IMO 2010 (LI International Mathematical Olympiad)

Functional equation on reals with floor function

Determine all functions $f:\mathbb{R}\to \mathbb{R}$ such that the equality $f(\lfloor x\rfloor y)=f(x)f(y)$ holds for all $x,y\in \mathbb{R}$. (Here $\lfloor z\rfloor$ denotes the greatest integer less than or equal to $z$.)

Topic: Equazioni funzionali, Algebra Metodo: Casework Abilita: Manipolazione algebrica, Lettura attenta, Casework accurato Area: Algebra e Analisi Fonte: apri PDF p.1

Functional equation on reals with floor function

Determine all functions $f:\mathbb{R}\to \mathbb{R}$ such that the equality $f(\lfloor x\rfloor y)=f(x)f(y)$ holds for all $x,y\in \mathbb{R}$. (Here $\lfloor z\rfloor$ denotes the greatest integer less than or equal to $z$.)

src_imho_2010__Q01

Incenter, circumcircle, angle condition, midpoint collinearity

Let $I$ be the incentre of triangle $ABC$ and let $\Gamma$ be its circumcircle. Let the line $AI$ intersect $\Gamma$ again at $D$. Let $E$ be a point on the arc $\widehat{BDC}$ and $F$ a point on the side $BC$ such that $\angle BAF=\angle CAE<\frac{1}{2}\angle BAC.$ Finally, let $G$ be the midpoint of the segment $IF$. Prove that the lines $DG$ and $EI$ intersect on $\Gamma$.

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

In the case of the ‘C’ range, the ‘C’ range shall be defined as the ‘C’ range, the ‘C’ range.

Let $I$ be the incenter of triangle $ABC$ and let $\Gamma$ be its circumcircle. Let the line $AI$ intersect $\Gamma$ again at $D$. Let $E$ be a point on the arc $\widehat{BDC}$ and $F$ a point on the side $BC$ such that $\angle BAF=\angle CAE<\frac{1}{2}\angle BAC.$ Finally, let $G$ be the midpoint of the segment $IF$. Prove that the lines $DG$ and $EI$ intersect on $\Gamma$.

src_imho_2010__Q02

Find all g: N→N making (g(m)+n)(m+g(n)) always a perfect square

Let $\mathbb{N}$ be the set of positive integers. Determine all functions $g:\mathbb{N}\to \mathbb{N}$ such that $(g(m)+n)(m+g(n))$ is a perfect square for all $m,n\in \mathbb{N}$.

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

*Find all g: N→N making (g(m)+n)

Let $\mathbb{N}$ be the set of positive integers. Determine all functions $g:\mathbb{N}\to \mathbb{N}$ such that $(g(m)+n)(m+g(n))$ is a perfect square for all $m,n\in \mathbb{N}$.

src_imho_2010__Q03

Point inside triangle, circumcircle tangent, SC=SP condition

Let $P$ be a point inside the triangle $ABC$. The lines $AP$, $BP$ and $CP$ intersect the circumcircle $\Gamma$ of triangle $ABC$ again at the points $K$, $L$ and $M$ respectively. The tangent to $\Gamma$ at $C$ intersects the line $AB$ at $S$. Suppose that $SC=SP$. Prove that $MK=ML$.

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

Point inside triangle, circumcircle tangent, SC=SP condition

Let $P$ be a point inside the triangle $ABC$. The lines $AP$, $BP$ and $CP$ intersect the circumcircle $\Gamma$ of triangle $ABC$ again at the points $K$, $L$ and $M$ respectively. The tangent to $\Gamma$ at $C$ intersects the line $AB$ at $S$. Suppose that $SC=SP$. Prove that $MK=ML$.

src_imho_2010__Q04

Coin operations on six boxes, finiteness of sequence reaching 2010^(2010^2010)

In each of six boxes $B_1,B_2,B_3,B_4,B_5,B_6$ there is initially one coin. There are two types of operation allowed:

\textit{Type 1}: Choose a nonempty box $B_j$ with $1\le j\le 5$. Remove one coin from $B_j$ and add two coins to $B_{j+1}$.

\textit{Type 2}: Choose a nonempty box $B_k$ with $1\le k\le 4$. Remove one coin from $B_k$ and exchange the contents of (possibly empty) boxes $B_{k+1}$ and $B_{k+2}$.

Determine whether there is a finite sequence of such operations that results in boxes $B_1,B_2,B_3,B_4,B_5$ being empty and box $B_6$ containing exactly $2010^{2010^{2010}}$ coins. (Note that $a^{b^c}=a^{(b^c)}$.)

Topic: Combinatoria, Teoria dei Numeri Metodo: Invarianti, Induzione, Ricorsione Abilita: Modellizzazione, Riconoscimento di pattern, Astrazione, Conteggio sistematico Area: Combinatoria, Logica e Probabilita, Aritmetica e Teoria dei Numeri Fonte: apri PDF p.1

*Coin operations on six boxes, finiteness of sequence reaching 2010^(2010^2010) *

In each of six boxes $B_1,B_2,B_3,B_4,B_5,B_6$ there is initially one coin. There are two types of operation allowed:

\textit{Type 1}: Choose a nonempty box $B_j$ with $1\le j\le 5$. Remove one coin from $B_j$ and add two coins to $B_{j+1}$.

\textit{Type 2}: Choose a nonempty box $B_k$ with $1\le k\le 4$. Remove one coin from $B_k$ and exchange the contents of (possibly empty) boxes $B_{k+1}$ and $B_{k+2}$.

Determine whether there is a finite sequence of such operations that results in boxes $B_1,B_2,B_3,B_4,B_5$ being empty and box $B_6$ containing exactly $2010^{2010^{2010}}$ coins. (Note that $a^{b^c}=a^{(b^c)}$.)

src_imho_2010__Q05

Sequence with max recurrence, find l≤s with a_n=a_{n+l} for large n

Let $a_1,a_2,a_3,\dots$ be a sequence of positive real numbers. Suppose that for some positive integer $s$, we have $a_n=\max \{a_k+a_{n-k}\mid 1\le k\le n-1\}$ for all $n>s$. Prove that there exist positive integers $\ell$ and $N$, with $\ell \le s$ and such that $a_n=a_{\ell }+a_{n-\ell }$ for all $n\ge N$.

Topic: Algebra, Combinatoria Metodo: Induzione, Estremalità, Ricorsione Abilita: Manipolazione algebrica, Riconoscimento di pattern, Lettura attenta, Astrazione Area: Algebra e Analisi, Combinatoria, Logica e Probabilita Fonte: apri PDF p.1

Sequence with max recurrence, find l≤s with a_n=a_{n+l} for large n

Let $a_1,a_2,a_3,\dots$ be a sequence of positive real numbers. Suppose that for some positive integer $s$, we have $a_n=\max \{a_k+a_{n-k}\mid 1\le k\le n-1\}$ for all $n>s$. Prove that there exist positive integers $\ell$ and $N$, with $\ell \le s$ and such that $a_n=a_{\ell }+a_{n-\ell }$ for all $n\ge N$.

src_imho_2010__Q06