Question 1
You and five friends need to raise dollars in donations for a charity, dividing the fundraising equally. How many dollars will each of you need to raise?
Solution
Question 2
For any three real numbers , , and , with , the operation is defined by: What is ?
Solution
Question 3
Alicia earns 20 dollars per hour, of which is deducted to pay local taxes. How many cents per hour of Alicia's wages are used to pay local taxes?
Solution
Question 4
What is the value of if ?
Solution
Question 5
A set of three points is randomly chosen from the grid shown. Each three point set has the same probability of being chosen. What is the probability that the points lie on the same straight line?
Solution
Question 6
Bertha has 6 daughters and no sons. Some of her daughters have 6 daughters, and the rest have none. Bertha has a total of 30 daughters and granddaughters, and no great-granddaughters. How many of Bertha's daughters and grand-daughters have no daughters?
Solution
Question 7
A grocer stacks oranges in a pyramid-like stack whose rectangular base is 5 oranges by 8 oranges. Each orange above the first level rests in a pocket formed by four oranges below. The stack is completed by a single row of oranges. How many oranges are in the stack?
Solution
Question 8
A game is played with tokens according to the following rule. In each round, the player with the most tokens gives one token to each of the other players and also places one token in the discard pile. The game ends when some player runs out of tokens. Players , , and start with 15, 14, and 13 tokens, respectively. How many rounds will there be in the game?
Solution
Question 9
In the figure, and are right angles. , and and intersect at . What is the difference between the areas of and ?
Solution
Question 10
Coin is flipped three times and coin is flipped four times. What is the probability that the number of heads obtained from flipping the two fair coins is the same?
Solution
Question 11
A company sells peanut butter in cylindrical jars. Marketing research suggests that using wider jars will increase sales. If the diameter of the jars is increased by without altering the volume, by what percent must the height be decreased?
Solution
Suppose old radius is 1 and old height is h, then:
Old volume = h * 1^2 = h.
Radius of new jar = 1 + 1/4
Area of new base = pi * (1 + 1/4) ^ 2
Suppose new height = x * old height.
Old Volume = New Volume = area of base * height
h = (1 + 1/4) ^ 2 * x * h
x = 1 / (1 + 1/4) ^ 2 = 16/25
Comparing x*h with h, we see the difference is 9/25, or 36%.
The key to not get confused is to understand that if a value x has changed by y%, then new value = x + y/100.
Question 12
Henry's Hamburger Heaven offers its hamburgers with the following condiments: ketchup, mustard, mayonnaise, tomato, lettuce, pickles, cheese, and onions. A customer can choose one, two, or three meat patties, and any collection of condiments. How many different kinds of hamburgers can be ordered?
Solution
Question 13
At a party, each man danced with exactly three women and each woman danced with exactly two men. Twelve men attended the party. How many women attended the party?
Solution
Question 14
The average value of all the pennies, nickels, dimes, and quarters in Paula's purse is cents. If she had one more quarter, the average would be cents. How many dimes does she have in her purse?
Solution
Question 15
Given that and , what is the largest possible value of ?
Solution
Question 16
The grid shown contains a collection of squares with sizes from to . How many of these squares contain the black center square?
Solution
4 of the 2*2 squares containing the black square,
9 of the 3*3 squares containing the black square,
4 of the 4*4 squares containing the black square,
1 of the 5*5 squares containing the black square.
Thus, the answer is 1+4+9+4+1=19
Question 17
Brenda and Sally run in opposite directions on a circular track, starting at diametrically opposite points. They first meet after Brenda has run 100 meters. They next meet after Sally has run 150 meters past their first meeting point. Each girl runs at a constant speed. What is the length of the track in meters?
Solution
Question 18
A sequence of three real numbers forms an arithmetic progression with a first term of 9. If 2 is added to the second term and 20 is added to the third term, the three resulting numbers form a geometric progression. What is the smallest possible value for the third term of the geometric progression?
Solution
Let d be the common difference and r be the common ratio. Then the arithmetic sequence is
9, 9+d, and 9+2d
The geometric sequence is
9, 11+d, and 29+2d
Since the middle term is the geometric mean of the other two terms, we have
(11+d)^2 = 9*(2d+29), rearranging and we get:
(d+14)(d-10) = 0. The smallest possible value occurs when d = -14, and at that time, the third term is 2(-14) + 29 = 1
Question 19
A white cylindrical silo has a diameter of 30 feet and a height of 80 feet. A red stripe with a horizontal width of 3 feet is painted on the silo, as shown, making two complete revolutions around it. What is the area of the stripe in square feet?
Solution
Question 20
Points and are located on square so that is equilateral. What is the ratio of the area of to that of ?
Solution
Question 21
Two distinct lines pass through the center of three concentric circles of radii 3, 2, and 1. The area of the shaded region in the diagram is of the area of the unshaded region. What is the radian measure of the acute angle formed by the two lines? (Note: radians is degrees.)
Solution
Question 22
Square has side length . A semicircle with diameter is constructed inside the square, and the tangent to the semicircle from intersects side at . What is the length of ?
Solution
Question 23
Circles , , and are externally tangent to each other and internally tangent to circle . Circles and are congruent. Circle has radius and passes through the center of . What is the radius of circle ?
Solution
Question 24
Let , be a sequence with the following properties.
- (i) , and
- (ii) for any positive integer .
What is the value of ?
Solution
Question 25
Three pairwise-tangent spheres of radius 1 rest on a horizontal plane. A sphere of radius 2 rests on them. What is the distance from the plane to the top of the larger sphere?
Solution
Answer Keys
Question 1: A
Question 2: B
Question 3: E
Question 4: D
Question 5: C
Question 6: E
Question 7: C
Question 8: B
Question 9: B
Question 10: D
Question 11: C
Question 12: C
Question 13: D
Question 14: A
Question 15: D
Question 16: D
Question 17: C
Question 18: A
Question 19: C
Question 20: D
Question 21: B
Question 22: D
Question 23: D
Question 24: D
Question 25: B