IMO 1995

Official IMO 1995 problems  ·  0/6 solved.

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

Problem 3

Determine 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 $1\le i<j<k\le n$, the area of $\triangle A_iA_jA_k$ is $r_i+r_j+r_k$.

Problem 4

The positive real numbers $x_0, x_1, x_2,.....x_{1994}, x_{1995}$ satisfy the relations

$x_0=x_{1995}$ and $x_{i-1}+\frac{2}{x_{i-1}}=2{x_i}+\frac{1}{x_i}$

for $i=1,2,3,....1995$

Find the maximum value that $x_0$ can have.

Problem 5

Let $ABCDEF$ be a convex hexagon with $AB=BC=CD$ and $DE=EF=FA$, such that $\angle BCD=\angle EFA=\frac{\pi}{3}$. Suppose $G$ and $H$ are points in the interior of the hexagon such that $\angle AGB=\angle DHE=\frac{2\pi}{3}$. 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 ${1,2,\ldots,2p}$ are there, the sum of whose elements is divisible by $p$?