IMO 2015 papers

Insiemi bilanciati e centre-free di n punti

We say that a finite set $\mathcal{S}$ of points in the plane is balanced if, for any two different points $A$ and $B$ in $\mathcal{S}$, there is a point $C$ in $\mathcal{S}$ such that $AC=BC$. We say that $\mathcal{S}$ is centre-free if for any three different points $A$, $B$ and $C$ in $\mathcal{S}$, there is no point of $\mathcal{S}$ equidistant from all three.

(a) Show that for all integers $n\ge 3$, there exists a balanced set having $n$ points.

(b) Determine all integers $n\ge 3$ for which there exists a balanced centre-free set having $n$ points.

Topic: Combinatoria, Geometria piana Metodo: Principio dei cassetti, Principio di estremalita Abilita: Conteggio sistematico Area: Combinatoria, Logica e Probabilita, Geometria Fonte: apri PDF p.1

*Balanced and centre-free sets of n points *

We say that a finite set $\mathcal{S}$ of points in the plane is balanced if, for any two different points $A$ and $B$ in $\mathcal{S}$, there is a point $C$ in $\mathcal{S}$ such that $AC=BC$. We say that $\mathcal{S}$ is centre-free if for any three different points $A$, $B$ and $C$ in $\mathcal{S}$, there is no point of $\mathcal{S}$ equidistant from all three.

(a) Show that for all integers $n\ge 3$, there exists a balanced set having $n$ points.

(b) Determine all integers $n\ge 3$ for which there exists a balanced centre-free set having $n$ points.

src_imo_2015__Q01

Triple (a,b,c) con ab-c,bc-a,ca-b potenze di 2

Determine all triples $(a,b,c)$ of positive integers such that each of the numbers $ab-c$, $bc-a$, $ca-b$ is a power of $2$.

(A power of 2 is an integer of the form $2^n$, where $n$ is a non-negative integer.)

Topic: Teoria dei Numeri Metodo: Analisi per casi, congruenze Abilita: Riconoscimento di pattern Area: Aritmetica e Teoria dei Numeri Fonte: apri PDF p.1

Triple (a,b,c) with ab-c,bc-a,ca-b powers of 2

Determine the triples $(a,b,c)$ of positive integers such that each of the numbers $ab-c$, $bc-a$, $ca-b$ is a power of $2$.

(A power of 2 is an integer of the form $2^n$, where $n$ is a non-negative integer.)

src_imo_2015__Q02

Circumcircle di KQH e FKM tangenti

Let $ABC$ be an acute triangle with $AB>AC$. Let $\Gamma$ be its circumcircle, $H$ its orthocentre, and $F$ the foot of the altitude from $A$. Let $M$ be the midpoint of $BC$. Let $Q$ be the point on $\Gamma$ such that $\angle HQA=90°$ and let $K$ be the point on $\Gamma$ such that $\angle HKQ=90°$. Assume that the points $A$, $B$, $C$, $K$ and $Q$ are all different and lie on $\Gamma$ in this order.

Prove that the circumcircles of triangles $KQH$ and $FKM$ are tangent to each other.

Topic: Geometria piana Metodo: Tecniche trigonometriche Abilita: Ragionamento geometrico Area: Geometria Fonte: apri PDF p.1

Circle of tangent KQH and FKM

Let $ABC$ be an acute triangle with $AB>AC$. Let $\Gamma$ be its circumcircle, $H$ its orthocentre, and $F$ the foot of the altitude from $A$. Let $M$ be the midpoint of $BC$. Let $Q$ be the point on $\Gamma$ such that $\angle HQA=90°$ and let $K$ be the point on $\Gamma$ such that $\angle HKQ=90°$. Assumes that the points $A$, $B$, $C$, $K$ and $Q$ are all different and lie on $\Gamma$ in this order.

