2006 UNCO Math Contest II Problems
UNIVERSITY OF NORTHERN COLORADO MATHEMATICS CONTEST FINAL ROUND January 28,2006.
For Colorado Students Grades 7-12.
If a dart is thrown at the target, what is the probability that it will hit the shaded area?
If and are positive integers, how many integers are strictly between the product and ? For example, there are 35 integers strictly between and
The first 14 integers are written in order around a circle.
Starting with 1, every fifth integer is underlined. (That is ). What is the number underlined?
Determine all positive integers such that divides evenly (without remainder) into ?
In the figure is parallel to and also is parallel to . The area of the larger triangle is . The area of the trapezoid is . Determine the area of triangle .
The sum of all of the positive integer divisors of is
(a) Determine a nice closed formula (i.e. without dots or the summation symbol) for the sum of all positive divisors of .
(b) Repeat for .
The five digits and of are such that and ; in addition, . Find another integer such that is also a five digit number that satisfies and .
Find all positive integers such that is a prime number. For each of your values of compute this cubic polynomial showing that it is, in fact, a prime.
Determine three positive integers and that simultaneously satisfy the following three conditions:
(ii) Each of and is the square of an integer, and
(iii) is as small as is possible.
How many triples of positive integers and are there with and .
Call the figure below a "-tableau" shape. Determine the number of rectangles of all sizes contained within this shape. Note that a square is considered a rectangle, and a rectangle is considered different from a . Express your answer as a binomial coefficient and explain the significance of your expression. Generalize, with proof, to an "-tableau" shape.
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