Probability or Combinatorics
A box contains a collection of triangular and square tiles. There are 25 tiles in the box, containing 84 edges total. If a random tile is pulled from the box, what is the probability that it is a square?
A) 3/25 B) 1/5 C) 7/25 D) 9/25 E) 11/24
D) 9/25
Two right circular cylinders have the same volume. The radius of the second cylinder is 10% more than the radius of the first. What is the relationship between the heights of the two cylinders?
A) The first height is 21% more than the second. B) The second height is 21% less than the first. C) The second height is 10% less than the first. D) The first height is 10% more than the second. E) The second height is 80% of the first
A) The first height is 21% more than the scond.
What is (2+4+6)/(1+3+5) - (1+3+5)/(2+4+6)?
A) -1 B) 5/36 C) 7/12 D) 49/20 E) 43/3
C) 7/12
Robert does three equally time-consuming tasks in a row without taking breaks. He begins the first task at 13:00 and finishes the second task at 14:40. When does she finish the third task?
A) 15:10 B) 15:30 C) 16:00 D) 16:10 E) 16:30
B) 15:30
Mr. Green measures his rectangular garden by walking two of the sides and finding that it is 15 steps by 20 steps. Each of Mr. Green's steps is 2 feet long. Mr. Green expects a half a pound of potatoes per square foot from his garden. How many pounds of potatoes does Mr. Green expect from his garden?
A) 600 B) 800 C) 1000 D) 1200 E)1400
A) 600
Let S be the set of sides and diagonals of a regular pentagon. A pair of elements of S are selected at random without replacement. What is the probability that two chosen segments have the same length?
A) 2/5 B) 4/9 C) 1/2 D) 5/9 E) 4/5
B) 4/9
Six points are equally spaced around a circle of radius 1. Three of these points are the vertices of a triangle that is neither equilateral nor isosceles. What is the area of this triangle?
A) (sqrt3)/3 B) (sqrt3)/2 C) 1 D) sqrt2 E) 2
B) (sqrt3)/2
When counting from 3 to 201, 53 is the 51st number counted. When counting backwards from 201 to 3, 53 is the nth number counted. What is n?
A) 146 B) 147 C) 148 D) 149 E) 150
Ray's car averages 40 miles per gallon of gasoline, and Tom's car averages 10 miles per gallon of gasoline. Ray and Tom each drive the same number of miles. What is the cars' combined rate of miles per gallon of gasoline?
A) 10 B) 16 C) 25 D) 30 E) 40
B) 16
Ann made a 3-step staircase using 18 toothpicks as shown in the figure. How many toothpicks does she need to add to complete a 5-step staircase?
A) 9 B) 18 C) 20 D) 22 E) 24
D) 22
How many rearrangements of abcd are there in which no two adjacent letters are also adjacent letters in the alphabet? For example, no such rearrangements could include either ab or ba.
A) 0 B) 1 C) 2 D) 3 E) 4
C) 2
Define AvB = A^2 * B - A * B^2. Which of the following describes the set of points (x,y) for which xvy = yvx?
A) a finite set of points B) one line C) two parallel lines D) two intersecting lines E) three lines
E) three lines
The positive integers are each greater than 1, have a product of 27000, and are pairwise relatively prime. What is their sum?
A) 100 B) 137 C) 156 D) 160 E) 165
D) 160
Billy, Bob, Joe, and 28 of their other classmates are playing a game of freeze tag. Every thirty seconds, Joe manages to freeze 3 other people. But each time immediately after those 30 seconds have elapsed, Billy and Bob unfreeze two people total. Assuming Joe will only freeze Billy and Bob when everyone else is frozen, and both Billy and Bob are capable of unfreezing two people by themselves should only one of them be frozen, how much time, in minutes, will it take Joe to freeze all 30 of his classmates?
