**UNIVERSITY OF MARYLAND MATHEMATICS COMPETITION
** **PART II, 1997
***Use SEPARATE sheets for different problems. Submit only the material
you want graded.
Each problem is worth 30 points. No calculators are allowed.
Proofs must be given for all answers.
*

1. Prove that for every point inside a regular polygon, the average of
the distances to the sides equals the radius of the inscribed circle.
The distance to a side means the shortest distance from the point to the
line obtained by extending the side.

2. Suppose we are given positive (not necessarily distinct) integers
*a*_{1}, *a*_{2},..., *a*_{1997 }. Show that it is possible to choose some numbers from
this list such that their sum is a multiple of 1997.

3. You have Blue blocks, Green blocks and Red blocks.
Blue blocks and green blocks are 2 inches thick.
Red blocks are 1 inch thick. In how many ways can you stack the
blocks into a vertical column that is exactly 12 inches high?
(For example, for height 3 there are 5 ways: RRR, RG, GR, RB, BR.)

4. There are 1997 nonzero real numbers written on the blackboard.
An operation consists of choosing any two of these numbers, *a* and *b*,
erasing them, and writing a+b/2 and b-a/2 instead of them.
Prove that if a sequence of such operations is performed, one can
never end up with the initial collection of numbers.

5. An *m*×*n* checkerboard (*m* and *n* are positive integers) is covered
by nonoverlapping tiles of sizes 2×2 and 1×4. One 2×2
tile is removed and replaced by a 1×4 tile.
Is it possible to rearrange the tiles so that they cover the checkerboard?