2006 iTest Problems
Contents
- 1 Multiple Choice Section
- 1.1 Problem 1
- 1.2 Problem 2
- 1.3 Problem 3
- 1.4 Problem 4
- 1.5 Problem 5
- 1.6 Problem 6
- 1.7 Problem 7
- 1.8 Problem 8
- 1.9 Problem 9
- 1.10 Problem 10
- 1.11 Problem 11
- 1.12 Problem 12
- 1.13 Problem 13
- 1.14 Problem 14
- 1.15 Problem 15
- 1.16 Problem 16
- 1.17 Problem 17
- 1.18 Problem 18
- 1.19 Problem 19
- 1.20 Problem 20
- 2 Short Answer Section
- 3 Solution
- 4 Ultimate Question
- 5 See Also
Multiple Choice Section
Problem 1
Find the number of positive integral divisors of 2006.
Problem 2
Find the harmonic mean of 10 and 20.
Problem 3
Let be distinct positive integers such that the product
. What is the largest possible value of the sum
?
Problem 4
Four couples go ballroom dancing one evening. Their first names are Henry, Peter, Louis, Roger, Elizabeth, Jeanne, Mary, and Anne. If Henry's wife is not dancing with her husband (but with Elizabeth's husband), Roger and Anne are not dancing, Peter is playing the trumpet, and Mary is playing the piano, and Anne's husband is not Peter, who is Roger's wife?
Problem 5
A line has y-intercept and forms a right angle to the line
. Find the x-intercept of the line.
Problem 6
What is the remainder when is divided by 7?
Problem 7
The sum of consecutive integers is
. Find the second largest integer.
Problem 8
The point is a point on a circle with center
. Perpendicular lines are drawn from
to perpendicular diameters,
and
, meeting them at points
and
, respectively. If the diameter of the circle is
, what is the length of
?
Problem 9
If and
is in the third quadrant, what is the absolute value of
?
Problem 10
Find the number of elements in the first rows of Pascal's Triangle that are divisible by
.
Problem 11
Find the radius of the inscribed circle of a triangle with sides of length ,
, and
.
Problem 12
What is the highest possible probability of getting of these
multiple choice questions correct, given that you don't know how to work any of them and are forced to blindly guess on each one?
Problem 13
Suppose that are three distinct prime numbers such that
. Find the maximum possible value for the product
.
Problem 14
Find , where
is the smallest positive integer such that
leaves a remainder of
when divided by
,
, and
.
Problem 15
How many integers between and
, inclusive, are perfect squares?
Problem 16
The Minnesota Twins face the New York Mets in the 2006 World Series. Assuming the two teams are evenly matched (each has a probability of winning any game) what is the probability that the World Series (a best of 7 series of games which lasts until one team wins four games) will require the full seven games to determine a winner?
Problem 17
Let . Find the numerical value of
.
Problem 18
Every even number greater than 2 can be expressed as the sum of two prime numbers.'
Name the mathematician for which this theorem was named, and then name the mathematician to whom he transmitted this theorem via letter in 1742.
Problem 19
Questions 19 and 20 are Sudoku-related questions. Sudoku is a puzzle game that has one and only one solution for each puzzle. Digits from 1 to 9 must go into each space on the grid such that every row, column, and
square contains one and only one of each digit.
Find the sum of by solving the Sudoku puzzle below.
1 _ _ | 3 5 8 | _ _ 6 4 _ _ | _ _ _ | _ x 8 _ _ 9 | _ 1 _ | 7 _ _ --------------------- _ z _ | 1 _ _ | _ 5 _ _ _ 3 | 2 _ 4 | 8 _ _ _ 2 _ | w _ 9 | _ _ _ --------------------- _ _ 6 | _ 2 _ | 9 _ _ 3 _ _ | _ y _ | _ _ 1 2 _ _ | 8 4 3 | _ _ 7
Problem 20
Sudoku is a puzzle game that has one and only one solution for each puzzle. Digits from 1 to 9 must go into each space on the grid such that every row, column, and
square contains one and only one of each digit.
