Difference between revisions of "2011 UNCO Math Contest II Problems"
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University of Northern Colorado MATHEMATICS CONTEST FINAL ROUND January 29, 2011 For Colorado Students Grades 712
• , read as n factorial, is computed as
• The factorials are
• The square integers are
Contents
Problem 1
The largest integer so that evenly divides is . Determine the largest integer so that evenly divides .
Problem 2
Let and be positive integers. List all the integers in the set that be written in the form . As an example, be so expressed since .
Problem 3
The two congruent rectangles shown have dimensions in. by in. What is the area of the shaded overlap region? Solution
Problem 4
Let be the set of all positive squares plus and be the set of all positive squares plus .
(a) What is the smallest number in both and ?
(b) Find all numbers that are in both and .
Problem 5
Determine the area of the square , with the given lengths along a zigzag line connecting and .
Problem 6
What is the remainder when is divided by ?
Problem 7
What is the of the first terms of the sequence that appeared on the First Round? Recall that a term in an even numbered position is twice the previous term, while a term in an odd numbered position is one more that the previous term.
Problem 8
The integer can be expressed as a sum of two squares as .
(a) Express as the sum of two squares.
(b) Express the product as the sum of two squares.
(c) Prove that the product of two sums of two squares, , can be represented as the sum of two squares.
Problem 9
Let be the number of ways of selecting three distinct numbers from so that they are the lengths of the sides of a triangle. As an example, ; the only possibilities are , and .
(a) Determine a recursion for T(n).
(b) Determine a closed formula for T(n).
Problem 10
The integers are written on the blackboard. Select any two, call them and and replace these two with the one number . Continue doing this until only one number remains and explain, with proof, what happens. Also explain with proof what happens in general as you replace with . As an example, if you select and you replace them with . If you select and , replace them with . You now have two ’s in this case but that’s OK.
Problem 11
Tie breaker – Generalize problem #2, and prove your statement.