2010 UNCO Math Contest II Problems
University of Northern Colorado MATHEMATICS CONTEST FINAL ROUND January 30, 2010.
For Colorado Students Grades 7-12.
• The ten digits are
• The positive integers are
• The prime numbers are
Find a -digit integer less than where each digit is odd and the sum of the cubes of the digits is the original number.
The rectangle has dimensions . The diagonal is divided into five segments of equal length. Find the total area of the shaded regions.
Suppose , and are three different positive integers and that their product is , i.e., What is the smallest possible value of the sum ?
(a) In the grid shown, four coins are randomly placed in different squares. What is the probability that no two coins lie in the same row or column?
(b) Generalize this to an grid.
is a -digit number . is a -digit number formed by augmenting with a on the right, i.e. .
is another -digit number formed by placing a on the left , i.e. . If is three times , what is the number ?
and are each -digit numbers (like and ), and all four digits are different. The sum is a -digit number made up of two more different digits ( is not necessarily prime). Further, the difference consists of yet two more different digits (again, is not necessarily prime). The number is a two digit number which uses the remaining two digits. What is ?
Simplify , using exponential notation to express your answer. Generalize this result.
(a) Find integers , and so that Express your answers in exponential form.
(b) Find integers and so that
Let where . What is the maximum number of elements in a subset of , which has at least three elements, such that for all in ? As an example, the subset of has the property that the sum of any two elements is strictly bigger than the third element, but the subset does not since is greater than . Since there is no subset of size satisfying these conditions, the answer for is .
(a) The square grid has dots equally spaced. How many squares (of all sizes) can you make using four of these dots as vertices? Two examples are shown.
(b) How many for a ?
(c) How many for a ?
(d) How many for an grid of dots?
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2009 UNCO Math Contest II
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