Find the sum of by solving the Sudoku puzzle below.
_ _ _ | _ 4 _ | _ z _ 1 _ 6 | _ _ _ | 7 _ 3 5 _ _ | 9 _ _ | _ _ 2 --------------------- _ 8 3 | w 2 _ | 5 _ _ 2 _ _ | 5 _ 9 | _ _ 7 _ _ 7 | _ 8 _ | 9 2 _ --------------------- 3 _ _ | _ _ 1 | _ _ 6 8 _ 9 | x _ _ | 3 _ 5 _ y _ | _ 3 _ | _ _ _
Short Answer Section
Problem 21
What is the last (rightmost) digit of ?
Problem 22
Triangle has sidelengths
,
, and
. Point
is the foot of the altitude from
, and
lies on segment
such that
. Find the area of the triangle
.
Problem 23
Jack and Jill are playing a chance game. They take turns alternately rolling a fair six sided die labeled with the integers 1 through 6 as usual (fair meaning the numbers appear with equal probability.) Jack wins if a prime number appears when he rolls, while Jill wins if when she rolls a number greater than 1 appears. The game terminates as soon as one of them has won. If Jack rolls first in a game, then the probability of that Jill wins the game can be expressed as where
and
are relatively prime positive integers. Compute
.
Problem 24
Points and
are chosen on side
of triangle
such that
is between
and
and
,
. If
, the area of
can be expressed as
, where
and
are relatively prime positive integers and
is a positive integer not divisible by the square of any prime. Compute
.
Problem 25
The expression reduces to
, where
and
are relatively prime positive integers. Find
.
Problem 26
A rectangle has area and perimeter
. The largest possible value of
can be expressed as
, where
and
are relatively prime positive integers. Compute
.
Problem 27
Line passes through
and into the interior of the equilateral triangle
.
and
are the orthogonal projections of
and
onto
respectively. If
and
, then the area of
can be expressed as
, where
and
are positive integers and
is not divisible by the square of any prime. Determine
.
Problem 28
The largest prime factor of is greater than
. Determine the remainder obtained when this prime factor is divided by
.
Problem 29
The altitudes in triangle have lengths 10, 12, and 15. The area of
can be expressed as
, where
and
are relatively prime positive integers and
is a positive integer not divisible by the square of any prime. Find
.
Problem 30
Triangle is equilateral. Points
and
are the midpoints of segments
and
respectively.
is the point on segment
such that
. Let
denote the intersection of
and
, The value of
can be expressed as
where
and
are relatively prime positive integers. Find
.
Problem 31
The value of the infinite series can be expressed as
where
and
are relatively prime positive numbers. Compute
.
Problem 32
Triangle is scalene. Points
and
are on segment
with
between
and
such that
,
, and
. If
and
trisect
, then
can be written uniquely as
, where
and
are relatively prime positive integers and
is a positive integer not divisible by the square of any prime. Determine
.
Problem 33
Six students sit in a group and chat during a complicated mathematical lecture. The professor, annoyed by the chatter, splits the group into two or more smaller groups. However, the smaller groups with at least two members continue to produce chatter, so the professor again chooses one noisy group and splits it into smaller groups. This process continues until the professor achieves the silence he needs to teach Algebraic Combinatorics. Suppose the procedure can be carried out in ways, where the order of group breaking matters (if A and B are disjoint groups, then breaking up group A and then B is considered different form breaking up group B and then A even if the resulting partitions are identical) and where a group of students is treated as an unordered set of people. Compute the remainder obtained when
is divided by
.
Problem 34
For each positive integer let
denote the set of positive integers
such that
is divisible by
. Define the function
by the rule
Let
be the least upper bound of
and let
be the number of integers
such that
and
. Compute the value of
.
