# 2011 UNCO Math Contest II Problems

University of Northern Colorado MATHEMATICS CONTEST FINAL ROUND January 29, 2011 For Colorado Students Grades 7-12

• , 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.