IMO 1992 - Shortlisted problems

(AUS 2) Prove that for any positive integer $m$ there exist an infinite number of pairs of integers $(x,y)$ such that (i) $x$ and $y$ are relatively prime; (ii) $y$ divides $x^2+m$; (iii) $x$ divides $y^2+m$.

(CHN 1) Let $\mathbb R^+$ be the set of all nonnegative real numbers. Given two positive real numbers $a$ and $b$, suppose that a mapping $f:\mathbb R^+\to\mathbb R^+$ satisfies the functional equation
$$f(f(x))+af(x)=b(a+b)x.$$
Prove that there exists a unique solution of this equation.

(CHN 2) The diagonals of a quadrilateral $ABCD$ are perpendicular: $AC\perp BD$. Four squares, $ABEF$, $BCGH$, $CDIJ$, $DAKL$, are erected externally on its sides. The intersection points of the pairs of straight lines $CL,DF$; $DF,AH$; $AH,BJ$; $BJ,CL$ are denoted by $P_1,Q_1,R_1,S_1$, respectively, and the intersection points of the pairs of straight lines $AI,BK$; $BK,CE$; $CE,DG$; $DG,AI$ are denoted by $P_2,Q_2,R_2,S_2$, respectively. Prove that $P_1Q_1R_1S_1\cong P_2Q_2R_2S_2$.

(CHN 3) IMO3. Given nine points in space, no four of which are coplanar, find the minimal natural number $n$ such that for any coloring with red or blue of $n$ edges drawn between these nine points there always exists a triangle having all edges of the same color.

(COL 3) Let $ABCD$ be a convex quadrilateral such that $AC=BD$. Equilateral triangles are constructed on the sides of the quadrilateral. Let $O_1,O_2,O_3,O_4$ be the centers of the triangles constructed on $AB,BC,CD,DA$ respectively. Show that $O_1O_3$ is perpendicular to $O_2O_4$.

(IND 2) IMO2. Find all functions $f:\mathbb R\to\mathbb R$ such that
$$f(x^2+f(y))=y+f(x)^2$$
for all $x,y\in\mathbb R$.

(IND 4) Circles $G,G_1,G_2$ are three circles related to each other as follows: Circles $G_1$ and $G_2$ are externally tangent to one another at a point $W$ and both these circles are internally tangent to the circle $G$. Points $A,B,C$ are located on the circle $G$ as follows: Line $BC$ is a direct common tangent to the pair of circles $G_1$ and $G_2$, and line $WA$ is the transverse common tangent at $W$ to $G_1$ and $G_2$, with $W$ and $A$ lying on the same side of the line $BC$. Prove that $W$ is the incenter of the triangle $ABC$.

(IND 5) Show that in the plane there exists a convex polygon of $1992$ sides satisfying the following conditions:
(i) its side lengths are $1,2,3,\ldots,1992$ in some order;
(ii) the polygon is circumscribable about a circle.

Alternative formulation. Does there exist a $1992$-gon with side lengths $1,2,3,\ldots,1992$ circumscribed about a circle? Answer the same question for a $1990$-gon.

(IRN 1) Let $f(x)$ be a polynomial with rational coefficients and $\alpha$ be a real number such that $\alpha^3-\alpha=f(\alpha)^3-f(\alpha)=33^{1992}$. Prove that for each $n\ge1$,
$$(f^{(n)}(\alpha))^3-f^{(n)}(\alpha)=33^{1992},$$
where $f^{(n)}(x)=f(f(\ldots f(x)))$, and $n$ is a positive integer.

(ITA 1) IMO5. Let $V$ be a finite subset of Euclidean space consisting of points $(x,y,z)$ with integer coordinates. Let $S_1,S_2,S_3$ be the projections of $V$ onto the $yz,xz,xy$ planes, respectively. Prove that
$$|V|^2\le |S_1||S_2||S_3|$$
($|X|$ denotes the number of elements of $X$).

