ is a prime and
\[a^3 + b^3 = p^n\mbox{.}\]
[i](2nd Benelux Mathematical Olympiad 2010, Problem 4)[/i]
2021 Benelux, 1
(a) Prove that for all $a, b, c, d \in \mathbb{R}$ with $a + b + c + d = 0$,
\[
\max(a, b) + \max(a, c) + \max(a, d) + \max(b, c) + \max(b, d) + \max(c, d) \geqslant 0.
\]
(b) Find the largest non-negative integer $k$ such that it is possible to replace $k$ of the six maxima in this inequality by minima in such a way that the inequality still holds for all $a, b, c, d \in \mathbb{R}$ with $a + b + c + d = 0$.
2021 Benelux, 3
A cyclic quadrilateral $ABXC$ has circumcentre $O$. Let $D$ be a point on line $BX$ such that $AD = BD$. Let $E$ be a point on line $CX$ such that $AE = CE$. Prove that the circumcentre of triangle $\triangle DEX$ lies on the perpendicular bisector of $OA$.