2006 UNCO Math Contest II Problems
UNIVERSITY OF NORTHERN COLORADO MATHEMATICS CONTEST FINAL ROUND January 28,2006.
For Colorado Students Grades 7-12.
Contents
Problem 1
If a dart is thrown at the target, what is the probability that it will hit the shaded area?
Problem 2
If and are positive integers, how many integers are strictly between the product and ? For example, there are 35 integers strictly between and
Problem 3
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?
Problem 4
Determine all positive integers such that divides evenly (without remainder) into ?
Problem 5
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 .
Problem 6
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 .
(c) Generalize.
Problem 7
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 .
Problem 8
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.
Problem 9
Determine three positive integers and that simultaneously satisfy the following three conditions:
(i)
(ii) Each of and is the square of an integer, and
(iii) is as small as is possible.
Problem 10
How many triples of positive integers and are there with and .
Problem 11
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.
See Also
2006 UNCO Math Contest II (Problems • Answer Key • Resources) | ||
Preceded by 2005 UNCO Math Contest II |
Followed by 2007 UNCO Math Contest II | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 | ||
All UNCO Math Contest Problems and Solutions |