Difference between revisions of "2007 UNCO Math Contest II Problems"
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decimal approximation). | decimal approximation). | ||
| + | <asy> | ||
| + | path T=polygon(3); | ||
| + | draw(unitcircle,black); | ||
| + | draw(scale(2)*T,black); | ||
| + | draw(shift(2/sqrt(3),-2/3)*scale(1/3)*unitcircle,black); | ||
| + | draw(shift(2/sqrt(3)/3,-2/9)*shift(2/sqrt(3),-2/3)*scale(1/9)*unitcircle,black); | ||
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[[2007 UNCO Math Contest II Problems/Problem 9|Solution]] | [[2007 UNCO Math Contest II Problems/Problem 9|Solution]] | ||
Revision as of 23:06, 19 October 2014
UNIVERSITY OF NORTHERN COLORADO MATHEMATICS CONTEST FINAL ROUND February 3,2007.
For Colorado Students Grades 7-12.
• The sequence of Fibonacci numbers is 
• The positive odd integers are 
• A regular decagon is a 
-sided figure all of whose sides are congruent.
Contents
Problem 1
Express the following sum as a whole number:
Problem 2
In Grants Pass, Oregon 
of the men are married to 
of the women.
What fraction of the adult population is married? Give a possible generalization.
Problem 3
State the general rule illustrated here and prove it:
Problem 4
If 
is a primitive cube root of one (this means that 
but 
) compute the value of
![\[x^{2006}+\frac{1}{x^{2006}}+x^{2007}+\frac{1}{x^{2007}}.\]](http://latex.artofproblemsolving.com/4/d/3/4d3fe43557d68d9c46f840e5241e58392fcc1dd9.png)
Problem 5
Ten different playing cards have the numbers
written on them
as shown. Three cards are selected at random
without replacement. What is the
probability that the sum of the
numbers on the three cards is divisible by 
?
![[asy] [/asy]](http://latex.artofproblemsolving.com/e/4/7/e47b8b9ccd3a0e41f210617f818a80ad8074143e.png)
Problem 6
(a) Demonstrate that every odd number 
can be expressed as a difference of two squares.
(b) Demonstrate which even numbers can be expressed as a difference of two squares.
Problem 7
(a) Express the infinite sum 
as a reduced fraction.
(b) Express the infinite sum 
as a reduced fraction. Here the denominators are powers of 
and the numerators 
are the Fibonacci numbers
where 
.
Problem 8
A regular decagon 
is drawn
in the coordinate plane with 
at 
and 
at 
. If 
denotes the point
, compute the numerical value of
the following product of complex numbers:
where 
as usual.
![[asy] draw(polygon(10),dot); draw((-2,0)--(2,0),black); draw((-5/3,-2)--(-5/3,2),black); MP("P_1",(-1,0),NW);MP("(2,0)",(-.9,0),SW); MP("P_6",(1,0),NE);MP("(8,0)",(.9,0),SE); [/asy]](http://latex.artofproblemsolving.com/a/c/d/acd58d961876928535be1ba8fa362bc000ac8b1c.png)
Problem 9
A circle is inscribed in an equilateral triangle whose side
length is 
. Then another circle is inscribed externally
tangent to the first circle but inside the triangle as shown.
And then another, and another. If this process continues
forever what is the total area of all the circles? Express
your answer as an exact multiple of 
(and not as a
decimal approximation).
![[asy] path T=polygon(3); draw(unitcircle,black); draw(scale(2)*T,black); draw(shift(2/sqrt(3),-2/3)*scale(1/3)*unitcircle,black); draw(shift(2/sqrt(3)/3,-2/9)*shift(2/sqrt(3),-2/3)*scale(1/9)*unitcircle,black); [/asy]](http://latex.artofproblemsolving.com/8/a/f/8af6a485cf6e58a89645057e34404328fd5cbaa7.png)
Problem 10
A quaternary “number” is an arrangement of digits, each of which is 
Some examples: 
(a) How many 
-digit quaternary numbers are there in which each of 
appear at least once?
(b) How many 
-digit quaternary numbers are there in which each of 
appear at least
once? Test your answer with 
(c) Generalize.
