DE-BUNDESWETTBEWERB 2004

At the beginning of a game, I write the numbers $1$ , $2$ , ..., $2004$ onto a desk. A move consists of

• selecting some numbers standing on the desk;
• calculating the rest of the sum of these numbers under division by $11$ and writing this rest onto the desk;
• deleting the selected numbers.

In such a game, after a number of moves, only two numbers remained on the desk. One of them was $1000$ . What was the other one?

Consider a triangle whose sidelengths $a$ , $b$ , $c$ are integers, and which has the property that one of its altitudes equals the sum of the two others.

Then, prove that $a^2+b^2+c^2$ is a perfect square.

Let $k$ be a positive integer. A natural number $m$ is called $k$ -typical if each divisor of $m$ leaves the remainder $1$ when being divided by $k$ .

Prove:

a) If the number of all divisors of a positive integer $n$ (including the divisors $1$ and $n$ ) is $k$ -typical, then $n$ is the $k$ -th power of an integer.

b) If $k > 2$ , then the converse of the assertion a) is not true.

Let $k$ be a positive integer. In a circle with radius $1$ , finitely many chords are drawn. You know that every diameter of the circle intersects at most $k$ of these chords.

Prove that the sum of the lengths of all these chords is less than $k \cdot \pi$ .

Given two circles $k_1$ and $k_2$ which intersect at two different points $A$ and $B$ . The tangent to the circle $k_2$ at the point $A$ meets the circle $k_1$ again at the point $C_1$ . The tangent to the circle $k_1$ at the point $A$ meets the circle $k_2$ again at the point $C_2$ . Finally, let the line $C_1C_2$ meet the circle $k_1$ in a point $D$ different from $C_1$ and $B$ .

Prove that the line $BD$ bisects the chord $AC_2$ .

Prove that there exist infinitely many pairs $\left(x; y\right)$ of different positive rational numbers, such that the numbers $\sqrt{x^2+y^3}$ and $\sqrt{x^3+y^2}$ are both rational.