Question 1
While eating out, Mike and Joe each tipped their server dollars. Mike tipped of his bill and Joe tipped of his bill. What was the difference, in dollars between their bills?
Solution
Question 2
For each pair of real numbers , define the operation as
.
What is the value of ?
Solution
Question 3
The equations and have the same solution . What is the value of ?
Solution
Question 4
A rectangle with a diagonal of length is twice as long as it is wide. What is the area of the rectangle?
Solution
Question 5
A store normally sells windows at $100 each. This week the store is offering one free window for each purchase of four. Dave needs seven windows and Doug needs eight windows. How many dollars will they save if they purchase the windows together rather than separately?
Solution
Question 6
The average (mean) of numbers is , and the average of other numbers is . What is the average of all numbers?
Solution
Question 7
Josh and Mike live miles apart. Yesterday Josh started to ride his bicycle toward Mike's house. A little later Mike started to ride his bicycle toward Josh's house. When they met, Josh had ridden for twice the length of time as Mike and at four-fifths of Mike's rate. How many miles had Mike ridden when they met?
Solution
Question 8
In the figure, the length of side of square is and . What is the area of the inner square ?
Solution
Question 9
Three tiles are marked and two other tiles are marked . The five tiles are randomly arranged in a row. What is the probability that the arrangement reads ?
Solution
Question 10
There are two values of for which the equation has only one solution for . What is the sum of those values of ?
Solution
Question 11
A wooden cube units on a side is painted red on all six faces and then cut into unit cubes. Exactly one-fourth of the total number of faces of the unit cubes are red. What is ?
Solution
Since each unit cube has 6 faces, and there are n^3 unit cubes, there are 6*n^3 little faces total.
Since one-fourth of the little faces are painted red, we have:
6n^2 / 6n^3 = 1/4
Solving above, we have n=4
Question 12
The figure shown is called a trefoil and is constructed by drawing circular sectors about the sides of the congruent equilateral triangles. What is the area of a trefoil whose horizontal base has length ?
Solution
Question 13
How many positive integers satisfy the following condition:
?
Solution
130n > n^2 > 16
Solving each part separately:
n^2 > 16 -> n > 4
130n > n^2 -> 130 > n
Therefore the answer is the number of positive integers over the interval (4,130) which is 125.
Question 14
How many three-digit numbers satisfy the property that the middle digit is the average of the first and the last digits?
Solution
If both the first digit and the last digit are odd, then we have 1, 3, 5, 7, or 9 as choices for each of these digits, and there are 5*5=25 numbers in this case.
If both the first and last digits are even, then we have 2, 4, 6, 8 as our choices for the first digit and 0, 2, 4, 6, 8 for the third digit. There are 4*5=20 numbers here.
The total number, then, is 20+25=45
Question 15
How many positive cubes divide ?
Solution
Therefore, a perfect cube that divides 3!*5!*7! must be in the form 2^a*3^b*5^c*7^d where a, b, c, and d are nonnegative multiples of 3 that are less than or equal to 8, 4, 2 and 1, respectively.
So:
a in {0,3,6} (3 possibilities)
b in {0,3} (2 possibilities)
c in {0} (1 possibility)
d in {0} (1 possibility)
So the number of perfect cubes that divide 3!*5!*7! is 3 * 2 * 1 * 1 = 6
Question 16
The sum of the digits of a two-digit number is subtracted from the number. The units digit of the result is . How many two-digit numbers have this property?
Solution
So (10a+b)-(a+b)=9a must have a units digit of 6
This is only possible if 9a=36, so a=4 is the only way this can be true.
So the numbers that have this property are 40, 41, 42, 43, 44, 45, 46, 47, 48, 49.
Therefore the answer is 10
Question 17
In the five-sided star shown, the letters , , , , and are replaced by the numbers , , , , and , although not necessarily in this order. The sums of the numbers at the ends of the line segments , , , , and form an arithmetic sequence, although not necessarily in this order. What is the middle term of the sequence?
Solution
Question 18
Team A and team B play a series. The first team to win three games wins the series. Each team is equally likely to win each game, there are no ties, and the outcomes of the individual games are independent. If team B wins the second game and team A wins the series, what is the probability that team B wins the first game?
Solution
Question 19
Three one-inch squares are placed with their bases on a line. The center square is lifted out and rotated 45 degrees, as shown. Then it is centered and lowered into its original location until it touches both of the adjoining squares. How many inches is the point from the line on which the bases of the original squares were placed?
Solution
Question 20
An equiangular octagon has four sides of length 1 and four sides of length , arranged so that no two consecutive sides have the same length. What is the area of the octagon?
Solution
Question 21
For how many positive integers does evenly divide ?
Solution
The sum of x1 + x2 + x3 + ... + xn is: ( x1 + xn ) * n / 2
6n/(( x1 + xn ) * n / 2) = 12/(n+1). When n=1,2,3,5,11 will it be an integer.
Question 22
Let be the set of the smallest positive multiples of , and let be the set of the smallest positive multiples of . How many elements are common to and ?
Solution
T = {6, 12, 18, 24, ..., 6*2005} = {6, 12, 18, 24, ..., 12030}
Least common multiples of 4 and 6 is 12.
All multiples of 12 in S are also in T.
In multiples of 4, every 3rd one is a multiple of 12. So in S, we have 2005/3 = 668 multiples of 12.
Question 23
Let be a diameter of a circle and let be a point on with . Let and be points on the circle such that and is a second diameter. What is the ratio of the area of to the area of ?
Solution
Question 24
For each positive integer , let denote the greatest prime factor of . For how many positive integers is it true that both and ?
Solution
Question 25
In we have , , and . Points and are on and respectively, with and . What is the ratio of the area of triangle to the area of the quadrilateral ?
Solution
Answer Keys
Question 1: D
Question 2: C
Question 3: B
Question 4: B
Question 5: A
Question 6: B
Question 7: B
Question 8: C
Question 9: B
Question 10: A
Question 11: B
Question 12: B
Question 13: E
Question 14: E
Question 15: E
Question 16: D
Question 17: D
Question 18: A
Question 19: D
Question 20: A
Question 21: B
Question 22: D
Question 23: C
Question 24: B
Question 25: D