IMO 1996 (XXXVII International Mathematical Olympiad)

Rectangle grid moves from A to B vertex

We are given a positive integer $r$ and a rectangular board $ABCD$ with dimensions $|AB|=20$, $|BC|=12$. The rectangle is divided into a grid of $20\times 12$ unit squares. The following moves are permitted on the board: one can move from one square to another only if the distance between the centers of the two squares is $\sqrt{r}$. The task is to find a sequence of moves leading from the square with $A$ as a vertex to the square with $B$ as a vertex.

(a) Show that the task cannot be done if $r$ is divisible by 2 or 3.

(b) Prove that the task is possible when $r=73$.

(c) Can the task be done when $r=97$?

Topic: Combinatoria, Teoria dei Numeri Metodo: Colorazione, Casework, Congruenze Abilita: Modellizzazione, Ragionamento geometrico, Casework accurato Area: Combinatoria, Logica e Probabilita, Aritmetica e Teoria dei Numeri Fonte: apri PDF p.1

Rectangle grid moves from A to B vertex

We are given a positive integer $r$ and a rectangular board $ABCD$ with dimensions $|AB|=20$, $|BC|=12$. The rectangle is divided into a grid of $20\times 12$ unit squares. The following moves are allowed on the board: one can move from one square to another only if the distance between the centers of the two squares is $\sqrt{r}$. The task is to find a sequence of moves leading from the square with $A$ as a vertex to the square with $B$ as a vertex.

(a) Show that the task cannot be done if $r$ is divisible by 2 or 3.

(b) Prove that the task is possible when $r=73$.

(c) Can the task be done when $r=97$?

src_imho_1996__Q01

Incenter concurrence in triangle with interior point

Let $P$ be a point inside triangle $ABC$ such that $\angle APB-\angle ACB=\angle APC-\angle ABC.$ Let $D$, $E$ be the incenters of triangles $APB$, $APC$, respectively. Show that $AP$, $BD$, $CE$ meet at a point.

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

Incenter concurrence in triangle with interior point

Let $P$ be a point inside triangle $ABC$ such that $\angle APB-\angle ACB=\angle APC-\angle ABC.$ Let $D$, $E$ be the incenters of triangles $APB$, $APC$, respectively. Show that$AP$,$BD$,$CE$ meet at a point.

src_imho_1996__Q02

Functional equation on nonneg integers with sum

Let $S$ denote the set of nonnegative integers. Find all functions $f$ from $S$ to itself such that $f(m+f(n))=f(f(m))+f(n)\forall m,n\in S.$

Topic: Insiemi e funzioni, Equazioni funzionali Metodo: Invarianti, Casework Abilita: Manipolazione algebrica, Riconoscimento di pattern, Lettura attenta Area: Algebra e Analisi Fonte: apri PDF p.1

Functional equation on nonnegative integers with sum

Let $S$ denotes the set of nonnegative integers. Find all functions $f$ from $S$ to itself such that $f(m+f(n))=f(f(m))+f(n)\forall m,n\in S.$

src_imho_1996__Q03

Least value of smaller of two squares from linear combos

The positive integers $a$ and $b$ are such that the numbers $15a+16b$ and $16a-15b$ are both squares of positive integers. What is the least possible value that can be taken on by the smaller of these two squares?

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

Least value of smaller of two squares from linear combos

The positive integers $a$ and $b$ are such that the numbers $15a+16b$ and $16a-15b$ are both squares of positive integers. What is the least possible value that can be taken on by the smaller of these two squares?

src_imho_1996__Q04

Circumradii sum inequality for hexagon with parallel sides

Let $ABCDEF$ be a convex hexagon such that $AB$ is parallel to $DE$, $BC$ is parallel to $EF$, and $CD$ is parallel to $FA$. Let $R_A,R_C,R_E$ denote the circumradii of triangles $FAB$, $BCD$, $DEF$, respectively, and let $P$ denote the perimeter of the hexagon. Prove that $R_A+R_C+R_E\ge \frac{P}{2}.$

Topic: Geometria piana, Disuguaglianze Metodo: Trigonometria, Disuguaglianze Abilita: Ragionamento geometrico, Manipolazione algebrica, Modellizzazione Area: Geometria, Algebra e Analisi Fonte: apri PDF p.1

Circumradii sum inequality for hexagon with parallel sides

Let $ABCDEF$ be a convex hexagon such that $AB$ is parallel to $DE$, $BC$ is parallel to $EF$, and $CD$ is parallel to $FA$. Let $R_A,R_C,R_E$ denote the circumradii of triangles $FAB$, $BCD$, $DEF$, respectively, and let $P$ denote the perimeter of the hexagon. Prove that $R_A+R_C+R_E\ge \frac{P}{2}.$

src_imho_1996__Q05

Equal values in integer sequence with two-step increments

Let $p$, $q$, $n$ be three positive integers with $p+q<n$. Let $(x_0,x_1,\dots ,x_n)$ be an $(n+1)$-tuple of integers satisfying the following conditions:

(a) $x_0=x_n=0$.

(b) For each $i$ with $1\le i\le n$, either $x_i-x_{i-1}=p$ or $x_i-x_{i-1}=-q$.

Show that there exist indices $i<j$ with $(i,j)\ne (0,n)$, such that $x_i=x_j$.

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

Equal values in integer sequence with two-step increments

Let $p$, $q$, $n$ be three positive integers with $p+q<n$. Let $(x_0,x_1,\dots ,x_n)$ be an $(n+1)$-tuple of integers satisfying the following conditions:

(a) $x_0=x_n=0$.

(b) For each $i$ with $1\le i\le n$, either $x_i-x_{i-1}=p$ or $x_i-x_{i-1}=-q$.

Show that there exist indices $i<j$ with $(i,j)\ne (0,n)$, such that $x_i=x_j$.

src_imho_1996__Q06