IMO 1993 (XXXIV International Mathematical Olympiad)

Prove f(x) not product of two nonconstant integer polynomials

Let $f(x)=x^n+5x^{n-1}+3$, where $n>1$ is an integer. Prove that $f(x)$ cannot be expressed as the product of two nonconstant polynomials with integer coefficients.

Topic: Algebra, Teoria dei Numeri Metodo: Fattorizzazione, Congruenze Abilita: Manipolazione algebrica, Ragionamento geometrico, Lettura attenta Area: Algebra e Analisi, Aritmetica e Teoria dei Numeri Fonte: apri PDF p.1

Prove f(x) not product of two nonconstant integer polynomials

Let $f(x)=x^n+5x^{n-1}+3$, where $n>1$ is an integer. Prove that $f(x)$ cannot be expressed as the product of two nonconstant polynomials with integer coefficients.

src_imho_1993__Q01

Ratios and perpendicular circumcircles in acute triangle ABC

Let $D$ be a point inside acute triangle $ABC$ such that $\angle ADB=\angle ACB+90^{\circ }$ and $AC\cdot BD=AD\cdot BC$.

(a) Calculate the ratio $(AB\cdot CD)/(AC\cdot BD)$.

(b) Prove that the tangents at $C$ to the circumcircles of $△ACD$ and $△BCD$ are perpendicular.

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

Ratios and perpendicular circumcircles in acute triangles ABC

Let $D$ be a point inside acute triangle $ABC$ such that $\angle ADB=\angle ACB+90^{\circ }$ and $AC\cdot BD=AD\cdot BC$.

(a) Calculate the ratio $(AB\cdot CD)/(AC\cdot BD)$.

(b) Prove that the tangents at $C$ to the circumcircles of $△ACD$ and $△BCD$ are perpendicular.

src_imho_1993__Q02

Chessboard game: find n for one piece remaining

On an infinite chessboard, a game is played as follows. At the start, $n^2$ pieces are arranged on an $n\times n$ block of adjoining squares, one piece in each square. A move in the game is a jump in a horizontal or vertical direction over an adjacent occupied square to an unoccupied square immediately beyond. The piece which has been jumped over is removed.

Find those values of $n$ for which the game can end with only one piece remaining on the board.

Topic: Combinatoria Metodo: Invarianti, Colorazione, Casework Abilita: Riconoscimento di pattern, Modellizzazione, Conteggio sistematico Area: Combinatoria, Logica e Probabilita Fonte: apri PDF p.1

Chessboard game: find n for one piece remaining

On an infinite chessboard, a game is played as follows. At the start, $n^2$ pieces are arranged on a $n\times n$ block of adjoining squares, one piece in each square. A move in the game is a jump in a horizontal or vertical direction over an adjacent occupied square to an unoccupied square immediately beyond. The piece which has been jumped over is removed.

Find those values of $n$ for which the game can end with only one piece remaining on the board.

src_imho_1993__Q03

Triangle inequality with altitudes of triangle PQR

For three points $P$, $Q$, $R$ in the plane, we define $m(PQR)$ as the minimum length of the three altitudes of $△PQR$. (If the points are collinear, we set $m(PQR)=0$.)

Prove that for points $A$, $B$, $C$, $X$ in the plane, $m(ABC)\le m(ABX)+m(AXC)+m(XBC).$

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

Triangle inequality with altitudes of triangle PQR

For three points $P$, $Q$, $R$ in the plane, we define $m(PQR)$ as the minimum length of the three altitudes of $△PQR$. (If the points are collinear, we set $m(PQR)=0$.)

Prove that for points $A$, $B$, $C$, $X$ in the plane, $m(ABC)\le m(ABX)+m(AXC)+m(XBC).$

src_imho_1993__Q04

Find all functions f: N→N with two given conditions

Does there exist a function $f:\mathbf{N}\to \mathbf{N}$ such that $f(1)=2$, $f(f(n))=f(n)+n$ for all $n\in \mathbf{N}$, and $f(n)<f(n+1)$ for all $n\in \mathbf{N}$?

Topic: Insiemi e funzioni, Teoria dei Numeri Metodo: Ricorsione, Induzione, Invarianti Abilita: Manipolazione algebrica, Riconoscimento di pattern, Astrazione Area: Algebra e Analisi, Aritmetica e Teoria dei Numeri Fonte: apri PDF p.1

Find all functions f: N→N with two given conditions

Does there exist a function $f:\mathbf{N}\to \mathbf{N}$ such that $f(1)=2$, $f(f(n))=f(n)+n$ for all $n\in \mathbf{N}$, and $f(n)<f(n+1)$ for all $n\in \mathbf{N}$?

src_imho_1993__Q05

Circle of lamps: show all return on after M(n) steps

There are $n$ lamps $L_0,\dots ,L_{n-1}$ in a circle ($n>1$), where we denote $L_{n+k}=L_k$. (A lamp at all times is either on or off.) Perform steps $s_0,s_1,\dots$ as follows: at step $s_i$, if $L_{i-1}$ is lit, switch $L_i$ from on to off or vice versa, otherwise do nothing. Initially all lamps are on. Show that:

(a) There is a positive integer $M(n)$ such that after $M(n)$ steps all the lamps are on again;

(b) If $n=2^k$, we can take $M(n)=n^2-1$;

(c) If $n=2^k+1$, we can take $M(n)=n^2-n+1$.

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

Circle of lamps: show all return on after M(n) steps

There are $n$ lamps $L_0,\dots ,L_{n-1}$ in a circle ($n>1$), where we denote $L_{n+k}=L_k$. (A lamp at all times is either on or off.) Perform steps $s_0,s_1,\dots$ as follows: at step $s_i$, if $L_{i-1}$ is lit, switch $L_i$ from on to off or vice versa, otherwise do nothing. Initially all the lamps are on. Show that:

(a) There is a positive integer $M(n)$ such that after $M(n)$ steps all the lamps are on again;

(b) If $n=2^k$, we can take $M(n)=n^2-1$;

(c) If $n=2^k+1$, we can take $M(n)=n^2-n+1$.

src_imho_1993__Q06