2006 iTest Problems/Ultimate Question
The following problems are from the Ultimate Question of the 2006 iTest, where solving this problem required the answer of a previous problem. When the problem is rewritten, the T-value is substituted.
Contents
[hide]Problem 41
Problem U1
Find the real number such that
Problem U2
Points and
lie on a circle centered at
such that
is right. Points
and
lie on radii
and
respectively such that
,
, and
. Determine the area of quadrilateral
.
Problem U3
When properly sorted, math books on a shelf are arranged in alphabetical order from left to right. An eager student checked out and read all of them. Unfortunately, the student did not realize how the books were sorted, and so after finishing the student put the books back on the shelf in a random order. If all arrangements are equally likely, the probability that exactly
of the books were returned to their correct (original) position can be expressed as
, where
and
are relatively prime positive integers. Compute
.
Problem 42
Problem U4
As ranges over the integers, the expression
evaluates to just one prime number. Find this prime.
Problem U5
In triangle , points
,
, and
are the feet of the angle bisectors of
,
,
respectively. Let point
be the intersection of segments
and
, and let
denote the perimeter of
. If
,
, and
, then the value of
can be expressed uniquely as
where
and
are positive integers such that
is not divisible by the square of any prime. Find
.
Problem U6
and
are nonzero real numbers such that
The smallest possible value of is equal to
where
and
are relatively prime positive integers. Find
.
Problem 43
Problem U8
Cyclic quadrilateral has side lengths
,
,
, and
. Let
and
be the midpoints of sides
and
. The diagonals
and
intersect
at
and
respectively.
can be expressed as
where
and
are relatively prime positive integers. Determine
.