Algebra
Geometry
Count/probability
Number theory
Random/any topic
100
At Zoom University, people’s faces appear as circles on a rectangular screen. The radius of one’s face is directly proportional to the square root of the area of the screen it is displayed on. Haydn’s face has a radius of 2 on a computer screen with area 36. What is the radius of his face on a 16 × 9 computer screen?
4
100
A Yule log is shaped like a right cylinder with height 10 and diameter 5. Freya cuts it parallel to its bases into 9 right cylindrical slices. After Freya cut it, how much did the combined surface area of the slices of the Yule log increase by?
100pi
100
How many permutations of the set {B, M, T, 2, 0} do not have B as their first element?
96
100
Using the digits 1, 2, 3, 4, 5, 6, 7, and 9, form 4 two-digit prime numbers, using each digit only once. What is the sum of the 4 prime numbers?
190
100
Justin is being served two different types of chips, A-chips, and B-chips. If there are 3 B-chips and 5 A-chips, and if Justin randomly grabs 3 chips, what is the probability that none of them are A-chips?
1/56
200
Let p be a polynomial with degree less than 4 such that p(x) attains a maximum at x = 1. If p(1) = p(2) = 5, find p(10).
5
200
Let A, B, C be unique collinear points AB = BC = 1/3 . Let P be a point that lies on the circle centered at B with radius 1/3 and the circle centered at C with radius 1/3 . Find the measure of angle PAC in degrees.
30
200
Johann has 64 fair coins. He flips all the coins. Any coin that lands on tails is tossed again. Coins that land on tails on the second toss are tossed a third time. What is the expected number of coins that are now heads?
56
200
The number 2564 * 6425 is the square of a positive integer N. In decimal representation, what is the sum of the digits of N?
14
200
Let points A=(0,0,0), B=(1,0,0), C=(0,2,0) and D=(0,0,3). Points E, F, G, and H are midpoints of line segments BD, AB, AC, and DC respectively. What is the area of EFGH?
3sqrt(5)/4
300
Let a, b, and c be integers that satisfy 2a + 3b = 52, 3b + c = 41, and bc = 60. Find a + b + c.
25
300
A square has coordinates at (0, 0),(4, 0),(0, 4), and (4, 4). Rohith is interested in circles of radius r centered at the point (1, 2). There is a range of radii a < r < b where Rohith’s circle intersects the square at exactly 6 points, where a and b are positive real numbers. What is b-a?
sqrt(5)-2
300
Haydn picks two different integers between 1 and 100, inclusive, uniformly at random. What is the probability that their product is divisible by 4 (in simplest form)?
1/2
300
How many four-digit numbers N have the property that the three-digit number obtained by removing the leftmost digit is one ninth of N?
7
300
If f(x + y) = f(xy) for all real numbers x and y, and f(2019) = 17, what is the value of f(17)?
17
400
If y+4=(x-2)2, x+4=(y-2)2, and x is not equal to y, what is the value of x2+y2?
15
400
3sqrt(33)/5
400
Four distinct circles are drawn in a plane. What is the maximum number of points where at least two of the circles intersect?
12
400
One morning each member of Angela's family drank an 8-ounce mixture of coffee with milk. The amounts of coffee and milk varied from cup to cup, but were never zero. Angela drank a quarter of the total amount of milk and a sixth of the total amount of coffee. How many people are in the family?
5
400
By default, iPhone passcodes consist of four base-10 digits. However, Freya decided to be unconventional and use hexadecimal (base-16) digits instead of base-10 digits! (Recall that 1016 = 1610.) She sets her passcode such that exactly two of the hexadecimal digits are prime. How many possible passcodes could she have set?
21600
500
Paula the painter and her two helpers each paint at constant, but different, rates. They always start at 8:00 AM, and all three always take the same amount of time to eat lunch. On Monday the three of them painted 50% of a house, quitting at 4:00 PM. On Tuesday, when Paula wasn't there, the two helpers painted only 24% of the house and quit at 2:12 PM. On Wednesday Paula worked by herself and finished the house by working until 7:12 P.M. How long, in minutes, was each day's lunch break?
48
500
A circle C with radius 3 has an equilateral triangle inscribed in it. Let D be a circle lying outside the equilateral triangle, tangent to C, and tangent to the equilateral triangle at the midpoint of one of its sides. What is the radius of D?
3/4
500
For some particular value of N, when (a+b+c+d+1)N is expanded and like terms are combined, the resulting expression contains exactly 1001 terms that include all four variables a, b, c, and d, each to some positive power. What is N?
14
500
For each positive integer n>1, let P(n) denote the greatest prime factor of n. For how many positive integers n is it true that both P(n) = sqrt(n) and P(n+48) = sqrt(n+48)??
1
500
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)