# 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 $\left{20 ,21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31\right}$ (Error compiling LaTeX. ! Missing delimiter (. inserted).) 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?

filldraw((0,0)--(13,0)--(12,5)--cycle,blue); pair A1=(0,0),B1=(25,0),C1=(25,5),D1=(0,5); draw(A1--B1--C1--D1--cycle,black); pair P=.1*unit(A1-D1)+D1,R=.1*unit(C1-D1)+D1; draw(P--(P+R-D)--R,black); pair A2=(0,0),B2=(25/13,-60/13),C2=(25,5),D2=(25-25/13,5+60/13); draw(A2--B2--C2--D2--cycle,black); P=.1*unit(A2-D2)+D2; R=.1*unit(C2-D2)+D2; draw(P--(P+R-D)--R,black); (Error compiling LaTeX. draw(P--(P+R-D)--R,black); ^ e3e47a918c489e98fe6f1264b4357c0d6143ea05.asy: 9.13: no matching function 'operator -(pair, <overloaded>)' draw(P--(P+R-D)--R,black); ^ e3e47a918c489e98fe6f1264b4357c0d6143ea05.asy: 9.14: use of variable 'D' is ambiguous)

## Problem 4

Let $A = \left{ 2,5,10,17,\cdots,n^2+1,\cdots\right}$ (Error compiling LaTeX. ! Missing delimiter (. inserted).) be the set of all positive squares plus and $B = \left{101, 104, 109, 116,\cdots,m^2 + 100,\cdots\right}$ (Error compiling LaTeX. ! Missing delimiter (. inserted).) 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 .

pair A=(0,4*sqrt(10)),B=(4*sqrt(10),4*sqrt(10)),C=(4*sqrt(10),0),D=(0,0); pair F=(sqrt(10),3*sqrt(10)),E=((9/5)*sqrt(10),(3/5)*sqrt(10)); draw(A--B--C--D--cycle,black); draw(D--E--F--B,black); pair P=.1*unit(D-E)+E,R=.1*unit(F-E)+E; draw(P--(P+R-E)--R,black); P=.1*unit(B-F)+F,R=.1*unit(E-F)+F; draw(P--(P+R-F)--R,black); MP("A",A,NW);MP("B",B,NE);MP("C",C,SE);MP("D",D,SW); MP("6",(D/2+E/2),NW);MP("8",(E/2+F/2),NE);MP("10",(F/2+B/2),S); (Error compiling LaTeX. P=.1*unit(B-F)+F,R=.1*unit(E-F)+F; ^ 8d1eaecbb1462f3788079defda2a11d8a43b97d5.asy: 11.17: syntax error error: could not load module '8d1eaecbb1462f3788079defda2a11d8a43b97d5.asy')

## 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 $\left{1, 2, 3,\cdots ,n\right}$ (Error compiling LaTeX. ! Missing delimiter (. inserted).) 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.