Algebra
The sums of three whole numbers taken in pairs are 12, 17, and 19. What is the middle number?
7
Four distinct points are arranged on a plane so that the segments connecting them have lengths a, a, a, a, 2a, and b. What is the ratio of b to a?
sqrt(3)
For which of the following integers b is the base-b number 2021b - 221b not divisible by 3?
3, 4, 6, 7, 8
8
Chloe chooses a real number uniformly at random from the interval [0, 2017]. Independently, Laurent chooses a real number uniformly at random from the interval [0, 4034]. What is the probability that Laurent's number is greater than Chloe's number?
3/4
Halfway through a 100-shot archery tournament, Chelsea leads by 50 points. For each shot a bullseye scores 10 points, with other possible scores being 8, 4, 2, and 0 points. Chelsea always scores at least 4 points on each shot. If Chelsea's next n shots are bullseyes she will be guaranteed victory. What is the minimum value for n?
42
Rectangle ABCD has AB = 6 and BC = 3. Point M is chosen on side AB so that angle AMD = angle CMD. What is the degree measure of angle AMD?
75 degrees
The largest divisor of 2,014,000,000 is itself. What is its fifth-largest divisor?
251,750,000
(or 2,014,000,000/8)
There are 20 students participating in an after-school program offering classes in yoga, bridge, and painting. Each student must take at least one of these three classes, but may take two or all three. There are 10 students taking yoga, 13 taking bridge, and 9 taking painting. There are 9 students taking at least two classes. How many students are taking all three classes?
3 students
A, B, C are three piles of rocks. The mean weight of the rocks in A is 40 pounds, the mean weight of the rocks in B is 50 pounds, the mean weight of the rocks in the combined piles A and B is 43 pounds, and the mean weight of the rocks in the combined piles A and C is 44 pounds. What is the greatest possible integer value for the mean in pounds of the rocks in the combined piles B and C?
59
On square ABCD, point E lies on side AD and point F lies on side BC, so that BE = EF = FD = 30. Find the area of the square ABCD.
810
Mary chose an even 4-digit number n. She wrote down all the divisors of n in increasing order from left to right: 1, 2, . . . n/2, n. At some moment Mary wrote 323 as a divisor of n. What is the smallest possible value of the next divisor written to the right of 323?
340
Eight people are sitting around a circular table, each holding a fair coin. All eight people flip their coins and those who flip heads stand while those who flip tails remain seated. What is the probability that no two adjacent people will stand?
47/256