Stats/Number Theory
The numbers 3, 5, 7, a, and b have an average (arithmetic mean) of 15. What is the average of a and b?
A) 0 B) 15 C) 30 D) 45 E) 60
If a fair coin is flipped six times, what is the probability of getting exactly two heads?
15/64
A box contains 5 red socks, 20 purple socks, 19 blue socks, and 12 green socks. What is the minimum number of socks that need to be picked in order to be certain that at least two purple socks were chosen?
A) 2 B) 20 C) 36 D) 37 E) 38
E) 38
What is the median of the following list of 4040 numbers?
1, 2, 3, 4, ..., 2020, 1^2, 2^2, 3^2, ....2020^2
1976.5
The faces of each of 7 standard dice are labeled with the integers from 1 to 6. Let p be the probability that when all 7 dice are rolled, the sum of the numbers on the top faces is 10. What is the other sum that occurs with the same probability?
A) 13 B) 26 C) 32 D) 39 E) 42
D) 39
Each piece of candy in a store costs a whole number of cents. Casper has exactly enough money to buy either 12 pieces of red candy, 14 pieces of green candy, 15 pieces of blue candy, or n pieces of purple candy. A piece of purple candy costs 20 cents. What is the smallest possible value of n?
A) 18 B) 21 C) 24 D) 25 E) 28
B) 21
A positive integer divisor of 12! is chosen at random. The probability that the divisor chosen is a perfect square can be expressed as m/n, where m and n are relatively prime positive integers. What is m+n?
A) 3 B) 5 C) 12 D) 18 E) 23
E) 23
When 7 fair standard 6-sided dice are thrown, the probability that the sum of the numbers on the top 10 faces 10 can be written as n/6^7, where n is a positive integer. What is n?
A) 42 B) 49 C) 56 D) 63 E) 84
E) 84
For a set of four distinct lines in a plane, there are exactly N distinct points that lie on two or more of the lines. What is the sum of all possible values of N?
A) 14 B) 16 C) 18 D) 19 E) 21
D) 19
A number m is randomly selected from the set 11, 13, 15, 17, 19, and a number n is randomly selected from the set 1999, 2000, 2001, ..., 2018. What is the probability that m^n has a unit's digit of 1?
A) 1/5 B) 1/4 C) 3/10 D) 7/20 E) 2/5
E) 2/5
Amelia has a coin that lands heads with probability 1/3, and Blaine has a coin that lands on heads with probability 2/5. Amelia and Blaine alternately toss their coins until someone gets a head; the first one to get a head wins. All coin tosses are independent. Amelia goes first. The probability that Amelia wins is p/q, where p and q are relatively prime positive integers. What is q-p?
A) 1 B) 2 C) 3 D) 4 E) 5
D) 4
The numbers 1, 2,...,9 are randomly placed into the 9 squares of a 3x3 grid. Each square gets one number, and each of the numbers is used once. What is the probability that the sum of the numbers in each row and each column is odd?
A) 1/21 B) 1/14 C) 5/63 D) 2/21 E) 1/7
B) 1/14