2008 UNCO Math Contest II Problems
UNIVERSITY OF NORTHERN COLORADO MATHEMATICS CONTEST FINAL ROUND February 2, 2008.
For Colorado Students Grades 7-12.
Determine the number of square arrays whose row and column sums are equal to , using as entries. Entries may be repeated, and not all of need be used as the two examples show.
Let be a set of four positive integers. If pairs of distinct elements of are added, the following six sums are obtained: Determine the values of , and [Hint: there are two possibilities.]
A rectangle is inscribed in a square creating four isosceles right triangles. If the total area of these four triangles is , what is the length of the diagonal of the rectangle?
In the figure there are line segments drawn from vertex to the base (not counting the segments or ).
(a) Determine the total number of triangles of all sizes.
(b) How many triangles are there if there are lines drawn from to interior points on ?
The sum of and is and the product of these five numbers is
(a) Determine the largest number which is the product of positive integers whose sum is .
(b) Determine the largest number which is the product of positive integers whose sum is .
Points and are on the same side of line in the plane. is units away from is units away from . The distance between and is . For all points on what is the smallest value of the sum of the distances from to and from to ?
Determine the value of so that the following fraction reduces to a quotient of two linear expressions:
Triangle has integer side lengths. One side is twice the length of a second side.
(a) If the third side has length what is the greatest possible perimeter?
(b) If the third side has length what is the greatest possible perimeter?
(c) Now suppose one side is three times the length of a second side and the third side has length of . What is the maximum perimeter?
(a) Prove that
(b) Prove that
(c) Prove that each term in the following sequence is a perfect square:
Let be the number of ways of splitting people into groups, each of size . As an example,
the people can be split into groups: and
(a) Compute and
(b) Conjecture a formula for
(c) Let be the number of ways of splitting into subsets of size . Compute and conjecture a formula for
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2007 UNCO Math Contest II
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