A) 13.5 B) 14 C) 14.5 D) 15 E) 15.5
B) 14
Two years ago, Pete was three times as old as his cousin Claire. 2 years before that, Pete was four times as old as Claire. In how many years will the ratio of their ages be 2:1?
A) 2 B) 4 C) 5 D) 6 E) 8
B) 4
The number 2013 has the property that its units digit is the sum of its other digits, that is 2+0+1=3. How many integers less than 2013 but greater than 1000 have this property?
A) 33 B) 34 C) 45 D) 46 E) 58
D) 46
Triangle ABC is equilateral with AB=1. Points E and G are on AC and points D and F are on AB such that both DE and FG are parallel to BC. Furthermore, triangle ADE and trapezoids DFGE and FBCG all have the same perimeter. What is DE + FG?
A) 1 B) 3/2 C) 21/13 D) 13/8 E) 5/3
C) 21/13
Points (sqrt(pi),a) and (sqrt(pi),b) are distinct points on the graph of y^2 + x^4 = 2x^2 * y + 1. What is |a-b|?
A) 1 B) pi/2 C) sqrt(1+pi) D) 1+sqrt(pi) E) 2
E) 2
Minnie rides on a flat road at 20 kilometers per hour (kph), downhill at 30 kph, and uphill at 5kph. Penny rides on a flat road at 30 kph, downhill at 40 kph, and uphill at 10 kph. Minnie goes from town A to town B, a distance of 10 km uphill, then from town B to town C, a distance of 15 km all downhill, and then back to town A, a distance of 20 km on the flat. Penny goes the other way around using the same route. How many more minutes does it take Minnie to complete the 45-km ride than it takes Penny?
A) 45 B) 60 C) 65 D) 90 E) 95
C) 65
Alex has 75 red tokens and 75 blue tokens. There is a booth where Alex can give two red tokens and receive in return a silver token and a blue token and another booth where Alex can give three blue tokens and receive in return a silver token and a red token. Alex continues to exchange tokens until no more exchanges are possible. How many silver tokens will Alex have at the end?
A) 62 B) 82 C) 83 D) 102 E) 103
The regular octagon ABCDEFGH has its center at J. Each of the vertices and the center are to be associated with one of the digits 1 through 9, with each digit used once, in such a way that the sums of the numbers on the lines AJE, BJF, CJG, and DJH are all equal. In how many ways can this be done?
A) 384 B) 576 C)1152 D) 1680 E) 3456
C) 1152
A line that passes through the origin intersects both the line x=1 and the line y=1 + (sqrt3)x/3. The three lines create an equilateral triangle. What is the perimeter of the triangle?
A) 2sqrt6 B) 2+2sqrt3 C) 6 D) 3+2sqrt3 E) 6+(sqrt3)/3
D) 3+2sqrt3
The real numbers c, b, a form an arithmetic sequence with a > b > c > 0. The quadratic ax^2 + bx + c has exactly one root. What is this root?
A) -2+sqrt3 B) -2-sqrt3 C) -1 D) -7-4sqrt3 E) -7+4sqrt3
A) -2+sqrt3
There are 10 horses, names Horse 1, Horse 2, ..., Horse 10. They get their names from how many minutes it takes them to run one lap around a circular race track: Horse k runs one lap in exactly k minutes. At time 0 all the horses are together at the starting point on the track. The horses start running in the same direction, and they keep running around the circular track at their constant speeds. The least time S > 0, in minutes, at which all 10 horses will again simultaneously be at the starting point is S = 2520. Let T > 0 be the least time, in minutes, such that at least 5 of the horses are again at the the starting point. What is the sum of the digits of T?
A) 2 B) 3 C) 4 D) 5 E) 6
B) 3
Consider the set of all fractions x/y, where x and y are relatively prime positive integers. How many of these fractions have the property that if both numerator and denominator are increased by 1, the value of the fraction is increased by 10%?
A) 0 B) 1 C) 2 D) 3 E) infinitely many
B) 1