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Difference between revisions of "2006 UNCO Math Contest II Problems"

(Created page with "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 <math>6\times 6</...")
(No difference)

Revision as of 21:40, 20 October 2014

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

Solution


Problem 6

Solution

==Problem 7

Solution

Problem 8

Solution

Problem 9

Solution

Problem 10

Solution


Problem 11

Solution

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