2008 UNCO Math Contest II Problems
UNIVERSITY OF NORTHERN COLORADO MATHEMATICS CONTEST FINAL ROUND February 2, 2008.
For Colorado Students Grades 7-12.
Contents
Problem 1
Determine the number of square arrays
whose row and column sums are equal to
,
using
as entries. Entries may be
repeated, and not all of
need be used as the
two examples show.
Problem 2
Let be a set of four positive integers. If pairs of distinct elements of
are added, the following six sums are obtained:
Determine the values of
, and
[Hint: there are two possibilities.]
Problem 3
A rectangle is inscribed in a square
creating four isosceles right triangles.
If the total area of these four triangles
is , what is the length of the diagonal
of the rectangle?
Problem 4
In the figure there are line segments drawn from
vertex
to the base
(not counting the segments
or
).
(a) Determine the total number of triangles of all sizes.
(b) How many triangles are there if there are lines
drawn from
to
interior points on
?
Problem 5
The sum of and
is
and the product of these five numbers is
(a) Determine the largest number which is the product of positive integers whose sum is .
(b) Determine the largest number which is the product of positive integers whose sum is .
Problem 6
Points and
are on the same side of
line
in the plane.
is
units away
from
is
units away from
.
The distance between
and
is
. For
all points
on
what is the smallest
value of the sum
of the distances
from
to
and from
to
?
Problem 7
Determine the value of so that the following fraction reduces to a quotient of two linear
expressions:
Problem 8
Triangle has integer side lengths.
One side is twice the length of a second side.
(a) If the third side has length what is the greatest possible perimeter?
(b) If the third side has length what is the greatest possible perimeter?
(c) Now suppose one side is three times the length of a second side and the third side has length
of . What is the maximum perimeter?
(d) Generalize
Problem 9
Let
(a) Prove that
(b) Prove that
(c) Prove that each term in the following sequence is a perfect square:
Problem 10
Let be the number of ways of splitting
people into
groups, each of size
. As an example,
the people
can be split into
groups:
and
Hence
(a) Compute and
(b) Conjecture a formula for
(c) Let be the number of ways of splitting
into
subsets of size
.
Compute
and conjecture a formula for