Counting
How many ways can you select two different integers between 1 and 10 (inclusive) which differ by no more than 2? Order does not matter.
17
A semicircle is attached to each side of a 3-4-5 triangle. What is the total area?
6 + 25pi/4
One of the roots of the polynomial 2x2 - x - 10 is -2. What is the other root?
5/2, or 2.5
How many proper divisors does 6! have?
29
What is the probability that 6 and 7 come before 8 and 9 in a uniformly random permutation of the first 100 natural numbers?
1/6
The midpoints of the four sides of a rectangle are (-3, 0), (2, 0), (5, 4) and (0, 4). What is the area of the rectangle?
40
What is the minimum value of (3x2 + 5x + 1) + (6x2 - 14x - 10)?
-45/4
What are the last two digits of 20262026?
76
How many ways can the letters in "ALPHABETA" be arranged so that no "A"s are adjacent?
25200
Two unit circles are placed with their centers at (0, 0) and at (0, 1). What is the area of their intersection?
2pi/3 - sqrt(3)/2
Fully factor 30x3 - 53x2 + 31x - 6.
(2x-1)(3x-2)(5x-3)
What is the smallest positive integer which has a remainder of 3 when divided by 5, has a remainder of 5 when divided by 7, and has a remainder of 7 when divided by 9?
313
Probabilly randomly arranges the integers 1 through 9 in a three by three grid. Then, he counts up the number of pairs of adjacent integers (vertically or horizontally, not diagonally) which also happen to differ by exactly 1. What is the expected value of this count?
8/3 = 2.66..
An equilateral triangle with side length 2 has 4 circles drawn in it: its incircle, and, in each of the 3 regions this creates, a smaller circle that is as large as possible. What is the radius of the smaller circles?
1/(3 sqrt(3)), or sqrt(3)/9
What is the sum of all real zeros of sin(ln(x)) which are no larger than 1?
1/(1-e^-pi)
What is the largest exponent k so that 15^k divides 30! (thirty factorial)?
7