IMO 1996 (37th IMO, Mumbai)

Knight-like moves of distance sqrt(r) across board corners

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 × 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 √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: parita, congruenze Area: Aritmetica e Teoria dei Numeri, Combinatoria, Logica e Probabilita Fonte: apri PDF p.1

Knight-like moves of distance sqrt(r) across board corners

We are given a positive integer r and a rectangular board ABCD with dimensions =AB = 20, The rectangle is divided into a grid of 20 × 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 √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_imo_1996__Q01

Show AP, BD, CE concurrent with incenter angle condition

Let P be a point inside triangle ABC such that ̸ APB −̸ ACB = ̸ APC −̸ 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: Sfruttamento della simmetria, Tecniche trigonometriche Abilita: Ragionamento geometrico Area: Geometria Fonte: apri PDF p.1

Show AP, BD, CE concurrent with incenter angle condition

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

src_imo_1996__Q02

Find all f with f(m+f(n))=f(f(m))+f(n)

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) ∀m, n ∈S. 37th International Mathematical Olympiad Mumbai, India Day II 9 a.m. - 1:30 p.m. July 11, 1996

Topic: successioni Metodo: monovarianti Abilita: generalizzazione Area: Algebra e Analisi Fonte: apri PDF p.1

*Find all f with f(m+f(n))=f(f(m))+f(n) *

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) ∀m, n ∈S. 37th International Mathematical Olympiad Mumbai, India Day II at 9 am - 1:30 p.m. July 11, 1996

src_imo_1996__Q03

Least square value with 15a+16b and 16a-15b both squares

I numeri interi positivi $a$ e $b$ sono tali che i numeri $15a+16b$ e $16a-15b$ siano entrambi quadrati di numeri interi positivi. Qual è il minimo valore possibile che può assumere il minore di questi due quadrati?

Topic: Teoria dei Numeri Metodo: congruenze Area: Aritmetica e Teoria dei Numeri Fonte: apri PDF p.2

Least square value with 15a+16b and 16a-15b both squares

The positive integers $a$ and $b$ are such that the numbers $15a+16b$ and $16a-15b$ are both squares of positive integers. What’s the smallest possible value that the least of these two squares can take?

src_imo_1996__Q04

circumradii sum at least half perimeter

Sia $ABCDEF$ un esagono convesso tale che $AB$ è parallelo a $DE$, $BC$ è parallelo a $EF$, e $CD$ è parallelo a $FA$. Siano $R_A,R_C,R_E$ i raggi dei cerchi circoscritti ai triangoli $FAB$, $BCD$, $DEF$, rispettivamente, e sia $P$ il perimetro dell’esagono. Dimostrare che $R_A+R_C+R_E\ge \frac{P}{2}.$

Topic: Geometria piana, Disuguaglianze Metodo: Disuguaglianze classiche Abilita: Ragionamento geometrico Area: Algebra e Analisi, Geometria Fonte: apri PDF p.2

circumradii sum at least half perimeter

If $ABCDEF$ is a convex hexagon such that $AB$ is parallel to $DE$, $BC$ is parallel to $EF$, and $CD$ is parallel to $FA$. The radii of the circles surrounding the triangles $FAB$, $BCD$, $DEF$, respectively, and $P$ are the perimeter of the hexagon. Show that $R_A+R_C+R_E\ge \frac{P}{2}.$

src_imo_1996__Q05

Lattice-path tuple has two indices with equal values

Siano $p,q,n$ tre numeri interi positivi con $p+q<n$. Sia $(x_0,x_1,\dots ,x_n)$ una $(n+1)$-upla di interi che soddisfa le seguenti condizioni:

(a) $x_0=x_n=0$.

(b) Per ogni $i$ con $1\le i\le n$, si ha $x_i-x_{i-1}=p$ oppure $x_i-x_{i-1}=-q$.

Dimostrare che esistono indici $i<j$ con $(i,j)\ne (0,n)$, tali che $x_i=x_j$.

Topic: Combinatoria Metodo: Principio dei cassetti Area: Combinatoria, Logica e Probabilita Fonte: apri PDF p.2

Lattice-path tuple has two indices with equal values

Three positive integers with $p+q<n$ are $p,q,n$. Whether $(x_0,x_1,\dots ,x_n)$ a $(n+1)$-upple of integers that satisfies the following conditions:

(a) $x_0=x_n=0$.

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

Demonstrate the existence of $i<j$ indexes with $(i,j)\ne (0,n)$, such as $x_i=x_j$.

src_imo_1996__Q06