JMO 1994

For any positive integer $n$, let $a_n$ denote the closest integer to $\sqrt{n}$, and let $b_n=n+a_n$. Determine the increasing sequence $(c_n)$ of positive integers which do not occur in the sequence $(b_n)$.

Five points, no three collinear, are given on the plane. Let $l_1,l_2,\ldots ,l_{10}$ be the lengths of the ten segments joining any two of the given points. Prove that if $l_1^2,\ldots ,l_9^2$ are rational numbers, then $l_{10}^2$ is also a rational number.

Let $P_0$ be a point in the plane of triangle $A_0A_1A_2$. Define $P_i\ (i=1,\ldots ,6)$ inductively as the point symmetric to $P_{i-1}$ with respect to $A_k$, where $k$ is the remainder when $i$ is divided by $3$.

a) Prove that $P_6\equiv P_1$. b) Find the locus of points $P_0$ for which $P_iP_{i+1}$ does not meet the interior of $\triangle A_0A_1A_2$ for $0\le i\le 5$.

In a triangle $ABC$, $M$ is the midpoint of $BC$. Given that $\angle MAC=15^{\circ}$, find the maximum possible value of $\angle ABC$.

In a deck of $N$ cards, the cards are denoted by $1$ to $N$. These cards are dealt to $N$ people twice. A person $X$ wins a prize if there is no person $Y$ who got a card with a smaller number than $X$ both times. Determine the expected number of prize winners.