(AUS 2) IMO2 Let $ABC$ be a triangle. The bisector of angle $A$ meets the circumcircle of triangle $ABC$ in $A_1$. Points $B_1$ and $C_1$ are defined similarly. Let $AA_1$ meet the lines that bisect the two external angles at $B$ and $C$ in point $A^0$. Define $B^0$ and $C^0$ similarly. If $S_{X_1X_2\ldots X_n}$ denotes the area of the polygon $X_1X_2\ldots X_n$, prove that
$$S_{A^0B^0C^0}=2S_{AC_1BA_1CB_1}\ge4S_{ABC}.$$
(AUS 3) Ali Barber, the carpet merchant, has a rectangular piece of carpet whose dimensions are unknown. Unfortunately, his tape measure is broken and he has no other measuring instruments. However, he finds that if he lays it flat on the floor of either of his storerooms, then each corner of the carpet touches a different wall of that room. If the two rooms have dimensions of 38 feet by 55 feet and 50 feet by 55 feet, what are the carpet dimensions?
(AUS 4) Ali Barber, the carpet merchant, has a rectangular piece of carpet whose dimensions are unknown. Unfortunately, his tape measure is broken and he has no other measuring instruments. However, he finds that if he lays it flat on the floor of either of his storerooms, then each corner of the carpet touches a different wall of that room. He knows that the sides of the carpet are integral numbers of feet and that his two storerooms have the same (unknown) length, but widths of 38 feet and 50 feet respectively. What are the carpet dimensions?
(BUL 3) Prove that for every integer $n\gt 1$ the equation
$$\frac{x^n}{n!}+\frac{x^{n-1}}{(n-1)!}+\cdots+\frac{x^2}{2!}+\frac{x}{1!}+1=0$$
has no rational roots.
(COL 1) Consider the polynomial $p(x)=x^n+nx^{n-1}+a_2x^{n-2}+\cdots+a_n$ having all real roots. If $r_1^{16}+r_2^{16}+\cdots+r_n^{16}=n$, where the $r_j$ are the roots of $p(x)$, find all such roots.
(CZS 1) For a triangle $ABC$, let $k$ be its circumcircle with radius $r$. The bisectors of the inner angles $A$, $B$, and $C$ of the triangle intersect respectively the circle $k$ again at points $A'$, $B'$, and $C'$. Prove the inequality
$$16Q^3\ge27r^4P,$$
where $Q$ and $P$ are the areas of the triangles $A'B'C'$ and $ABC$ respectively.
(FIN 1) Show that any two points lying inside a regular $n$-gon $E$ can be joined by two circular arcs lying inside $E$ and meeting at an angle of at least $\left(1-\frac2n\right)\pi$.
(FRA 2) Let $R$ be a rectangle that is the union of a finite number of rectangles $R_i$, $1\le i\le n$, satisfying the following conditions:
(i) The sides of every rectangle $R_i$ are parallel to the sides of $R$.
(ii) The interiors of any two different $R_i$ are disjoint.
(iii) Every $R_i$ has at least one side of integral length.
Prove that $R$ has at least one side of integral length.
(FRA 4) For all integers $n$, $n\ge0$, there exist uniquely determined integers $a_n,b_n,c_n$ such that
$$\left(1+4\sqrt[3]2-4\sqrt[3]4\right)^n=a_n+b_n\sqrt[3]2+c_n\sqrt[3]4.$$
Prove that $c_n=0$ implies $n=0$.
(GRE 3) Let $g:\mathbb C\to\mathbb C$, $w\in\mathbb C$, $a\in\mathbb C$, $w^3=1$ $(w\ne1)$. Show that there is one and only one function $f:\mathbb C\to\mathbb C$ such that
$$f(z)+f(wz+a)=g(z),\qquad z\in\mathbb C.$$
Find the function $f$.
(HUN 3) At $n$ distinct points of a circular race course there are $n$ cars ready to start. Each car moves at a constant speed and covers the circle in an hour. On hearing the initial signal, each of them selects a direction and starts moving immediately. If two cars meet, both of them change directions and go on without loss of speed.
Show that at a certain moment each car will be at its starting point.
(ICE 3) IMO4 The quadrilateral $ABCD$ has the following properties:
(i) $AB=AD+BC$;
(ii) there is a point $P$ inside it at a distance $x$ from the side $CD$ such that $AP=x+AD$ and $BP=x+BC$.
Show that
$$\frac1{\sqrt x}\ge\frac1{\sqrt{AD}}+\frac1{\sqrt{BC}}.$$
(IND 2) A bicentric quadrilateral is one that is both inscribable in and circumscribable about a circle. Show that for such a quadrilateral, the centers of the two associated circles are collinear with the point of intersection of the diagonals.
(IRE 1) Let $a,b,c,d,m,n$ be positive integers such that $a^2+b^2+c^2+d^2=1989$, $a+b+c+d=m^2$, and the largest of $a,b,c,d$ is $n^2$. Determine, with proof, the values of $m$ and $n$.
(ISR 1) The set $\{a_0,a_1,\ldots,a_n\}$ of real numbers satisfies the following conditions:
(i) $a_0=a_n=0$;
(ii) for $1\le k\le n-1$,
$$a_k=c+\sum_{i=k}^{n-1}a_{i-k}(a_i+a_{i+1}).$$
Prove that $c\le\frac1{4n}$.
(MON 1) Given seven points in the plane, some of them are connected by segments so that:
(i) among any three of the given points, two are connected by a segment;
(i) the number of segments is minimal.
How many segments does a figure satisfying (i) and (ii) contain? Give an example of such a figure.
(MON 4) Given a convex polygon $A_1A_2\ldots A_n$ with area $S$, and a point $M$ in the same plane, determine the area of polygon $M_1M_2\ldots M_n$, where $M_i$ is the image of $M$ under rotation $R_{A_i}^{\alpha}$ around $A_i$ by $\alpha$, $i=1,2,\ldots,n$.
(MON 6) A positive integer is written in each square of an $m\times n$ board. The allowed move is to add an integer $k$ to each of two adjacent numbers in such a way that no negative numbers are obtained. (Two squares are adjacent if they have a common side.) Find a necessary and sufficient condition for it to be possible for all the numbers to be zero by a finite sequence of moves.
(NET 1) IMO3 Given a set $S$ in the plane containing $n$ points and satisfying the conditions:
(i) no three points of $S$ are collinear,
(ii) for every point $P$ of $S$ there exist at least $k$ points in $S$ that have the same distance to $P$,
prove that the following inequality holds:
$$k\lt \frac12+\sqrt{2n}.$$