PL-OM 1989

An even number of politicians are sitting at a round table. After a break, they come back and sit down again in arbitrary places. Show that there must be two people with the same number of people sitting between them as before the break..

Additional problem: Solve the problem when the number of people is in a form $6k+3$ .

$k_1, k_2, k_3$ are three circles. $k_2$ and $k_3$ touch externally at $P$ , $k_3$ and $k_1$ touch externally at $Q$ , and $k_1$ and $k_2$ touch externally at $R$ . The line $PQ$ meets $k_1$ again at $S$ , the line $PR$ meets $k_1$ again at $T$ . The line $RS$ meets $k_2$ again at $U$ , and the line $QT$ meets $k_3$ again at $V$ . Show that $P, U, V$ are collinear.

The edges of a cube are labeled from $1$ to $12$ . Show that there must exist at least eight triples $(i, j, k)$ with $1 \leq i < j < k \leq 12$ so that the edges $i, j, k$ are consecutive edges of a path. Also show that there exists labeling in which we cannot find nine such triples.

$n, k$ are positive integers. $A_0$ is the set $\{1, 2, ... , n\}$ . $A_i$ is a randomly chosen subset of $A_{i-1}$ (with each subset having equal probability). Show that the expected number of elements of $A_k$ is $\dfrac{n}{2^k}$

Three circles of radius $a$ are drawn on the surface of a sphere of radius $r$ . Each pair of circles touches externally and the three circles all lie in one hemisphere. Find the radius of a circle on the surface of the sphere which touches all three circles.

Show that for positive reals $a, b, c, d$ we have $$\left(\dfrac{ab + ac + ad + bc + bd + cd}{6} \right)^3 \geq \left(\dfrac{abc + abd + acd + bcd}{4}\right)^2$$