Counting
How many ways are there to arrange the letters in ADDITION?
10,080
A group of 84 people attended the zoo field trip. For the trip to the zoo, 63 people rode the bus and the remaining people were in cars with 3 people in each car. On the return trip, the same cars carried 4 people each, and the remaining people rode the bus. How many people rode the bus on the return trip?
56
If the degree measures of the angles of a triangle can be expressed as x, 3x – 10 and x – 10, what is the measure of the largest angle?
110 degrees
If 2^2 · 4^4 = 2k , what is the value of k?
512
At a factory, a machine puts a cap and a label on a bottle. The caps, labels and bottles are each available in the same four colors. Calculate the probability that the cap, label and bottle will all be different colors. Express your answer as common fraction.
3/8
After a hockey game, each member of the losing team shook hands with each member of the winning team. Afterwards, each member of the winning team gave a fist-bump to each of her teammates. Each team has 20 players. If n handshakes occurred and m fist-bumps occurred, what is the value of n + m?
590
If the median of the data set {x + 2, x + 3, x − 4, x − 1, x + 1} is 6, what is the value of x?
5
The area of a rectangle with integer side lengths is 32 cm^2. What is the least possible perimeter of the rectangle?
24
If A represents a digit such that the sum of the two-digit numbers 2A, 3A and 4A is the three-digit number 10A, what is the value of A?
5
A fair tetrahedral die, whose faces are numbered 1, 2, 3 and 4 is rolled three times. What is the probability that the sum of the numbers rolled is 7? Express your answer as a common fraction
3/16
The town of Heterochromia has 1200 residents. Each resident has two eyes, and each eye is green, blue or brown. If 400 residents have at least one green eye, 600 residents have at least one blue eye, and 900 residents have at least one brown eye, how many residents have two eyes of the same color?
500
Margaret holds tea parties every Tuesday afternoon for the purpose of using her collection of 100 teacups. If she invites n people, she will use n + 1 teacups: one for each invited guest and one for herself. If she has already had 24 tea parties, each with two guests, how many tea parties with three guests should she host to ensure each teacup is used exactly once?
7
A line with slope 2 intersects a line with slope 6 at the point (40, 30). What is the distance between the x-intercepts of these two lines?
10
Aditya writes down some of his favorite numbers. Every number on his list is a prime number, and no two numbers on his list share any digits in common. What is the greatest possible number of terms in Aditya’s list?
6
Ms. Pauling’s chemistry class has 5 lab benches, each of which seats 2 students. If 6 students enter her otherwise empty classroom, and each student picks a random available open seat, what is the probability that at least one of the lab benches is completely empty?
13/21
How many 3-digit positive integers have digits whose product equals 24?
21
If f is a function such that f(f(x)) = x^2 – 1, what is f(f(f(f(3))))?
63
Right triangle DEF has one side of length 2023 meters and is similar to right triangle ABC where AB=40, BC=75, and angle ABC=90 degrees. What is the area of triangle DEF if all of its sides have integer lengths?
849,660
An abundant number is a number for which the sum of its positive proper factors is greater than the number itself. For example, because the sum of the positive proper factors of 24 is 36, it follows that 24 is an abundant number. What is the least abundant number greater than 24?
30
Diana has two fair spinners. The sectors of the first are numbered with the prime numbers less than 10. The sectors of the second are numbered with the positive perfect squares less than 40. On each of the spinners, all sectors have equal area. What is the probability that if both spinners are spun, the selected numbers on the two are not relatively prime? Express your answer as a common fraction.
1/4
Dak has a quarter, a dime, a nickel, and a penny. How many different amounts can be obtained by using one or more of the coins in Dak's collection?
15
A bug crawls a distance of n miles at a speed of n + 1 miles per hour one day and then crawls 2n + 1 miles at a speed of n^2 + n miles per hour the next day. If the total time for the trip was 6 hours, what was the bug’s average speed? Express your answer as a common fraction.
4/15
For positive integers n and m, each exterior angle of a regular n-sided polygon is 45 degrees larger than each exterior angle of a regular m-sided polygon. One example is n = 4 and m = 8 because the measures of each exterior angle of a square and a regular octagon are 90 degrees and 45 degrees, respectively. What is the greatest of all possible values of m?
56
If a, b and c are all positive integers greater than 1 for which a + ab + abc = 415, what is the value of c?
40
Yu has 12 coins, consisting of 5 pennies, 4 nickels and 3 dimes. He tosses them all in the air. What is the probability that the total value of the coins that land heads-up is exactly 30 cents? Express your answer as a common fraction.
23/2048