IMO 2021 (LXII International Mathematical Olympiad)

Two piles of cards always share a perfect square sum

Let $n\ge 100$ be an integer. Ivan writes the numbers $n,n+1,\dots ,2n$ each on different cards. He then shuffles these $n+1$ cards, and divides them into two piles. Prove that at least one of the piles contains two cards such that the sum of their numbers is a perfect square.

Topic: Teoria dei Numeri, Combinatoria Metodo: Principio dei cassetti, Casework Abilita: Ragionamento geometrico, Riconoscimento di pattern, Conteggio sistematico Area: Aritmetica e Teoria dei Numeri, Combinatoria, Logica e Probabilita Fonte: apri PDF p.1

Two piles of cards always share a perfect square sum

Let $n\ge 100$ be an integer. Ivan writes the numbers $n,n+1,\dots ,2n$ each on different cards. He then shuffles these$n+1$ cards, and divides them into two piles. Prove that at least one of the piles contains two cards such that the sum of their numbers is a perfect square.

src_imho_2021__Q01

Double sum inequality with square roots of absolute values

Show that the inequality ${\sum }_{i=1}^n{\sum }_{j=1}^n\sqrt{|x_i-x_j|}\le {\sum }_{i=1}^n{\sum }_{j=1}^n\sqrt{|x_i+x_j|}$ holds for all real numbers $x_1,\dots ,x_n$.

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

Double sum inequality with square roots of absolute values

Show that the inequality ${\sum }_{i=1}^n{\sum }_{j=1}^n\sqrt{|x_i-x_j|}\le {\sum }_{i=1}^n{\sum }_{j=1}^n\sqrt{|x_i+x_j|}$ holds for all real numbers $x_1,\dots ,x_n$.

src_imho_2021__Q02

Concurrency of BC, EF, O1O2 in acute triangle configuration

Let $D$ be an interior point of the acute triangle $ABC$ with $AB>AC$ so that $\angle DAB=\angle CAD$. The point $E$ on the segment $AC$ satisfies $\angle ADE=\angle BCD$, the point $F$ on the segment $AB$ satisfies $\angle FDA=\angle DBC$, and the point $X$ on the line $AC$ satisfies $CX=BX$. Let $O_1$ and $O_2$ be the circumcentres of the triangles $ADC$ and $EXD$, respectively. Prove that the lines $BC$, $EF$, and $O_1{O}_2$ are concurrent.

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

Concurrency of BC, EF, O1O2 in acute triangle configuration

Let $D$ be an interior point of the acute triangle $ABC$ with $AB>AC$ so that $\angle DAB=\angle CAD$. The point $E$ on the segment $AC$ satisfies $\angle ADE=\angle BCD$, the point $F$ on the segment $AB$ satisfies $\angle FDA=\angle DBC$, and the point $X$ on the line $AC$ satisfies $CX=BX$. Let $O_1$ and $O_2$ be the circumcentres of the triangles $ADC$ and $EXD$, respectively. Prove that the lines $BC$, $EF$, and $O_1{O}_2$ are concurrent.

src_imho_2021__Q03

Circle tangent to convex quadrilateral sides yields equal sum

Let $\Gamma$ be a circle with centre $I$, and $ABCD$ a convex quadrilateral such that each of the segments $AB$, $BC$, $CD$ and $DA$ is tangent to $\Gamma$. Let $\Omega$ be the circumcircle of the triangle $AIC$. The extension of $BA$ beyond $A$ meets $\Omega$ at $X$, and the extension of $BC$ beyond $C$ meets $\Omega$ at $Z$. The extension of $AD$ beyond $D$ meets $\Omega$ at $Y$, and the extension of $CD$ beyond $D$ meets $\Omega$ at $T$. Prove that $AD+DT+TX+XA=CD+DY+YZ+ZC.$

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

Circle tangent to convex quadrilateral sides yields equal sum

Let $\Gamma$ be a circle with center $I$, and $ABCD$ a convex quadrilateral such that each of the segments $AB$, $BC$, $CD$ and $DA$ is tangent to $\Gamma$. Let $\Omega$ be the circumcircle of the triangle $AIC$. The extension of $BA$ beyond $A$ meets $\Omega$ at $X$, and the extension of $BC$ beyond $C$ meets $\Omega$ at $Z$. The extension of $AD$ beyond $D$ meets $\Omega$ at $Y$, and the extension of $CD$ beyond $D$ meets $\Omega$ at $T$. Prove that $AD+DT+TX+XA=CD+DY+YZ+ZC.$

src_imho_2021__Q04

Jumpy’s walnut swaps always allow finding adjacent walnuts

Two squirrels, Bushy and Jumpy, have collected 2021 walnuts for the winter. Jumpy numbers the walnuts from 1 through 2021, and digs 2021 little holes in a circular pattern in the ground around their favourite tree. The next morning Jumpy notices that Bushy had placed one walnut into each hole, but had paid no attention to the numbering. Unhappy, Jumpy decides to reorder the walnuts by performing a sequence of 2021 moves. In the $k$-th move, Jumpy swaps the positions of the two walnuts adjacent to walnut $k$.

Prove that there exists a value of $k$ such that, on the $k$-th move, Jumpy swaps some walnuts $a$ and $b$ such that $a<k<b$.

Topic: Combinatoria Metodo: Invarianti, Casework, Estremalità Abilita: Ragionamento geometrico, Riconoscimento di pattern, Lettura attenta, Astrazione Area: Combinatoria, Logica e Probabilita Fonte: apri PDF p.1

Jumpy’s walnut swaps always allow finding adjacent walnuts

Two squirrels, Bushy and Jumpy, have collected 2021 walnuts for the winter. Jump numbers the walnuts from 1 through 2021, and digs 2021 little holes in a circular pattern in the ground around their favorite tree. The next morning Jumpy notices that Bushy had placed one walnut into each hole, but had paid no attention to the numbering. Unhappy, Jumpy decides to reorder the walnuts by performing a sequence of 2021 moves. In the $k$-th move, Jumpy swaps the positions of the two walnuts adjacent to walnut $k$.

Prove that there exists a value of $k$ such that, on the $k$-th move, Jumpy swaps some walnuts $a$ and $b$ such that $a<k<b$.

src_imho_2021__Q05

Subset sums bound forces large intersecting subset

Let $m\ge 2$ be an integer, $A$ be a finite set of (not necessarily positive) integers, and $B_1,B_2,\dots ,B_m$ be subsets of $A$. Assume that for each $k=1,2,\dots ,m$ the sum of the elements of $B_k$ is $m^k$. Prove that $A$ contains at least $m/2$ elements.

Topic: Combinatoria, Teoria dei Numeri Metodo: Induzione, Estremalità, Doppio conteggio Abilita: Astrazione, Manipolazione algebrica, Conteggio sistematico, Stima Area: Combinatoria, Logica e Probabilita, Aritmetica e Teoria dei Numeri Fonte: apri PDF p.1

Subset sums bound forces large intersecting subset

Let $m\ge 2$ be an integer, $A$ be a finite set of (not necessarily positive) integers, and $B_1,B_2,\dots ,B_m$ be subsets of $A$. Assumes that for each $k=1,2,\dots ,m$ the sum of the elements of $B_k$ is $m^k$. Prove that $A$ contains at least $m/2$ elements.

src_imho_2021__Q06