RIOPLATENSE 1990

How many positive integer solutions does the equation have $$\left\lfloor\frac{x}{10}\right\rfloor= \left\lfloor\frac{x}{11}\right\rfloor + 1?$$ ( $\lfloor x \rfloor$ denotes the integer part of $x$ , for example $\lfloor 2\rfloor = 2$ , $\lfloor \pi\rfloor = 3$ , $\lfloor \sqrt2 \rfloor =1$ )

Some of the people attending a meeting greet each other. Let $n$ be the number of people who greet an odd number of people. Prove that $n$ is even.

Let $ABCD$ be a trapezium with bases $AB$ and $CD$ such that $AB = 2 CD$ . From $A$ the line $r$ is drawn perpendicular to $BC$ and from $B$ the line $t$ is drawn perpendicular to $AD$ . Let $P$ be the intersection point of $r$ and $t$ . From $C$ the line $s$ is drawn perpendicular to $BC$ and from $D$ the line $u$ perpendicular to $AD$ . Let $Q$ be the intersection point of $s$ and $u$ . If $R$ is the intersection point of the diagonals of the trapezium, prove that points $P, Q$ and $R$ are collinear.