Algebra
Geometry
Number Theory
Counting/Probability
General
100
After an evening of trick or treating, three brothers, Albert, Brian and Colin share their Halloween candies as follows: First, Albert takes one candy along with one third of the remaining candies. Then, Brian takes half of the remaining candies and two more candies after that. Finally, Colin takes the last three candies. How many candies did they collect in total?
16
100
If an equilateral triangle has an area of 6, what is the area of a square that has the same side lengths as the triangle? (fully rationalized and simplified)
8 sqrt3
100
What is the units digit of 72913 * 22348 - 3451?
5
100
In a science classroom, 19 students have a brother, 15 students have a sister, 7 students have both a brother and a sister, and 6 students don’t have any siblings at all. How many students are in the classroom?
33
100
Let {x} denote the sum of the positive divisors of x. For example {6} = 1 + 2 + 3 + 6 = 12. What is {{{4}}}?
15
200
The equation x^x^x^x… all the way to infinity is equal to 5. What value of x satisfies this equation?
5^(1/5)
200
What is the area enclosed by the graph of |5x|+|7y| = 35
70
200
What is the value of 10112 + 110012 expressed in base 5?
121
200
12 athletes, no two of whom are the same height, try out for the basketball team. One at a time, they draw a wristband at random, without replacement, from a bag containing 4 blue bands, 4 red bands, and 4 green bands. They are divided into a blue group, a red group, and a green group. The tallest member of each group is named the group captain. What is the probability that the group captains are the three tallest athletes?
16/55
200
Regular hexagon ABCDEF has side length 1. What is the area of the set of points inside the hexagon that are within sqrt(3) / 4 away from at least one of the sides?
9sqrt(3)/8
300
How many ordered pairs of positive integers (x, y) satisfy 2(xy + 60) = (x + 8)(y + 8)?
6
300
How many isosceles triangles are there with positive area whose side lengths are all positive integers and whose longest side has length 1025.
1537
300
What is the largest positive integer n such that 2^n divides 7^8 - 3^8?
5
300
A traffic light runs repeatedly through the following cycle: green for 20 seconds, then yellow for 3 seconds, and then red for 20 seconds. Victoria picks a random three-second time interval to watch the light. What is the probability that the color changes while she is watching?
9/43
300
It is the first day of math camp, and Nathaniel does not know the names of any of the instructors. However, he knows that one of them is Victoria, some of them are named Peter, and the rest are named Colin. The probability that a randomly chosen instructor is named Colin is 2/3. The probability that two randomly chosen instructors are both named Colin is 3/7. What is the probability that three randomly chosen instructors are all named Colin?
24/91
400
Suppose x ≠ 1. What is the product of all real numbers x satisfying the equation log2x − 1 − logx4 = 0?
2
400
The triangle ABC with sides AB = 6 and AC = 9, point D is given on side AC such that AD = 4. If BD and BC both have integer lengths, what is the sum of all possible lengths for BC?
27
400
How many ordered pairs of integers (m,n) are there such that 100 ≥ m ≥ 1, 100 ≥ n ≥ 1, and m^n * n^m leaves a remainder of 1 when divided by 4?
1250
400
In a game, Alex, Bella, and Carlos take turns rolling a fair six-sided die. Alex rolls first, followed by Bella, Carlos, Alex, and so on. The game ends when a player rolls a 1 for the first time, and that player is the winner. What is the probability that Bella wins?
30/91
400
A rectangular 3x5 index card is folded so that one set of diagonally opposite corners meet. What is the area of the new shape?
9.9
500
Suppose x^3 − 5x + 2 = 0 has 3 distinct complex roots a, b, c. Compute a^4 + b^4 + c^4.
50
500
The line 
divides the square region defined by 
and 
into an upper and lower region. The line 
divides the lower region into two regions of equal area. Then 
can be written as 
, where 
and 
are positive integers. What is 
?
20
500
Leo wrote 6 distinct numbers on each side of 3 cards, and laid the cards out on a table. The three numbers facing up are 44, 59, and 38. The sum of the two numbers on each of the three cards are equal and the three hidden numbers are prime numbers. What is the average of the hidden numbers?
14
500
Carlos uses a 4-digit passcode to unlock his computer. In his passcode, exactly one digit is even, exactly one (possibly different) digit is prime, and no digit is 0. How many 4-digit passcodes satisfy these conditions?
464
500
a is chosen randomly on the interval [0, 4], and b is chosen randomly on the interval [0, 2]. What is the probability that 1 ≤ |a − b| ≤ 2?
5/16