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 $60=3*4*5.$