2009 UNCO Math Contest II Problems
University of Northern Colorado MATHEMATICS CONTEST FINAL ROUND January 31,2009.
For Colorado Students Grades 7-12.
How many positive -digit numbers are there such that For example, and have this property but and do not.
(a) Let . For how many between and inclusive is a multiple of ?
(b) For how many between and inclusive is a multiple of 5?
An army of ants is organizing a march to the Obama inauguration. If they form columns of ants there are left over. If they form columns of or ants there are left over. What is the smallest number of ants that could be in the army?
The two large isosceles right triangles are congruent. If the area of the inscribed square is square units, what is the area of the inscribed square ?
Let each of distinct points on the positive -axis be joined to each of distinct points on the positive -axis. Assume no three segments are concurrent (except at the axes). Obtain with proof a formula for the number of interior intersection points. The diagram shows that the answer is when and
A polynomial has a remainder of when divided by and a remainder of when divided by What is the remainder when is divided by ?
Two diagonals are drawn in the trapezoid forming four triangles. The areas of two of the triangles are and as shown. What is the total area of the trapezoid?
A square is divided into three pieces of equal area by two parallel lines as shown. If the distance between the two parallel lines is what is the area of the square?
Let . Determine the number of subsets of such that contains at least two elements and such that no two elements of differ by when
(c) generalize for any .
If the following triangular array of numbers is continued using the pattern established, how many numbers (not how many digits) would there be in the row? As an example, the row has numbers. Use exponent notation to express your answer.
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