(JAP 2) In a triangle $ABC$, let $D$ and $E$ be the intersections of the bisectors of $\angle ABC$ and $\angle ACB$ with the sides $AC,AB$, respectively. Determine the angles $\angle A,\angle B,\angle C$ if
$$\angle BDE=24^\circ,\qquad\angle CED=18^\circ.$$

(NET 1) Let $f,g$, and $a$ be polynomials with real coefficients, $f$ and $g$ in one variable and $a$ in two variables. Suppose
$$f(x)-f(y)=a(x,y)(g(x)-g(y))$$
for all $x,y\in\mathbb R$.
Prove that there exists a polynomial $h$ with $f(x)=h(g(x))$ for all $x\in\mathbb R$.

(NZL 1) IMO1. Find all integer triples $(p,q,r)$ such that $1\lt p\lt q\lt r$ and $(p-1)(q-1)(r-1)$ is a divisor of $(pqr-1)$.

(POL 2) For any positive integer $x$ define $g(x)$ to be the greatest odd divisor of $x$; $f(x)=x/2+x/g(x)$ if $x$ is even, and $f(x)=2^{(x+1)/2}$ if $x$ is odd.
Construct the sequence $x_1=1$, $x_{n+1}=f(x_n)$. Show that the number $1992$ appears in this sequence, determine the least $n$ such that $x_n=1992$, and determine whether $n$ is unique.

(PRK 2) Does there exist a set $M$ with the following properties?
(i) The set $M$ consists of $1992$ natural numbers.
(ii) Every element in $M$ and the sum of any number of elements have the form $m^k$ ($m,k\in\mathbb N$, $k\ge2$).

(KOR 2) Prove that $N=\frac{5^{125}-1}{5^{25}-1}$ is a composite number.

(SWE 1) Let $\alpha(n)$ be the number of digits equal to one in the binary representation of a positive integer $n$. Prove that:
(a) the inequality $\alpha(n^2)\le\frac12\alpha(n)(\alpha(n)+1)$ holds;
(b) the above inequality is an equality for infinitely many positive integers;
(c) there exists a sequence $(n_i)_1^\infty$ such that $\alpha(n_i^2)/\alpha(n_i)\to0$ as $i\to\infty$.
Alternative parts: Prove that there exists a sequence $(n_i)_1^\infty$ such that $\alpha(n_i^2)/\alpha(n_i)$ tends to
(d) $\infty$;
(e) an arbitrary real number $\gamma\in(0,1)$;
(f) an arbitrary real number $\gamma\ge0$.

(USA 2) Let $[x]$ denote the greatest integer less than or equal to $x$. Pick any $x_1$ in $[0,1)$ and define the sequence $x_1,x_2,x_3,\ldots$ by $x_{n+1}=0$ if $x_n=0$ and $x_{n+1}=1/x_n-[1/x_n]$ otherwise. Prove that
$$x_1+x_2+\cdots+x_n\lt \frac{F_1}{F_2}+\frac{F_2}{F_3}+\cdots+\frac{F_n}{F_{n+1}},$$
where $F_1=F_2=1$ and $F_{n+2}=F_{n+1}+F_n$ for $n\ge1$.

(IRE 1) Let $f(x)=x^8+4x^6+2x^4+28x^2+1$. Let $p\gt 3$ be a prime and suppose there exists an integer $z$ such that $p$ divides $f(z)$. Prove that there exist integers $z_1,z_2,\ldots,z_8$ such that if
$$g(x)=(x-z_1)(x-z_2)\cdots(x-z_8),$$
then all coefficients of $f(x)-g(x)$ are divisible by $p$.

(FRA 1) IMO4. In the plane, let there be given a circle $C$, a line $l$ tangent to $C$, and a point $M$ on $l$. Find the locus of points $P$ that have the following property: There exist two points $Q$ and $R$ on $l$ such that $M$ is the midpoint of $QR$ and $C$ is the incircle of $PQR$.