NO-ABEL 2007

We consider the sum of the digits of a positive integer. For example, the sum of the digits of $2007$ is equal to $9$ , since $2 + 0 + 0 + 7 = 9$ . (a) How many integers $n$ , where $0 < n < 100 000$ , have an even sum of digits? (b) How many integers $n$ , where $0 < n < 100 000$ , have a sum of digits that is less than or equal to $22$ ?

The vertices of a convex pentagon $ABCDE$ lie on a circle $\gamma_1$ . The diagonals $AC , CE, EB, BD$ , and $DA$ are tangents to another circle $\gamma_2$ with the same centre as $\gamma_1$ . (a) Show that all angles of the pentagon $ABCDE$ have the same size and that all edges of the pentagon have the same length. (b) What is the ratio of the radii of the circles $\gamma_1$ and $\gamma_2$ ? (The answer should be given in terms of integers, the four basic arithmetic operations and extraction of roots only.)

(a) Let $x$ and $y$ be two positive integers such that $\sqrt{x} +\sqrt{y}$ is an integer. Show that $\sqrt{x}$ and $\sqrt{y}$ are both integers. (b) Find all positive integers $x$ and $y$ such that $\sqrt{x} +\sqrt{y}=\sqrt{2007}$ .

Let $a, b$ and $c$ be integers such that $a + b + c = 0$ . (a) Show that $a^4 + b^4 + c^4$ is divisible by $a^2 + b^2 + c^2$ . (b) Show that $a^{100} + b^{100} + c^{100}$ is divisible by $a^2 + b^2 + c^2$ . .