Q1. 3/8 of all students at Social High are in all three of the following clubs: Albanian, Bardic, and Checkmate.  1/2 of all students are in Albanian,  5/8 are in Bardic, and  3/4 are in Checkmate. If every student is in at least one club, what fraction of the student body is in exactly 2 clubs?

A) 1/8 B) 1/4 C) 3/8 D) 1/2 E) 5/8

Q2. Every trading day, the price of CF Corp stock either goes up by $1 or goes down by $1 with equal likelihood. At the end of 5 trading days, what is the probability that the price of CF Corp stock is up by exactly $3 from its initial price?

A) 1/16 B) 1/8 C) 5/32 D) 9/32 E) 3/8

Q3. The harmonic mean of two numbers x and y, symbolized as h(x, y), is defined as 2 divided by the sum of the reciprocals of x and y, whereas the geometric mean g(x, y) is defined as the square root of the product of x and y (when this square root exists), and the arithmetic mean m(x, y) is defined as (x + y)/2 . For which of the following pairs of values for x and y is g(x, y) equal to the arithmetic mean of h(x, y) and m(x, y) ?

A) x = -2, y = -1 B) x = -1, y = 2 C) x = 2, y = 8D) x = 8, y = 8 E) x = 8, y = 64

Q4. A total of j dollars is given to Abby, Bill, and Carla and divided equally among them. Then Abby gives Bill k dollars, Bill gives Carla 2k dollars, and Carla gives Abby 3k dollars. Afterwards, Abby has exactly half of all the dollars given to the three people. In terms of k, how much money was given originally to Abby, Bill, and Carla?

A) 4k B) 6k C) 8k D) 12k E) 16k

Q5. What is the 99th digit after the decimal point in the decimal expansion of 2/9 + 3/11?

A) 1 B) 2 C) 4 D) 7 E) 9

Q6. If v is not equal to zero, is |w| < |v|? B

1) w/v < 12) w2/v2 < 1

Q7. In the xy-plane, region Q consists of all points (x, y) such that x2 + y2 ≤ 100. Is the point (a, b) in region Q? (E)

1) a + b = 142) a > b

Q8. The three digit positive integer n can be written as ABC, in which A, B, and C stand for the unknown digits of n. What is the remainder when n is divided by 37? (D)

1) A + B/10 + C/100 = B + C/10 + A/1002) A + B/10 + C/100 = C + A/10 + B/100

Q9. Positive integer n leaves a remainder of 4 after division by 6 and a remainder of 3 after division by 5. If n is greater than 30, what is the remainder that  leaves after division by 30?

A) 3 B) 12 C) 18 D) 22 E) 28

Q10. For which of the following functions f(x) is the relation f(f(x)) = f(f(f(f(x)))) NOT true for at least some values of x not equal to zero?

A) f(x) = -|x| B) f(x) = 2 – x C) f(x) = 3x D) f(x) = 4/xE) f(x) = 5

Q11. A number of eggs dyed various colors were hidden for an egg hunt. How many eggs in total were hidden?(E)

1) The number of red eggs hidden is the square of an integer, while the total number of eggs hidden is 24 times that integer.2) Exactly 143 of the eggs hidden were not red.

Q12. If ab ≠ 0 and a ≠ -3b, what is the value of (4a + 6b)/(a + 3b)?(B)

1) a – 3b = 62) 2a/(a + 3b) = 4

Q13. x is replaced by 1 – x everywhere in the expression 1/x – 1/(1 - x), with x ≠ 0 and x ≠ 1. If the result is then multiplied x2 – x, the outcome equals

A) x + 1B) x – 1C) 1 – x2 D) 2x – 1 E) 1 – 2x

Q14. A circle is inscribed in an equilateral triangle, such that the two figures touch at exactly 3 points, one on each side of the triangle. Which of the following is closest to the percent of the area of the triangle that lies within the circle?