Prove that the circumcircles of triangles $KQH$ and $FKM$ are tangent to each other.

src_imo_2015__Q03

X giace sulla retta AO (cerchi tangenti)

Triangle $ABC$ has circumcircle $\Omega$ and circumcentre $O$. A circle $\Gamma$ with centre $A$ intersects the segment $BC$ at points $D$ and $E$, such that $B$, $D$, $E$ and $C$ are all different and lie on line $BC$ in this order. Let $F$ and $G$ be the points of intersection of $\Gamma$ and $\Omega$, such that $A$, $F$, $B$, $C$, $G$ lie on $\Omega$ in this order. Let $K$ be the second point of intersection of the circumcircle of triangle $BDF$ and the segment $AB$. Let $L$ be the second point of intersection of the circumcircle of triangle $CGE$ and the segment $AC$.

Suppose that the lines $FK$ and $GL$ are different and intersect at the point $X$. Prove that $X$ lies on the line $AO$.

Topic: Geometria piana Metodo: Sfruttamento della simmetria Abilita: Ragionamento geometrico Area: Geometria Fonte: apri PDF p.2

*X lies on the straight AO (tangent circles) *

Triangle $ABC$ has circumcircle $\Omega$ and circumcentre $O$. A circle $\Gamma$ with centre $A$ intersects the segment $BC$ at points $D$ and $E$, such that $B$, $D$, $E$ and $C$ are all different and lie on line $BC$ in this order. Let $F$ and $G$ be the points of intersection of $\Gamma$ and $\Omega$, such that $A$, $F$, $B$, $C$, $G$ lie on $\Omega$ in this order. Let $K$ be the second point of intersection of the circumcircle of triangle $BDF$ and the segment $AB$. Let $L$ be the second point of intersection of the circumcircle of triangle $CGE$ and the segment $AC$.

Suppose that the lines $FK$ and $GL$ are different and intersect at the point $X$. Prove that $X$ lies on the line $AO$.

src_imo_2015__Q04

Equazione funzionale f(x+f(x+y))+f(xy)=…

Let $\mathbb{R}$ denote the set of real numbers. Determine all functions $f:\mathbb{R}\to \mathbb{R}$ satisfying the equation $f(x+f(x+y))+f(xy)=x+f(x+y)+yf(x)$ for all real numbers $x$ and $y$.

Topic: successioni Metodo: Analisi per casi Abilita: Manipolazione algebrica Area: Algebra e Analisi Fonte: apri PDF p.2

This is the functional equation f(x+f(x+y))+f(xy) =…*

Let $\mathbb{R}$ denotes the set of real numbers. Determine the functions $f:\mathbb{R}\to \mathbb{R}$ satisfying the equation $f(x+f(x+y))+f(xy)=x+f(x+y)+yf(x)$ for all real numbers $x$ and $y$.

src_imo_2015__Q05

Somme limitate di successione con k+ak distinti

The sequence $a_1,a_2,\dots$ of integers satisfies the conditions:

(i) $1\le a_j\le 2015$ for all $j\ge 1$;

(ii) $k+a_k\ne \ell +a_{\ell }$ for all $1\le k<\ell$.

Prove that there exist two positive integers $b$ and $N$ such that $\left|{\sum }_{j=m+1}^n(a_j-b)\right|\le 1007^2$ for all integers $m$ and $n$ satisfying $n>m\ge N$.

Topic: Combinatoria, Teoria dei Numeri Metodo: Principio di estremalita, monovarianti Abilita: Conteggio sistematico Area: Aritmetica e Teoria dei Numeri, Combinatoria, Logica e Probabilita Fonte: apri PDF p.2

Limited succession sums with distinct k+ak

The sequence $a_1,a_2,\dots$ of integers satisfies the conditions:

(i) $1\le a_j\le 2015$ for all $j\ge 1$;

(ii) $k+a_k\ne \ell +a_{\ell }$ for all $1\le k<\ell$.

Prove that there exist two positive integers $b$ and $N$ such that $\left|{\sum }_{j=m+1}^n(a_j-b)\right|\le 1007^2$ for all integers $m$ and $n$ satisfying $n>m\ge N$.

src_imo_2015__Q06