OMM 2001

Find all $7$ -digit numbers which are multiples of $21$ and which have each digit $3$ or $7$ .

Given some colored balls (at least three different colors) and at least three boxes. The balls are put into the boxes so that no box is empty and we cannot find three balls of different colors which are in three different boxes. Show that there is a box such that all the balls in all the other boxes have the same color.

$ABCD$ is a cyclic quadrilateral. $M$ is the midpoint of $CD$ . The diagonals meet at $P$ . The circle through $P$ which touches $CD$ at $M$ meets $AC$ again at $R$ and $BD$ again at $Q$ . The point $S$ on $BD$ is such that $BS = DQ$ . The line through $S$ parallel to $AB$ meets $AC$ at $T$ . Show that $AT = RC$ .

For positive integers $n, m$ define $f(n,m)$ as follows. Write a list of $2001$ numbers $a_i$ , where $a_1 = m$ , and $a_{k+1}$ is the residue of $a_k^2$ $mod n$ (for $k = 1, 2,..., 2000$ ). Then put $f(n,m) = a_1-a_2 + a_3 -a_4 + a_5- ... + a_{2001}$ . For which $n \ge 5$ can we find m such that $2 \le m \le n/2$ and $f(m,n) > 0$ ?

$ABC$ is a triangle with $AB < AC$ and $\angle A = 2 \angle C$ . $D$ is the point on $AC$ such that $CD = AB$ . Let L be the line through $B$ parallel to $AC$ . Let $L$ meet the external bisector of $\angle A$ at $M$ and the line through $C$ parallel to $AB$ at $N$ . Show that $MD = ND$ .

A collector of rare coins has coins of denominations $1, 2,..., n$ (several coins for each denomination). He wishes to put the coins into $5$ boxes so that: (1) in each box there is at most one coin of each denomination; (2) each box has the same number of coins and the same denomination total; (3) any two boxes contain all the denominations; (4) no denomination is in all $5$ boxes. For which $n$ is this possible?