A) 50% B) 55% C) 60% D) 65% E) 70%

Q15. Set S consists of 5 values, not necessarily in ascending order: {4, 8, 12, 16, x}. For how many values of x does the mean of set S equal the median of set S?

A) Zero B) One C) Two D) Three E) More than three

Q16. If n is a positive integer greater than 1, then p(n) represents the product of all the prime numbers less than or equal to n. The second smallest prime factor of  p(12) + 11 is

A) 2 B) 11 C) 13 D) 17 E) 211

Q17. Two different primes may be said to"rhyme" around an integer if they are the same distance from the integer on the number line. For instance, 3 and 7 rhyme around 5. What integer between 1 and 20, inclusive, has the greatest number of distinct rhyming primes around it?

A) 12 B) 15 C) 17 D) 18 E) 20

Q18. If 1/(x – 2) = 1/(x + 2) + 1/(x – 1), which of the following is a possible value of x?

A) -2 B) -1 C) 0 D) 1 E) 2

Q19. Several friends in a dinner group decide to contribute equally to the purchase of a $36 gift. How many people are in the group?(D)

1) The number of people in the group is equal to the size of each person’s contribution, in dollars.2) If three more people joined the group, each person’s individual contribution would fall by $2.

Q20. If a, b, and c are positive integers, what is the remainder when b – a is divided by 3?(C)

1) a = c3

2) b = (c + 1)3

Q21. If 2x + 2y = x2 + y2, where x and y are nonnegative integers, what is the greatest possible value of |x – y|?

A) 0 B) 1 C) 2 D) 3 E) 4

Q22. How many factors does x have, if x is a positive integer?(C)

1) x = pn, where p is a prime number2) nn = n + n, where n is a positive integer

Q23. The function g(x) is defined as the greatest integer less than or equal to x, while the function h(x) is defined as the least integer greater than or equal to x. What is the product g(1.7) * h(2.3) * g(-1.7) *h(-2.3)?

A) 6 B) 9 C) 12 D) 16 E) 24

Q24. If p, q, r, and s are consecutive integers, with p < q < r < s, is pr < qs? (A)

1) pq < rs2) ps < qr

Q25. The product of the digits of the four-digit number h is 36. No two digits of  are identical. How many different numbers are possible values of h?

A) 6 B) 12 C) 24 D) 36 E) 48

Q26. Positive integers a, b, c, m, n, and p are defined as follows: m = 2a3b, n = 2c, and p = 2m/n, is p odd?(B)

1) a < b2) a < c

Q27. A rectangular solid box is x inches long, y inches wide and z inches tall, where x, y, and z are positive integers, exactly two of which are equal. What is the total surface area of the box?(E)

1) One face of the box has an area of 9 square inches.2) One face of the box has an area of 81 square inches.

Q28. Max(x, y) is defined as the maximum of x and y, and Min (x, y) is defined as the minimum of  x and y. What is the average of Max(x, 60) and Min(40, x)?(C)

1) Min(x, 60) = x2) Max(40, x) = x

Q29. Given that x4 – 25x2 = -144, which of the following is NOT a sum of two possible values of z?

A) -7 B) -1 C) 0 D) 3 E) 7

Q30. Is (x – 2)2 > x2?(C)

1) x2 > x2) 1/x > 0

Q31. Each of the cucumbers in 100 pounds of cucumbers is composed of 99% water, by weight. After some of the water evaporates, the cucumbers are now 98% water by weight. What is the new weight of the cucumbers, in pounds?

A) 2 B) 50 C) 92 D) 96 E) 98

Q32. While on straight, parallel tracks, train A and train B are traveling at different constant rates. If train A is currently 2 miles behind train B, how many minutes from now will train A be 4 miles behind train B?(D)

1) Three minutes ago, train A was 1 mile behind train B.2) Train A is traveling at 80 miles per hour, and train B is traveling at 100 miles per hour.

