This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 10

2020 IMO Shortlist, N3
Tags: IMO 2020 , IMO , number theory , arithmetic mean , geometric mean , IMO Shortlist , IMO Shortlist 2020
A deck of $n > 1$ cards is given. A positive integer is written on each card. The deck has the property that the arithmetic mean of the numbers on each pair of cards is also the geometric mean of the numbers on some collection of one or more cards. For which $n$ does it follow that the numbers on the cards are all equal? [i]Proposed by Oleg Košik, Estonia[/i]

2020 IMO Shortlist, G9
Tags: IMO 2020 , IMO , combinatorial geometry , combinatorics , geometry , IMO Shortlist , IMO Shortlist 2020
Prove that there exists a positive constant $c$ such that the following statement is true: Consider an integer $n > 1$, and a set $\mathcal S$ of $n$ points in the plane such that the distance between any two different points in $\mathcal S$ is at least 1. It follows that there is a line $\ell$ separating $\mathcal S$ such that the distance from any point of $\mathcal S$ to $\ell$ is at least $cn^{-1/3}$. (A line $\ell$ separates a set of points S if some segment joining two points in $\mathcal S$ crosses $\ell$.) [i]Note. Weaker results with $cn^{-1/3}$ replaced by $cn^{-\alpha}$ may be awarded points depending on the value of the constant $\alpha > 1/3$.[/i] [i]Proposed by Ting-Feng Lin and Hung-Hsun Hans Yu, Taiwan[/i]

2020 IMO Shortlist, C6
Tags: IMO , combinatorics , IMO 2020 , graph theory , petersen , IMO Shortlist , IMO Shortlist 2020
There are $4n$ pebbles of weights $1, 2, 3, \dots, 4n.$ Each pebble is coloured in one of $n$ colours and there are four pebbles of each colour. Show that we can arrange the pebbles into two piles so that the following two conditions are both satisfied: [list] [*]The total weights of both piles are the same. [*] Each pile contains two pebbles of each colour. [/list] [i]Proposed by Milan Haiman, Hungary and Carl Schildkraut, USA[/i]

2020 IMO, 2
Tags: inequalities , IMO 2020 , IMO , algebra , IMO Shortlist 2020 , Mount Inquality , Hi
The real numbers $a, b, c, d$ are such that $a\geq b\geq c\geq d>0$ and $a+b+c+d=1$. Prove that \[(a+2b+3c+4d)a^ab^bc^cd^d<1\] [i]Proposed by Stijn Cambie, Belgium[/i]

2020 IMO Shortlist, A4
Tags: inequalities , IMO 2020 , IMO , algebra , IMO Shortlist 2020 , Mount Inquality , Hi
The real numbers $a, b, c, d$ are such that $a\geq b\geq c\geq d>0$ and $a+b+c+d=1$. Prove that \[(a+2b+3c+4d)a^ab^bc^cd^d<1\] [i]Proposed by Stijn Cambie, Belgium[/i]

2020 IMO, 6
Tags: IMO 2020 , IMO , combinatorial geometry , combinatorics , geometry , IMO Shortlist , IMO Shortlist 2020
Prove that there exists a positive constant $c$ such that the following statement is true: Consider an integer $n > 1$, and a set $\mathcal S$ of $n$ points in the plane such that the distance between any two different points in $\mathcal S$ is at least 1. It follows that there is a line $\ell$ separating $\mathcal S$ such that the distance from any point of $\mathcal S$ to $\ell$ is at least $cn^{-1/3}$. (A line $\ell$ separates a set of points S if some segment joining two points in $\mathcal S$ crosses $\ell$.) [i]Note. Weaker results with $cn^{-1/3}$ replaced by $cn^{-\alpha}$ may be awarded points depending on the value of the constant $\alpha > 1/3$.[/i] [i]Proposed by Ting-Feng Lin and Hung-Hsun Hans Yu, Taiwan[/i]

2020 IMO, 3
Tags: IMO , combinatorics , IMO 2020 , graph theory , petersen , IMO Shortlist , IMO Shortlist 2020
There are $4n$ pebbles of weights $1, 2, 3, \dots, 4n.$ Each pebble is coloured in one of $n$ colours and there are four pebbles of each colour. Show that we can arrange the pebbles into two piles so that the following two conditions are both satisfied: [list] [*]The total weights of both piles are the same. [*] Each pile contains two pebbles of each colour. [/list] [i]Proposed by Milan Haiman, Hungary and Carl Schildkraut, USA[/i]

2020 IMO, 4
Tags: combinatorics , IMO 2020 , IMO , pigeonhole principle , IMO Shortlist , graph theory , IMO Shortlist 2020
There is an integer $n > 1$. There are $n^2$ stations on a slope of a mountain, all at different altitudes. Each of two cable car companies, $A$ and $B$, operates $k$ cable cars; each cable car provides a transfer from one of the stations to a higher one (with no intermediate stops). The $k$ cable cars of $A$ have $k$ different starting points and $k$ different finishing points, and a cable car which starts higher also finishes higher. The same conditions hold for $B$. We say that two stations are linked by a company if one can start from the lower station and reach the higher one by using one or more cars of that company (no other movements between stations are allowed). Determine the smallest positive integer $k$ for which one can guarantee that there are two stations that are linked by both companies. [i]Proposed by Tejaswi Navilarekallu, India[/i]

2020 IMO, 5
Tags: IMO 2020 , IMO , number theory , arithmetic mean , geometric mean , IMO Shortlist , IMO Shortlist 2020
A deck of $n > 1$ cards is given. A positive integer is written on each card. The deck has the property that the arithmetic mean of the numbers on each pair of cards is also the geometric mean of the numbers on some collection of one or more cards. For which $n$ does it follow that the numbers on the cards are all equal? [i]Proposed by Oleg Košik, Estonia[/i]

2020 IMO Shortlist, C3
Tags: combinatorics , IMO 2020 , IMO , pigeonhole principle , IMO Shortlist , graph theory , IMO Shortlist 2020
There is an integer $n > 1$. There are $n^2$ stations on a slope of a mountain, all at different altitudes. Each of two cable car companies, $A$ and $B$, operates $k$ cable cars; each cable car provides a transfer from one of the stations to a higher one (with no intermediate stops). The $k$ cable cars of $A$ have $k$ different starting points and $k$ different finishing points, and a cable car which starts higher also finishes higher. The same conditions hold for $B$. We say that two stations are linked by a company if one can start from the lower station and reach the higher one by using one or more cars of that company (no other movements between stations are allowed). Determine the smallest positive integer $k$ for which one can guarantee that there are two stations that are linked by both companies. [i]Proposed by Tejaswi Navilarekallu, India[/i]