(LUX) IMO1. A convex quadrilateral $ABCD$ has perpendicular diagonals. The perpendicular bisectors of $AB$ and $CD$ meet at a unique point $P$ inside $ABCD$. Prove that $ABCD$ is cyclic if and only if triangles $ABP$ and $CDP$ have equal areas.
(POL) Let $ABCD$ be a cyclic quadrilateral. Let $E$ and $F$ be variable points on the sides $AB$ and $CD$, respectively, such that $AE:EB=CF:FD$. Let $P$ be the point on the segment $EF$ such that $PE:PF=AB:CD$. Prove that the ratio between the areas of triangles $APD$ and $BPC$ does not depend on the choice of $E$ and $F$.
(UKR) IMO5. Let $I$ be the incenter of triangle $ABC$. Let $K,L$, and $M$ be the points of tangency of the incircle of $ABC$ with $AB,BC$, and $CA$, respectively. The line $t$ passes through $B$ and is parallel to $KL$. The lines $MK$ and $ML$ intersect $t$ at the points $R$ and $S$. Prove that $\angle RIS$ is acute.
(ARM) Let $M$ and $N$ be points inside triangle $ABC$ such that
$$\angle MAB=\angle NAC\qquad\angle MBA=\angle NBC.$$
Prove that
$$\frac{AM\cdot AN}{AB\cdot AC}+\frac{BM\cdot BN}{BA\cdot BC}+\frac{CM\cdot CN}{CA\cdot CB}=1.$$
(FRA) Let $ABC$ be a triangle, $H$ its orthocenter, $O$ its circumcenter, and $R$ its circumradius. Let $D$ be the reflection of $A$ across $BC$, $E$ that of $B$ across $CA$, and $F$ that of $C$ across $AB$. Prove that $D,E$, and $F$ are collinear if and only if $OH=2R$.
(POL) Let $ABCDEF$ be a convex hexagon such that $\angle B+\angle D+\angle F=360^\circ$ and
$$\frac{AB}{BC}\cdot\frac{CD}{DE}\cdot\frac{EF}{FA}=1.$$
Prove that
$$\frac{BC}{CA}\cdot\frac{AE}{EF}\cdot\frac{FD}{DB}=1.$$
(GBR) Let $ABC$ be a triangle such that $\angle ACB=2\angle ABC$. Let $D$ be the point on the side $BC$ such that $CD=2BD$. The segment $AD$ is extended to $E$ so that $AD=DE$. Prove that
$$\angle ECB+180^\circ=2\angle EBC.$$
(IND) Let $ABC$ be a triangle such that $\angle A=90^\circ$ and $\angle B\lt \angle C$. The tangent at $A$ to its circumcircle $\omega$ meets the line $BC$ at $D$. Let $E$ be the reflection of $A$ across $BC$, $X$ the foot of the perpendicular from $A$ to $BE$, and $Y$ the midpoint of $AX$. Let the line $BY$ meet $\omega$ again at $Z$. Prove that the line $BD$ is tangent to the circumcircle of triangle $ADZ$.
(MON) Let $a_1,a_2,\ldots,a_n$ be positive real numbers such that $a_1+a_2+\cdots+a_n\lt 1$. Prove that
$$\frac{a_1a_2\cdots a_n[1-(a_1+a_2+\cdots+a_n)]}{(a_1+a_2+\cdots+a_n)(1-a_1)(1-a_2)\cdots(1-a_n)}\le\frac1{n^{n+1}}.$$
(AUS) Let $r_1,r_2,\ldots,r_n$ be real numbers greater than or equal to $1$. Prove that
$$\frac1{r_1+1}+\frac1{r_2+1}+\cdots+\frac1{r_n+1}\ge\frac{n}{\sqrt[n]{r_1r_2\cdots r_n}+1}.$$
(RUS) Let $x,y$, and $z$ be positive real numbers such that $xyz=1$. Prove that
$$\frac{x^3}{(1+y)(1+z)}+\frac{y^3}{(1+z)(1+x)}+\frac{z^3}{(1+x)(1+y)}\ge\frac34.$$
(POL) Let $n\ge k\ge0$ be integers. The numbers $c(n,k)$ are defined as follows:
$$c(n,0)=c(n,n)=1\qquad(n\ge0);$$
$$c(n+1,k)=2^kc(n,k)+c(n,k-1)\qquad(n\ge k\ge1).$$
Prove that $c(n,k)=c(n,n-k)$ for all $n\ge k\ge0$.
(BUL) IMO6. Determine the least possible value of $f(1998)$, where $f$ is a function from the set $\mathbb N$ of positive integers into itself such that for all $m,n\in\mathbb N$,
$$f(n^2f(m))=m[f(n)]^2.$$
(AUS) Determine all pairs $(a,b)$ of real numbers such that $a\lfloor bn\rfloor=b\lfloor an\rfloor$ for all positive integers $n$. (Note that $\lfloor x\rfloor$ denotes the greatest integer less than or equal to $x$.)
(UKR) Determine the smallest integer $n\ge4$ for which one can choose four different numbers $a,b,c$, and $d$ from any $n$ distinct integers such that $a+b-c-d$ is divisible by $20$.
(GBR) A sequence of integers $a_1,a_2,a_3,\ldots$ is defined as follows: $a_1=1$, and for $n\ge1$, $a_{n+1}$ is the smallest integer greater than $a_n$ such that $a_i+a_j\ne3a_k$ for any $i,j,k$ in $\{1,2,\ldots,n+1\}$, not necessarily distinct. Determine $a_{1998}$.
(BLR) IMO3. For any positive integer $n$, let $\tau(n)$ denote the number of its positive divisors (including $1$ and itself). Determine all positive integers $m$ for which there exists a positive integer $n$ such that $\frac{\tau(n^2)}{\tau(n)}=m$.
(ARG) Prove that for each positive integer $n$, there exists a positive integer with the following properties:
(i) It has exactly $n$ digits.
(ii) None of the digits is $0$.
(iii) It is divisible by the sum of its digits.