Q33. Is abc =2?(E)

1) ab = 22) bc = 2

Q34. A circular table has a glass center with a diameter of 4x inches, which is surrounded by a metal ring with a width of 2 inches. In terms of x, what fraction of the table's surface is made up by the metal ring?

A) 1/(x + 1) B) 1/x C) x2/(x + 1)2 D) (2x + 1)/(x + 1)2

E) x/(x + 1)

Q35. Is x > y?(C)

1) 6x > 5y2) xy < 0

Q36. Each of the numbers a, b, c, d is equal to -1, 0, or 1. What is the value of a + b + c + d?

1) a/2 + b/4 + c/8 + d/16 = 1/82) a/4 + b/16 + c/64 + d/256 = 11/64

Q37. At a delivery store, seven packages have an average (arithmetic mean) weight of 225 pounds and a median weight of 270 pounds. What is the maximum possible weight, in pounds, of the lightest package?

A) 25 B) 165 C) 195 D) 225 E) 270

Q38. Which of the following inequalities has a solution set that, when graphed on a number line, is a single, finite line segment?

A) x ≥ 4 B) x2 ≥ 4 C) x3 ≤ 64 D) |x|≥ 4 E) |x| ≤ 4

Q39. A cylindrical tank has the radius of the base equal to 1 meters and the height equal to 2 meters. The tank is half full of water. If all the water from that tank is poured into another cylindrical tank with the radius of the base equal to 2 meters and the height equal to 1 meters, then what percent of this new tank will be filled with water?

A) 20% B) 25% C) 32% D) 35% E) 40%

Q40. Some part of a 50% solution of acid was replaced with an equal amount of 30% solution of acid. If, as a result, 40% solution of acid was obtained, what part of the original solution was replaced?

A) 1/5 B) 1/4 C) 1/2 D) 3/4 E) 4/5

Q41. If x and y are prime numbers, what is the value of x + y?(E)

1) x - y is a prime integer.2) y < x < 6 

Q42. A bowl contains red and green chips and no chips of any other color. If two chips are drawn from the basket at random without replacement, what is the probability that both chips will be green?(E)

1) 20% of all chips in the basket are green.2) The ratio of the number of red chips to the number of green chips is 4:1.

Q43. What is the product of a certain pair of consecutive odd integers?(A)

1) Exactly one of the integers is negative.2) At least one of the integers is positive.

Q44. If function f(x) satisfies f(x) = f(x2) for all x, which of the following must be true?

A) f(4) = f(2)*f(2) B) f(16) – f(-2) = 0 C) f(-2) + f(4) = 0D) f(3) = 3f(3) E) f(0) = 0

Q45. If the equation |x/2| + |y/2| = 5 encloses a certain region on the coordinate plane, what is the area of this region?

A) 20 B) 50 C) 100 D) 200 E) 400

Q46. The ratio of the number of employees of three companies X, Y and Z is 3:4:8, respectively. Is the average age of all employees in these companies less than 40 years?(B)

1) The total age of all the employees in these companies is 600.2) The average age employees in X, Y, and Z, is 40, 20, and 50, respectively.

Q47. Jack can complete the task in 8 hours. Tom can complete the same task in 12 hours. If Jack works on the task alone for 4 hours and then Tom starts to help him, how many hours in total will Jack have worked by the time the task is finished?

A) 5 B) 5.2 C) 5.6 D) 6.4 E) 7.2

Q48. Between 12:24 and 14:36 the hour hand of the clock turns by

A) 60 degrees B) 66 degrees C) 72 degrees D) 74 degreesE) 76 degrees

Q49. Set T consists of a certain number of even integers divisible by 3. Is standard deviation of T positive?(B)(1) All elements of set T are positive.(2) The range of set T is 0.

Q50. How many integers are divisible by 3 between 10! and 10! + 20 inclusive?

A) 6 B) 7 C) 8 D) 9 E) 10