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
Problem 41
Problem U1
Find the real number 
such that
![\[\sqrt{x-9} + \sqrt{x-6} = \sqrt{x-1}.\]](http://latex.artofproblemsolving.com/e/6/b/e6b65bac4a7c29faa1f80320d9f69fa1767bb77d.png)
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 
.
![[asy] draw(circle((0,0),10)); draw((0,10)--(0,0)--(10,0)--(0,10)); draw((0,3)--(4,0)); label("O",(0,0),SW); label("C",(0,3),W); label("A",(0,10),N); label("D",(4,0),S); label("B",(10,0),E); [/asy]](http://latex.artofproblemsolving.com/a/b/e/abee71171256aa5b98954710a59865eb1b5dac8a.png)
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 .
