Found problems: 11
2017 Benelux, 3
In the convex quadrilateral $ABCD$ we have $\angle B = \angle C$ and $\angle D = 90^{\circ}.$ Suppose that $|AB| = 2|CD|.$ Prove that the angle bisector of $\angle ACB$ is perpendicular to $CD.$
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.
2017 Benelux, 4
A [i]Benelux n-square[/i] (with $n\geq 2$) is an $n\times n$ grid consisting of $n^2$ cells, each of them containing a positive integer, satisfying the following conditions:
$\bullet$ the $n^2$ positive integers are pairwise distinct.
$\bullet$ if for each row and each column we compute the greatest common divisor of the $n$ numbers in that row/column, then we obtain $2n$ different outcomes.
(a) Prove that, in each Benelux n-square (with $n \geq 2$), there exists a cell containing a number which is at least $2n^2.$
(b) Call a Benelux n-square [i]minimal[/i] if all $n^2$ numbers in the cells are at most $2n^2.$ Determine all $n\geq 2$ for which there exists a minimal Benelux n-square.
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.
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 $\omega_A$.
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 $AH$.
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 $49$.
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 $m$.
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]