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```        UNIVERSITY OF MARYLAND MATHEMATICS COMPETITION

PART II,  1998

No calculators are allowed.

PROOFS MUST BE GIVEN FOR ALL ANSWERS

1. Four positive numbers are placed at the vertices of a rectangle.
Each number is at least as large as the average of the two numbers
at the adjacent vertices.  Prove that all four numbers are equal.

2. The sum 498+499+500+501=1998 is one way of expressing 1998
as a sum of consecutive positive integers. Find all ways of expressing
1998 as a sum of two or more consecutive positive integers. Prove

3. An infinite strip (two parallel lines and the region between them)
has a width of 1 inch. What is the largest value of A such that every
triangle with area A square inches can be placed on this strip?

4. A plane divides space into two regions. Two planes that intersect
in a line divide space into four regions. Now suppose that twelve planes
are given in space so that a) every two of them intersect in a line,
b) every three of them intersect in a point, and c) no four of them have
a common point. Into how many regions is space divided?