2006 SMT/Team Problems
Contents
[hide]Problem 1
Given , where
is at
,
is at
, and
is on the positive
axis. Cone
is formed when
is rotated about the
axis, and cone
is formed when
is rotated about the
axis. If the volume of cone
minus the volume fo cone
is
, find the length of
.
Problem 2
In a given sequence , for terms
,
. For example, if the first two elements are
and
, respectively, the third entry would be
, and the fourth would be
, and so on. Given that a sequence of integers having this form starts with
, and the
element is
, what is the second element?
Problem 3
A triangle has altitudes of lengths and
. What is the maximum length of the third altitude?
Problem 4
Let and
. The expression
can be written as a polynomial in terms of
and
. What is this polynomial?
Problem 5
There exist two positive numbers such that
. Find the product of the two possible
.
Problem 6
The expression is equivalent to the expression
for all positive integers
where
and
are functions and
is constant. Find
.
Problem 7
Let be the set of all
-tuples
that satisfy
and
. If one of these
-tuples is chosen at random, what's the probability that
or
is greater than or equal to
?
Problem 8
Evaluate:
Problem 9
has
. Points
and
are midpoints of
and
, respectively. The medians
and
intersect at a right angle. Find
.
Problem 10
Find the smallest positive integer for which there are at least
even and
odd positive integers
so that
is an integer.
Problem 11
Polynomial has roots
. The coefficients satisfy
for all pairs of integers
. Given that
, determine
.
(Note: The original problem asked for , but the official solution makes it clear that the actual desired sum is
.)
Problem 12
Find the total number of -tuples
of positive integers so that
for each
, and
regular polygons with numbers of sides
respectively will fit into a tesselation at a point. That is, the sum of one interior angle from each of the polygons is
.
Problem 13
A ray is drawn from the origin tangent to the graph of the upper part of the hyperbola in the first quadrant. This ray makes an angle of
with the positive
axis. Compute
.
Problem 14
Find the smallest nonnegative integer for which
is divisible by
.
Problem 15
Let denote the
composite integer so that
. Compute
(Hint: )
(Note: The original hint stated that .)