SE-SKOLORNAS 1997

Let $AC$ be a diameter of a circle and $AB$ be tangent to the circle. The segment $BC$ intersects the circle again at $D$ . Show that if $AC = 1$ , $AB = a$ , and $CD = b$ , then $$\frac{1}{a^2+ \frac12 }< \frac{b}{a}< \frac{1}{a^2}$$

Let $D$ be the point on side $AC$ of a triangle $ABC$ such that $BD$ bisects $\angle B$ , and $E$ be the point on side $AB$ such that $3\angle ACE = 2\angle BCE$ . Suppose that $BD$ and $CE$ intersect at a point $P$ with $ED = DC = CP$ . Determine the angles of the triangle.

Let $A$ and $B$ be integers with an odd sum. Show that every integer can be written in the form $x^2 -y^2 +Ax+By$ , where $x,y$ are integers.

Players $A$ and $B$ play the following game. Each of them throws a dice, and if the outcomes are $x$ and $y$ respectively, a list of all two digit numbers $10a + b$ with $a,b\in \{1,..,6\}$ and $10a + b \le 10x + y$ is created. Then the players alternately reduce the list by replacing a pair of numbers in the list by their absolute difference, until only one number remains. If the remaining number is of the same parity as the outcome of $A$ 's throw, then $A$ is proclaimed the winner. What is the probability that $A$ wins the game?

Let $s(m)$ denote the sum of (decimal) digits of a positive integer $m$ . Prove that for every integer $n > 1$ not equal to $10$ there is a unique integer $f(n) \ge 2$ such that $s(k)+s(f(n)-k) = n$ for all integers $k$ with $0 < k < f(n)$ .

Assume that a set $M$ of real numbers is the union of finitely many disjoint intervals with the total length greater than $1$ . Prove that $M$ contains a pair of distinct numbers whose difference is an integer.