IMO 1993 (34th IMO)

Prove x^n+5x^(n-1)+3 is irreducible over integers

Let f(x) = xn + 5xn−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 Metodo: manipolazione algebrica Area: Algebra e Analisi Fonte: apri PDF p.1

Prove x^n+5x^(n-1) +3 is irreducible over integers

Let f(x) = xn + 5xn−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_imo_1993__Q01

Compute ratio and prove tangents perpendicular for interior D

Let D be a point inside acute triangle ABC such that ̸ ADB = ̸ ACB + π/2 and AC · BD = AD · BC. (a) Calculate the ratio (AB · CD)/(AC · BD). (b) Prove that the tangents at C to the circumcircles of △ACD and △BCD are perpendicular.

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

Compute ratio and prove tangents perpendicular to interior D

Let D be a point inside acute triangle ABC such that ADB = ACB + π/2 and AC · BD = AD · BC. (a) Calculate the ratio (AB · CD)/(AC · BD). (b) Prove that the tangents at C to the circumcircles of △ACD and △BCD are perpendicular.

src_imo_1993__Q02

which n leave one piece

On an infinite chessboard, a game is played as follows. At the start, n2 pieces are arranged on the chessboard in an n by 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. Second Day July 19, 1993 Time Limit: 41 2 hours

Topic: Combinatoria Metodo: parita, monovarianti Area: Combinatoria, Logica e Probabilita Fonte: apri PDF p.1

which n leave one piece

On an infinite chessboard, a game is played as follows. At the start, n2 pieces are arranged on the chessboard in an n by 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. Second Day July 19, 1993 Time limit: 41 2 hours

src_imo_1993__Q03

Minimum-altitude function triangle inequality for four points

Per tre punti $P$, $Q$, $R$ nel piano, sia $m(PQR)$ la lunghezza minima delle tre altezze del triangolo $△PQR$. (Se i punti sono collineari, poniamo $m(PQR)=0$.)

Dimostrare che per i punti $A$, $B$, $C$, $X$ nel piano vale: $m(ABC)\le m(ABX)+m(AXC)+m(XBC).$

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

Minimum-altitude function triangle inequality for four points

For three points $P$, $Q$, $R$ in the plane, the minimum length of the three heights of the triangle $△PQR$ shall be $m(PQR)$. (If the points are hilly, we put $m(PQR)=0$.)

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

src_imo_1993__Q04

Existence of f with f(1)=2, f(f(n))=f(n)+n, increasing

Esiste una funzione $f:\mathbf{N}\to \mathbf{N}$ tale che $f(1)=2,f(f(n))=f(n)+n\text{ per ogni }n\in \mathbf{N},f(n)<f(n+1)\text{ per ogni }n\in \mathbf{N}?$

Topic: successioni Metodo: Ricorsione Area: Algebra e Analisi Fonte: apri PDF p.1

Existence of f with f(1)=2, f(f(n))=f(n)+n, increasing

There is a function $f:\mathbf{N}\to \mathbf{N}$ such that $f(1)=2,f(f(n))=f(n)+n\text{ per ogni }n\in \mathbf{N},f(n)<f(n+1)\text{ per ogni }n\in \mathbf{N}?$

src_imo_1993__Q05

return-to-all-on step counts

Ci sono $n$ lampade $L_0,\dots ,L_{n-1}$ disposte in cerchio ($n>1$), dove si intende $L_{n+k}=L_k$. (Una lampada è sempre accesa o spenta.) Si eseguono i passi $s_0,s_1,\dots$ come segue: al passo $s_i$, se $L_{i-1}$ è accesa, si commuta $L_i$ (da accesa a spenta o viceversa); altrimenti non si fa nulla. Inizialmente tutte le lampade sono accese. Dimostrare che:

(a) Esiste un intero positivo $M(n)$ tale che dopo $M(n)$ passi tutte le lampade siano di nuovo accese;

(b) Se $n=2^k$, si può prendere $M(n)=n^2-1$;

(c) Se $n=2^k+1$, si può prendere $M(n)=n^2-n+1$.

Topic: Combinatoria Metodo: monovarianti Abilita: Riconoscimento di pattern Area: Combinatoria, Logica e Probabilita Fonte: apri PDF p.2

This is the total number of steps to be taken to achieve the desired results.

There are $n$ lamps $L_0,\dots ,L_{n-1}$ arranged in a circle ($n>1$), where $L_{n+k}=L_k$ is meant. (A lamp is always switched on or off.) The steps $s_0,s_1,\dots$ are performed as follows: at step $s_i$, if $L_{i-1}$ is switched on, switch to $L_i$ (from switched on or vice versa); otherwise nothing is done. Initially, all the lamps are lit. Demonstrate that:

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

(b) If $n=2^k$, $M(n)=n^2-1$ may be taken;

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

src_imo_1993__Q06