IMO 1986 (XXVII International Mathematical Olympiad)

Show ab-1 not a perfect square for some pair in set

Let $d$ be any positive integer not equal to $2$, $5$, or $13$. Show that one can find distinct $a$, $b$ in the set $\{2,5,13,d\}$ such that $ab-1$ is not a perfect square.

Topic: Teoria dei Numeri Metodo: Casework, Fattorizzazione Abilita: Casework accurato, Manipolazione algebrica, Lettura attenta Area: Aritmetica e Teoria dei Numeri Fonte: apri PDF p.1

Show ab-1 not a perfect square for some pair in set

Let $d$ be any positive integer not equal to $2$, $5$, or $13$. Show that one can find distinct $a$, $b$ in the set $\{2,5,13,d\}$ such that $ab-1$ is not a perfect square.

src_imho_1986__Q01

Rotation sequence returning to start implies equilateral triangle

A triangle $A_1{A}_2{A}_3$ and a point $P_0$ are given in the plane. We define $A_s=A_{s-3}$ for all $s\ge 4$. We construct a set of points $P_1$, $P_2$, $P_3$, $\dots$, such that $P_{k+1}$ is the image of $P_k$ under a rotation with center $A_{k+1}$ through angle $120^{\circ }$ clockwise (for $k=0,1,2,\dots$). Prove that if $P_{1986}=P_0$, then the triangle $A_1{A}_2{A}_3$ is equilateral.

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

Rotation sequence returning to start implies equilateral triangle

A triangle $A_1{A}_2{A}_3$ and a point $P_0$ are given in the plane. We define $A_s=A_{s-3}$ for all $s\ge 4$. We construct a set of points $P_1$, $P_2$, $P_3$, $\dots$, such that $P_{k+1}$ is the image of $P_k$ under a rotation with center $A_{k+1}$ through angle $120^{\circ }$ clockwise (for $k=0,1,2,\dots$). Prove that if $P_{1986}=P_0$, then the triangle $A_1{A}_2{A}_3$ is equilateral.

src_imho_1986__Q02

Pentagon integer operation with negative number terminates

To each vertex of a regular pentagon an integer is assigned in such a way that the sum of all five numbers is positive. If three consecutive vertices are assigned the numbers $x$, $y$, $z$ respectively and $y<0$ then the following operation is allowed: the numbers $x$, $y$, $z$ are replaced by $x+y$, $-y$, $z+y$ respectively. Such an operation is performed repeatedly as long as at least one of the five numbers is negative. Determine whether this procedure necessarily comes to an end after a finite number of steps.

Topic: Combinatoria, Algebra Metodo: Invarianti, method_invarianti Abilita: Astrazione, Riconoscimento di pattern, Modellizzazione Area: Combinatoria, Logica e Probabilita, Algebra e Analisi Fonte: apri PDF p.1

Pentagon integer operation with negative number terminates

To each vertex of a regular pentagon an integer is assigned in such a way that the sum of all five numbers is positive. If three consecutive vertices are assigned the numbers $x$, $y$, $z$ respectively and $y<0$ then the following operation is allowed: the numbers $x$, $y$, $z$ are replaced by $x+y$, $-y$, $z+y$ respectively. Such an operation is performed repeatedly as long as at least one of the five numbers is negative. Determine whether this procedure necessarily comes to an end after a finite number of steps.

src_imho_1986__Q03

Locus of vertex X of triangle sliding around regular n-gon

Let $A$, $B$ be adjacent vertices of a regular $n$-gon ($n\ge 5$) in the plane having center at $O$. A triangle $XYZ$, which is congruent to and initially coincides with $OAB$, moves in the plane in such a way that $Y$ and $Z$ each trace out the whole boundary of the polygon, $X$ remaining inside the polygon. Find the locus of $X$.

Topic: Geometria piana Metodo: Coordinate, Simmetria Abilita: Ragionamento geometrico, Modellizzazione, Astrazione Area: Geometria Fonte: apri PDF p.1

Locus of vertex X of triangle sliding around regular n-gon

Let $A$, $B$ be adjacent vertices of a regular $n$-gon ($n\ge 5$) in the plane having center at $O$. A triangle $XYZ$, which is congruent to and initially coincides with $OAB$, moves in the plane in such a way that $Y$ and $Z$ each trace out the entire boundary of the polygon, $X$ remaining inside the polygon. Find the locus of $X$.

src_imho_1986__Q04

Find all functions on nonneg reals satisfying functional equation

Find all functions $f$, defined on the non-negative real numbers and taking non-negative real values, such that: \begin{itemize} \item[(i)] $f(xf(y))f(y)=f(x+y)$ for all $x,y\ge 0$, \item[(ii)] $f(2)=0$, \item[(iii)] $f(x)\ne 0$ for $0\le x<2$. \end{itemize}

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

Find all functions on nonneg real satisfying functional equation

Find all functions $f$, defined on the non-negative real numbers and taking non-negative real values, such that: \begin{itemize} \item[(i)] $f(xf(y))f(y)=f(x+y)$ for all $x,y\ge 0$, \item[(ii) $f(2)=0$, \item[(iii)] $f(x)\ne 0$ for $0\le x<2$. I’m going to tell you.

src_imho_1986__Q05

Color integer-coordinate points so axis-parallel lines differ by at most 1

One is given a finite set of points in the plane, each point having integer coordinates. Is it always possible to color some of the points in the set red and the remaining points white in such a way that for any straight line $L$ parallel to either one of the coordinate axes the difference (in absolute value) between the numbers of white points and red points on $L$ is not greater than $1$?

Topic: Combinatoria Metodo: Biiezione, Grafi, Colorazione Abilita: Astrazione, Modellizzazione, Ragionamento geometrico Area: Combinatoria, Logica e Probabilita Fonte: apri PDF p.1

Color integer-coordinate points so axis-parallel lines differ by at most 1

One is given a finite set of points in the plane, each point having integer coordinates. Is it always possible to color some of the points in the set red and the remaining points white in such a way that for any straight line $L$ parallel to either of the coordinate axes the difference (in absolute value) between the numbers of white points and red points on $L$ is not greater than $1$?

src_imho_1986__Q06