Question 1
The ratio is:
Solution
Question 2
For the nonzero numbers and define Find .
Solution
Question 3
The arithmetic mean of the nine numbers in the set is a -digit number , all of whose digits are distinct. The number does not contain the digit
Solution
Question 4
What is the value of
when ?
Solution
Question 5
Circles of radius and are externally tangent and are circumscribed by a third circle, as shown in the figure. Find the area of the shaded region.
Solution
Question 6
For how many positive integers is a prime number?
Solution
Question 7
Let be a positive integer such that is an integer. Which of the following statements is not true?
Solution
Question 8
Suppose July of year has five Mondays. Which of the following must occur five times in the August of year ? (Note: Both months have days.)
Solution
Question 9
Using the letters , , , , and , we can form five-letter "words". If these "words" are arranged in alphabetical order, then the "word" occupies position
Solution
Question 10
Suppose that and are nonzero real numbers, and that the equation has solutions and . what is the pair ?
Solution
Question 11
The product of three consecutive positive integers is times their sum. What is the sum of their squares?
Solution
Question 12
For which of the following values of does the equation have no solution for ?
Solution
Question 13
Find the value(s) of such that is true for all values of .
Solution
Question 14
The number is the square of a positive integer . In decimal representation, the sum of the digits of is
Solution
Question 15
The positive integers , , , and are all prime numbers. The sum of these four primes is
Solution 1
Solution 2
Since A+B is odd, one of A, B is odd and the other is even, ie prime even 2.
Since A+B > A > A-B, it follows that A is odd and B = 2.
So sum of 4 primes = 3 * A + 2
It cannot be even, cannot be divided by 3, cannot be divided by 5, cannot be divided by 7, so the only right choice is E.
Question 16
For how many integers is the square of an integer?
Solution
Question 17
A regular octagon has sides of length two. Find the area of .
Solution
Question 18
Four distinct circles are drawn in a plane. What is the maximum number of points where at least two of the circles intersect?
Solution
Question 19
Suppose that is an arithmetic sequence with What is the value of
Solution
Question 20
Let and be real numbers such that and Then is
Solution
Question 21
Andy's lawn has twice as much area as Beth's lawn and three times as much as Carlos' lawn. Carlos' lawn mower cuts half as fast as Beth's mower and one third as fast as Andy's mower. If they all start to mow their lawns at the same time, who will finish first?
Solution
Question 22
Let be a right-angled triangle with . Let and be the midpoints of the legs and , respectively. Given and , find .
Solution
Question 23
Let be a sequence of integers such that and for all positive integers and Then is
Solution
Question 24
Riders on a Ferris wheel travel in a circle in a vertical plane. A particular wheel has radius feet and revolves at the constant rate of one revolution per minute. How many seconds does it take a rider to travel from the bottom of the wheel to a point vertical feet above the bottom?
Solution
Question 25
When is appended to a list of integers, the mean is increased by . When is appended to the enlarged list, the mean of the enlarged list is decreased by . How many integers were in the original list?
Solution
Answer Keys
Question 1: E
Question 2: C
Question 3: A
Question 4: D
Question 5: E
Question 6: B
Question 7: E
Question 8: D
Question 9: D
Question 10: C
Question 11: B
Question 12: E
Question 13: D
Question 14: B
Question 15: E
Question 16: D
Question 17: C
Question 18: D
Question 19: C
Question 20: B
Question 21: B
Question 22: B
Question 23: D
Question 24: D
Question 25: A