Problem 35
Compute the of ordered quadruples
of complex numbers (not necessarily nonreal) such that the following system is satisfied:
Problem 36
Let denote
. The recursive sequence
satisfies
and, for all positive integers
,
Suppose that the series
can be expressed uniquely as
, where
and
are coprime positive integers and
is not divisible by the square of any prime. Find the value of
.
Problem 37
The positive reals ,
,
satisfy the relations
The value of
can be expressed uniquely as
, where
,
,
,
are positive integers such that
is not divisible by the square of any prime and no prime dividing
divides both
and
. Compute
.
Solution
To begin, let's number the equations as follows
In order to solve for , we begin by subtracting the first equation from the second
equation:
Similarily, subtracting the second eqaution from the third equation:
Subtracting the factored equations we see that
Therefore, we can conclude that either , or that
. As we are told that
has a unique value, we can use either option.
Continuing with the second option, we can plug into the first equation:
Multiplying each side of the resulting equation by four:
We can then subtract twice our original second equation, , from the equation from the previous step, yielding:
Rearranging the new equation, we see that:
We can ignore the negative root of the RHS because both y and z are positive, and because which is a result of (3) being greater than (1).
Continuing to manipulate the newest equation yields:
Plugging this back into (2)
Solving for yields that
Therefore, the solution is
Problem 38
Segment is a diameter of circle
. Point
lies in the interior of segment
such that
, and
is a point on
such that
. Segment
is a diameter of the circle
. A third circle,
, is drawn internally tangent to
, externally tangent to
, and tangent to segment
. If
is centered on the opposite side of
as
, then the radius of
can be expressed as
, where
and
are relatively prime positive integers. Compute
.
Problem 39
is a regular dodecagon. The number 1 is written at the vertex A, and 0's are written at each of the other vertices. Suddenly and simultaneously, the number at each vertex is replaced by the arithmetic mean of the two numbers appearing at the adjacent vertices. If this procedure is repeated a total of
times, then the resulting number at A can be expressed as
, where
and
are relatively prime positive integers. Compute the remainder obtained when
is divided by
.
Problem 40
Acute triangle satisfies
and
. Tetrahedron
is formed by choosing points
,
, and
on the segments
,
, and
(respectively) and folding
,
,
, over
,
, and
(respectively) to the common point
. Let
denote the circumradius of
. Compute the smallest positive integer
for which we can be certain that
. It may be helpful to use
.
Ultimate Question
In the next 2 problems, the problem after will require the answer of the current problem. TNFTPP stands for the number from the previous problem. Problem 41 requires the answer to the third problem. Problem 42 requires the answer to the seventh problem. Problem 43, however, requires the sum of the answers to all ten questions.
For those who want to try these problems without having to find the T-values of the previous problem, a link will be here. Also, all solutions will have the T-values substituted.
Problem 41
Problem U1
Find the real number such that
Problem U2
Let . 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
Let . 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
Let . As
ranges over the integers, the expression
evaluates to just one prime number. Find this prime.
Problem U5
Let , and let
be the sum of the digits of
. 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
Let .
and
are nonzero real numbers such that
The smallest possible value of is equal to
where
and
are relatively prime positive integers. Find
.
Problem U7
Let . Triangle
has integer side lengths, including
, and a right angle,
. Let
and
denote the inradius and semiperimeter of
respectively. Find the perimeter of the triangle ABC which minimizes
.
Problem 43
Problem U8
Let , and let
be the sum of the digits of
. 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
.
Problem U9
Let . Determine the number of 5 element subsets
of
such that the sum of the elements of
is divisible by 5.
Problem U10
Let and let
be the sum of the digits of
. Point
in the interior of triangle
satisfies
,
, and
. If the sides of ABC satisfy
then the area of triangle can be expressed as
, where
and
are relatively prime positive integers. Compute the remainder obtained when
is divided by
.
Recall that you are turning in the sum of all ten answers, NOT the answer to this problem.
See Also
2006 iTest (Problems, Answer Key) | ||
Preceded by: 2005 iTest |
Followed by: 2007 iTest | |
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