CAPS 2006

There are $n$ children around a round table. Erika is the oldest among them and she has $n$ candies, while no other child has any candy. Erika decided to distribute the candies according to the following rules. In every round, she chooses a child with at least two candies and the chosen child sends a candy to each of his/her two neighbors. (So in the first round Erika must choose herself). For which $n \ge 3$ is it possible to end the distribution after a finite number of rounds with every child having exactly one candy?

The sum of four real numbers is $9$ and the sum of their squares is $21$ . Prove that these numbers can be denoted by $a, b, c, d$ so that $ab-cd \ge 2$ holds.

Show that for every integer $k \ge 1$ there is a positive integer $n$ such that the decimal representation of $2^n$ contains a block of exactly $k$ zeros, i.e. $2^n = \dots a00 \dots 0b \cdots$ with $k$ zeros and $a, b \ne 0$ .

Find the number of sequences $(a_n)_{n=1}^\infty$ of integers satisfying $a_n \ne -1$ and $$a_{n+2} =\frac{a_n + 2006}{a_{n+1} + 1}$$ for each $n \in \mathbb{N}$ .

Find out if there is a convex pentagon $A_1A_2A_3A_4A_5$ such that, for each $i = 1, \dots , 5$ , the lines $A_iA_{i+3}$ and $A_{i+1}A_{i+2}$ intersect at a point $B_i$ and the points $B_1,B_2,B_3,B_4,B_5$ are collinear. (Here $A_{i+5} = A_i$ .)