IMO 1995

Problem 1. Let $A,B,C,D$ be four distinct points on a line, in that order. The circles with diameters $AC$ and $BD$ intersect at $X$ and $Y$. The line $XY$ meets $BC$ at $Z$. Let $P$ be a point on the line $XY$ other than $Z$. The line $CP$ intersects the circle with diameter $AC$ at $C$ and $M$, and the line $BP$ intersects the circle with diameter $BD$ at $B$ and $N$. Prove that the lines $AM,DN,XY$ are concurrent.

Problem 2. Let $a,b,c$ be positive real numbers such that $abc=1$. Prove that

$\frac{1}{a^3(b+c)}+\frac{1}{b^3(c+a)}+\frac{1}{c^3(a+b)}\ge \frac{3}{2}.$

Problem 3. Determine all integers $n>3$ for which there exist $n$ points $A_1,\dots ,A_n$ in the plane, no three collinear, and real numbers $r_1,\dots ,r_n$ such that for $1\le i<j<k\le n$, the area of $△A_i{A}_j{A}_k$ is $r_i+r_j+r_k$.

Problem 4. Find the maximum value of $x_0$ for which there exists a sequence $x_0,x_1\dots ,x_{1995}$ of positive reals with $x_0=x_{1995}$, such that for $i=1,\dots ,1995$,

$x_{i-1}+\frac{2}{x_{i-1}}=2x_i+\frac{1}{x_i}.$

Problem 5. Let $ABCDEF$ be a convex hexagon with $AB=BC=CD$ and $DE=EF=FA$, such that $\angle BCD=\angle EFA=\pi /3$. Suppose $G$ and $H$ are points in the interior of the hexagon such that $\angle AGB=\angle DHE=2\pi /3$. Prove that $AG+GB+GH+DH+HE\ge CF$.

Problem 6. Let $p$ be an odd prime number. How many $p$-element subsets $A$ of $\{1,2,\dots 2p\}$ are there, the sum of whose elements is divisible by $p$?