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
[hide]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
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?
Solution
Problem 4
Let be the set of all positive squares plus
and
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
.
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
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.
See Also
2011 UNCO Math Contest II (Problems • Answer Key • Resources) | ||
Preceded by 2010 UNCO Math Contest II |
Followed by 2012 UNCO Math Contest II | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 | ||
All UNCO Math Contest Problems and Solutions |