International Mathematical Olympiad · 1995

Problems

Problem 1Let $A,B,C,D$ 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 $AM$ , $DN$ , and $XY$ are concurrent.Solutions: 1
Problem 2Let $a,b,c$ be positive real numbers with $abc=1$ . Prove that $\frac1{a^3(b+c)}+\frac1{b^3(c+a)}+\frac1{c^3(a+b)}\ge\frac32$ .Solutions: 1
Problem 3Determine all integers $n>3$ for which there exist $n$ points $A_1,\ldots,A_n$ in the plane, no three collinear, and real numbers $r_1,\ldots,r_n$ such that for any distinct $i,j,k$ , the area of triangle $A_iA_jA_k$ is $r_i+r_j+r_k$ .Solutions: 1
Problem 4Find the maximum value of $x_0$ for which there exists a sequence $x_0,x_1,\ldots,x_{1995}$ of positive reals with $x_0=x_{1995}$ such that for $i=1,\ldots,1995$ , $x_{i-1}+\frac2{x_{i-1}}=2x_i+\frac1{x_i}$ .Solutions: 1
Problem 5Let $ABCDEF$ be a convex hexagon with $AB=BC=CD$ and $DE=EF=FA$ , such that $\angle BCD=\angle EFA=60^\circ$ . Suppose that $G$ and $H$ are points in the interior of the hexagon such that $\angle AGB=\angle DHE=120^\circ$ . Prove that $AG+GB+GH+DH+HE\ge CF$ .Solutions: 1
Problem 6Let $p$ be an odd prime number. How many $p$ -element subsets $A$ of $\{1,2,\ldots,2p\}$ are there, the sum of whose elements is divisible by $p$ ?Solutions: 1