IMO 2020 (LXI International Mathematical Olympiad)

Angle bisectors and perpendicular bisector meet at a point

Consider the convex quadrilateral $ABCD$. The point $P$ is in the interior of $ABCD$. The following ratio equalities hold: $\angle PAD:\angle PBA:\angle DPA=1:2:3=\angle CBP:\angle BAP:\angle BPC.$ Prove that the following three lines meet in a point: the internal bisectors of angles $\angle ADP$ and $\angle PCB$ and the perpendicular bisector of segment $AB$.

Topic: Geometria piana Metodo: Simmetria Abilita: Ragionamento geometrico, Lettura attenta, Modellizzazione Area: Geometria Fonte: apri PDF p.1

Angle bisectors and perpendicular bisector meet at a point

Consider the convex quadrilateral $ABCD$. The point $P$ is in the interior of $ABCD$. The following ratio equalities hold: $\angle PAD:\angle PBA:\angle DPA=1:2:3=\angle CBP:\angle BAP:\angle BPC.$ Prove that the following three lines meet in a point: the internal bisectors of angles $\angle ADP$ and $\angle PCB$ and the perpendicular bisector of segment $AB$.

src_imho_2020__Q01

Inequality for reals summing to 1 with a≥b≥c≥d≥0

The real numbers $a,b,c,d$ are such that $a\ge b\ge c\ge d>0$ and $a+b+c+d=1$. Prove that $(a+2b+3c+4d)a^a{b}^b{c}^c{d}^d<1.$

Topic: Disuguaglianze Metodo: Disuguaglianze Abilita: Manipolazione algebrica, Stima, Lettura attenta Area: Algebra e Analisi Fonte: apri PDF p.1

Inequality for reals summing to 1 with a≥b≥c≥d≥0*

The real numbers $a,b,c,d$ are such that $a\ge b\ge c\ge d>0$ and $a+b+c+d=1$. Prove that $(a+2b+3c+4d)a^a{b}^b{c}^c{d}^d<1.$

src_imho_2020__Q02

Pebbles of weights 1 to 4n partitioned into two equal piles

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: \begin{itemize} \item The total weights of both piles are the same. \item Each pile contains two pebbles of each colour. \end{itemize}

Topic: Combinatoria Metodo: Invarianti, Casework Abilita: Modellizzazione, Conteggio sistematico, Ragionamento geometrico Area: Combinatoria, Logica e Probabilita Fonte: apri PDF p.1

Pebbles of weights 1 to 4n divided into two equal piles

There are $4n$ pebbles of weights $1,2,3,\dots ,4n$. Each pebble is colored in one of $n$ colors and there are four pebbles of each color. Show that we can arrange the pebbles into two piles so that the following two conditions are both satisfied: \begin{itemize} \item Each pile contains two pebbles of each color. I’m going to tell you.

src_imho_2020__Q03

Cable cars on a mountain; find n so all cards equal

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 cable 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.

Topic: Combinatoria Metodo: Estremalità, Casework, Grafi Abilita: Modellizzazione, Ragionamento geometrico, Lettura attenta, Conteggio sistematico Area: Combinatoria, Logica e Probabilita Fonte: apri PDF p.1

Cable cars on a mountain; find n know all cards equal

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 cable 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.

src_imho_2020__Q04

Deck of cards: arithmetic mean equals geometric mean on pairs

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?

Topic: Teoria dei Numeri, Algebra Metodo: Casework, Fattorizzazione Abilita: Ragionamento geometrico, Manipolazione algebrica, Riconoscimento di pattern, Lettura attenta Area: Aritmetica e Teoria dei Numeri, Algebra e Analisi Fonte: apri PDF p.1

Deck of cards: arithmetic mean equals geometric mean on pairs

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?

src_imho_2020__Q05

Positive constant c so a line separates S from each point

Prove that there exists a positive constant $c$ such that the following statement is true:

Consider an integer $n>1$, and a set $S$ of $n$ points in the plane such that the distance between any two different points in $S$ is at least 1. It follows that there is a line $\ell$ separating $S$ such that the distance from any point of $S$ to $\ell$ is at least $c{n}^{-1/3}$.

(A line $\ell$ separates a set of points $S$ if some segment joining two points in $S$ crosses $\ell$.)

\textit{Note.} Weaker results with $c{n}^{-1/3}$ replaced by $c{n}^{-\alpha }$ may be awarded points depending on the value of the constant $\alpha >1/3$.

Topic: Combinatoria, Geometria piana Metodo: Estremalità, Casework Abilita: Modellizzazione, Stima, Ragionamento geometrico, Lettura attenta Area: Combinatoria, Logica e Probabilita, Geometria Fonte: apri PDF p.1

Positive constant c so a line separates S from each point

Prove that there exists a positive constant $c$ such that the following statement is true:

Consider an integer $n>1$, and a set $S$ of $n$ points in the plane such that the distance between any two different points in $S$ is at least 1. It follows that there is a line $\ell$ separating $S$ such that the distance from any point of $S$ to $\ell$ is at least $c{n}^{-1/3}$.

(A line $\ell$ separates a set of points $S$ if some segment joining two points in $S$ crosses $\ell$.)

\textit{Note.} Weaker results with $c{n}^{-1/3}$ replaced by $c{n}^{-\alpha }$ may be awarded points depending on the value of the constant $\alpha >1/3$.

src_imho_2020__Q06