BY-MO 1999

Evaluate the product $\prod_{k=0}^{2^{1999}}(4\sin^2 \frac{k\pi}{2^{2000}}-3)$

Let $m, n$ be positive integers. Starting with all positive integers written in a line, we can form a list of numbers in two ways: $(1)$ Erasing every $m$ -th and then, in the obtained list, erasing every $n$ -th number; $(2)$ Erasing every $n$ -th number and then, in the obtained list, erasing every $m$ -th number.

A pair $(m,n)$ is called good if, whenever some positive integer $k$ occurs in both these lists, then it occurs in both lists on the same position.

(a) Show that the pair $(2, n)$ is good for any $n\in \mathbb{N}$ . (b) Is there a good pair $(m, n)$ with $2<m<n$ ?

A sequence of numbers $a_1,a_2,...,a_{1999}$ is given. In each move it is allowed to choose two of the numbers, say $a_m,a_n$ , and replace them by the numbers $$\frac{a_n^2}{a_m^2}-\frac{n}{m}\left(\frac{a_m^2}{a_n}-a_m\right), \frac{a_m^2}{a_n^2}-\frac{m}{n}\left(\frac{a_n^2}{a_m}-a_n\right)$$ respectively. Starting with the sequence $a_i = 1$ for $20 \nmid i$ and $a_i =\frac{1}{5}$ for $20 \mid i$ , is it possible to obtain a sequence whose all terms are integers?

A circle is inscribed in the trapezoid ABCD. Let K, L, M, N be the points of tangency of this circle with the diagonals AC and BD, respectively (K is between A and L, and M is between B and N). Given that $AK\cdot LC=16$ and $BM\cdot ND=\frac94$ , find the radius of the circle.

[Moderator edit: A solution of this problem can be found on http://www.ajorza.org/math/mathfiles/scans/belarus.pdf , page 20 (the statement of the problem is on page 6). The author of the problem is I. Voronovich.]

Let O be the center of circleW. Two equal chords AB and CDofWintersect at Lsuch that AL>LBand DL>LC. Let MandNbe points on AL and DL respectively such that (ALC)=2*(MON). Prove that the chord of W passing through Mand N is equal to AB and CD.