2007 iTest Problems/Ultimate Question
The following questions are from the "Ultimate Question" of the 2007 iTest, but with the T-values substituted. This is for people who want to work on the problems without having to solve the previous problems.
Contents
[hide]Problem 51
Find the highest point (largest possible -coordinate) on the parabola
Problem 52
Let be the region consisting of points
of the Cartesian plane satisfying both
and
. Find the area of region
.
Problem 53
Three distinct positive Fibonacci numbers, all greater than , are in arithmetic progression. Let
be the smallest possible value of their sum. Find the remainder when
is divided by
.
Problem 54
Consider the sequence . Inserting the difference between
and
between them, we get the sequence
. Repeating the process of inserting differences between numbers, we get the sequence
. A third iteration of this process results in
. A total of
iterations produces a sequence with
terms. If the integer
appears a total of
times among these
terms, find the remainder when
gets divided by
.
Problem 55
Let . Let
be the smallest real solution of
. Find the value of
.
Problem 56
In the binary expansion of , how many of the first
digits to the right of the radix point are
's?
Problem 57
How many positive integers are within of exactly
perfect squares? (Note:
is considered a perfect square.)
Problem 58
For natural numbers , we define
Compute the value of
.
Problem 59
Fermi and Feynman play the game in which Fermi wins with probability
, where
and
are relatively prime positive integers such that
. The rest of the time Feynman wins (there are no ties or incomplete games). It takes a negligible amount of time for the two geniuses to play
so they play many many times. Assuming they can play infinitely many games (eh, they're in Physicist Heaven, we can bend the rules), the probability that they are ever tied in total wins after they start (they have the same positive win totals) is
. Find the value of
.
Problem 60
Triangle has
and
. Point
is on
so that
bisects angle
. The circle through
, and
has center
and intersects line
again at
, and likewise the circle through
, and
has center
and intersects line
again at
. If the four points
, and
lie on a circle, find the length of
.