(PHI 3) Let $ABC$ be any triangle and $P$ any point in its interior. Let $P_1,P_2$ be the feet of the perpendiculars from $P$ to the two sides $AC$ and $BC$. Draw $AP$ and $BP$, and from $C$ drop perpendiculars to $AP$ and $BP$. Let $Q_1$ and $Q_2$ be the feet of these perpendiculars. Prove that the lines $Q_1P_2$, $Q_2P_1$, and $AB$ are concurrent.
(JAP 5) For an acute triangle $ABC$, $M$ is the midpoint of the segment $BC$, $P$ is a point on the segment $AM$ such that $PM=BM$, $H$ is the foot of the perpendicular line from $P$ to $BC$, $Q$ is the point of intersection of segment $AB$ and the line passing through $H$ that is perpendicular to $PB$, and finally, $R$ is the point of intersection of the segment $AC$ and the line passing through $H$ that is perpendicular to $PC$.
Show that the circumcircle of $\triangle QHR$ is tangent to the side $BC$ at point $H$.
(PRK 1) Let $S$ be any point on the circumscribed circle of $\triangle PQR$. Then the feet of the perpendiculars from $S$ to the three sides of the triangle lie on the same straight line. Denote this line by $l(S,PQR)$. Suppose that the hexagon $ABCDEF$ is inscribed in a circle. Show that the four lines $l(A,BDF)$, $l(B,ACE)$, $l(D,ABF)$, and $l(E,ABC)$ intersect at one point if and only if $CDEF$ is a rectangle.
(FRA 2) IMO5 Let $ABC$ be a triangle and $M$ an interior point in $ABC$. Show that at least one of the angles $\angle MAB$, $\angle MBC$, and $\angle MCA$ is less than or equal to $30^\circ$.
(SPA 4) In the triangle $ABC$, with $\angle A=60^\circ$, a parallel $IF$ to $AC$ is drawn through the incenter $I$ of the triangle, where $F$ lies on the side $AB$. The point $P$ on the side $BC$ is such that $3BP=BC$. Show that $\angle BFP=\angle B/2$.
(USS 4) IMO1 Prove for each triangle $ABC$ the inequality
$$\frac14\lt \frac{IA\cdot IB\cdot IC}{l_A l_B l_C}\le\frac8{27},$$
where $I$ is the incenter and $l_A,l_B,l_C$ are the lengths of the angle bisectors of $ABC$.
(CHN 2) Let $O$ be the center of the circumsphere of a tetrahedron $ABCD$. Let $L,M,N$ be the midpoints of $BC,CA,AB$ respectively, and assume that $AB+BC=AD+CD$, $BC+CA=BD+AD$, and $CA+AB=CD+BD$. Prove that $\angle LOM=\angle MON=\angle NOL$.
(NET 1) Let $S$ be a set of $n$ points in the plane. No three points of $S$ are collinear. Prove that there exists a set $P$ containing $2n-5$ points satisfying the following condition: In the interior of every triangle whose three vertices are elements of $S$ lies a point that is an element of $P$.
(FRA 3) In the plane we are given a set $E$ of 1991 points, and certain pairs of these points are joined with a path. We suppose that for every point of $E$, there exist at least 1593 other points of $E$ to which it is joined by a path. Show that there exist six points of $E$ every pair of which are joined by a path.
Alternative version. Is it possible to find a set $E$ of 1991 points in the plane and paths joining certain pairs of the points in $E$ such that every point of $E$ is joined with a path to at least 1592 other points of $E$, and in every subset of six points of $E$ there exist at least two points that are not joined?
(USA 5) IMO4 Suppose $G$ is a connected graph with $n$ edges. Prove that it is possible to label the edges of $G$ from 1 to $n$ in such a way that in every vertex $v$ of $G$ with two or more incident edges, the set of numbers labeling those edges has no common divisor greater than 1.
(CHN 3) IMO3 Let $S=\{1,2,3,\ldots,280\}$. Find the minimal natural number $n$ such that in any $n$-element subset of $S$ there are five numbers that are pairwise relatively prime.
(POL 4) Given any integer $n\ge2$, assume that the integers $a_1,a_2,\ldots,a_n$ are not divisible by $n$ and, moreover, that $n$ does not divide $a_1+a_2+\cdots+a_n$. Prove that there exist at least $n$ different sequences $(e_1,e_2,\ldots,e_n)$ consisting of zeros or ones such that $e_1a_1+e_2a_2+\cdots+e_na_n$ is divisible by $n$.
(POL 3) Let $a,b,c$ be integers and $p$ an odd prime number. Prove that if $f(x)=ax^2+bx+c$ is a perfect square for $2p-1$ consecutive integer values of $x$, then $p$ divides $b^2-4ac$.
(USS 2) Let $a_n$ be the last nonzero digit in the decimal representation of the number $n!$. Does the sequence $a_1,a_2,\ldots,a_n,\ldots$ become periodic after a finite number of terms?
(ROM 1) IMO2 Let $n>6$ and $a_1\lt a_2\lt \cdots\lt a_k$ be all natural numbers that are less than $n$ and relatively prime to $n$. Show that if $a_1,a_2,\ldots,a_k$ is an arithmetic progression, then $n$ is a prime number or a natural power of two.
(IRE 5) Let $a$ be a rational number with $0\lt a\lt 1$ and suppose that
$$\cos3\pi a+2\cos2\pi a=0.$$
(Angle measurements are in radians.) Prove that $a=2/3$.
(IRE 3) Let $\alpha$ be the positive root of the equation $x^2=1991x+1$. For natural numbers $m,n$ define
$$m*n=mn+[\alpha m][\alpha n],$$
where $[x]$ is the greatest integer not exceeding $x$. Prove that for all natural numbers $p,q,r$,
$$(p*q)*r=p*(q*r).$$