2006 UNCO Math Contest II Problems

UNIVERSITY OF NORTHERN COLORADO MATHEMATICS CONTEST FINAL ROUND January 28,2006.

For Colorado Students Grades 7-12.

Problem 1

If a dart is thrown at the $6\times 6$ target, what is the probability that it will hit the shaded area?

$[asy] filldraw((2,2)--(4,6)--(6,6)--(6,4)--cycle,blue); filldraw((2,2)--(6,2)--(6,1)--cycle,blue); filldraw((2,2)--(0,0)--(0,1)--cycle,blue); filldraw((2,2)--(0,4)--(0,6)--cycle,blue); for(int i=0;i<7;++i){ draw((0,i)--(6,i),black); draw((i,0)--(i,6),black); } dot((2,2));dot((0,4));dot((0,6));dot((4,6));dot((6,6)); dot((6,4));dot((6,2));dot((6,1));dot((0,0));dot((0,1)); [/asy]$

Solution

Problem 2

If $a,b$ and $c$ are positive integers, how many integers are strictly between the product $abc$ and $(a+1)(b+1)(c+1)$ ? For example, there are 35 integers strictly between $24=2*3*4$ and $60=3*4*5.$

Solution

Problem 3

The first 14 integers are written in order around a circle.

Starting with 1, every fifth integer is underlined. (That is $1,6,11,2,7,\ldots$ ). What is the $2006^{th}$ number underlined?

$[asy] draw(unitcircle,black); pair A; for(int j=1;j<15;++j){ A=dir(90-(j-1)*(360/14)); MP(string(j),A,A); } [/asy]$

Solution

Problem 4

Determine all positive integers $n$ such that $n^2+3$ divides evenly (without remainder) into $n^4-3n^2+10$ ?

Solution

Problem 5

In the figure $BD$ is parallel to $AE$ and also $BF$ is parallel to $DE$ . The area of the larger triangle $ACE$ is $128$ . The area of the trapezoid $BDEA$ is $78$ . Determine the area of triangle $ABF$ .

$[asy] draw((0,0)--(1,2)--(4,0)--cycle,black); draw((1/2,1)--(2.5,1)--(2,0),black); MP("A",(4,0),SE);MP("C",(1,2),N);MP("E",(0,0),SW); MP("D",(.5,1),W);MP("B",(2.5,1),NE);MP("F",(2,0),S); [/asy]$

Solution

Problem 6

The sum of all of the positive integer divisors of $6^2=36$ is $1+2+3+4+6+9+12+18+36=91$

(a) Determine a nice closed formula (i.e. without dots or the summation symbol) for the sum of all positive divisors of $6^n$ .

(b) Repeat for $12^n$ .

(c) Generalize.

Solution

Problem 7

The five digits $a,b,c,d$ and $e$ of $55225$ are such that $a=b=e$ and $c=d$ ; in addition, $55225=235^2=(235)(235)$ . Find another integer $m$ such that $m^2$ is also a five digit number $abcde$ that satisfies $a=b=e$ and $c=d$ .

Solution

Problem 8

Find all positive integers $n$ such that $n^3-12n^2+40n-29$ is a prime number. For each of your values of $n$ compute this cubic polynomial showing that it is, in fact, a prime.