You have a fruit skewer that can hold 7 different fruits, which you also have. You put the fruits through the skewer by randomly choosing a fruit and randomly adding it to the left or the right side. What is the probability that the piece of pineapple and strawberry don't touch?

There are 49 knaves who always lie and 1 knight who always tells the truth standing in a random order. The first person tells you that the knight is within the first 20 people, the last person tells you that the knight is within the last 20 people, and the person in the middle tells you that they are the knight. What are the possible positions that the knight is in?

Austin starts with a $7$ x $7$ grid. He chooses a $5$ x $5$ grid with in it, then chooses a $3$ x $3$ grid within the $5$ x $5$ grid, and finally chooses a $1$ x $1$ cell in the $3$ x $3$ grid. What is the probability, if Austin selects each grid randomly, that the final $1$ x $1$ cell is the center cell of the $7$ x $7$ grid?

TJ has 100 cats, each of them conveniently numbered from 1 to 100. He has this weird petting routine, where he selects a random cat that hasn't been petted, and pets it if it is the lowest numbered cat that hasn't been petted. He then repeats this process until all cats have been petted. What is the expected number of cats that TJ pets?

Starting with a $30^{\circ}$ angle, draw a line bisecting it. Then bisect the upper $15^{\circ}$ angle, then bisect the lower $7.5^{\circ}$ angle, and keep repeating this process, alternating between choosing the upper and lower angle to bisect. Repeating this process will get you near a line that starts to become relatively stationary. What are the two angles this line divides the original $30^{\circ}$ angle into?

"Super Rounding" a number consists of the following. Start from the second to last decimal digit and round with respect to that digit. Then keep repeating this process until it is an integer. An example of "Super Rounding" the number $1.3377761684$: $$1.337776168$$$$1.33777617$$$$1.3377762$$$$1.337776$$$$1.33778$$$$1.3379$$$$1.338$$$$1.34$$$$1.3$$$$1$$ Given a random number between 0 and 1, what is the probability that, after squaring the number and "Super Rounding" it, you will get 1?

Consider a circle with radius 4. Inside the circle, there are two perpendicular chords that intersect at a point within the circle. The lengths of these chords are 4 and 6 units. Find the distance between the intersection point of the two chords and the center of the circle.

Kellan chooses 3 random integers from 1 to 2024: $a, b, c$. He then solves for $f = a \oplus b \oplus c$ where $\oplus$ is the bitwise OR function. What is the expected value of $f$?

You are trying to find the sum of three numbers, but both of your calculators are broken. Your first calculator outputs the sum of the squares of each number while your second calculator outputs the sum of the pairwise products of the numbers (like $xy + yz + zx$ if $x, y, $ and $z$ were the 3 numbers). Can you still find the sum of the 3 numbers?

Lisa and Ann are playing a game with dice. They each have their own fair twenty-sided die. In the game, they each roll their die at the same time and a player wins if and only if they land a number bigger than their opponent. What is the probability that Lisa wins?

It is now the year 2024, which excites Mathew because 2024 holds the following special properties:

1. All of its digits are even
2. It is not divisible by 3, but the number formed by the last 2 digits is divisible by 3
3. It is divisible by exactly 3 distinct primes

When is the next time Mathew can be excited by another year that follows these special properties?

Santa wants to find the traitor among his elves who is plotting to kill him. Of the 1000 glasses of milk he has, one of them is poisoned. The poison takes about 24 hours to work. Santa has ten prisoner grinches whom he'll use as testers. He needs to identify the poisoned glass of milk in exactly 24 hours. How can he do this using only the prisoners and without risking the life of his own?

In a perfectly reflective 1 x 1 square, you have a laser pointer in the top right corner. Your goal is to make the laser shine into the bottom left corner, but the laser can bounce at most 2023 times off the walls. What is the length of the longest path it can take to the bottom left corner?

Find all ordered pairs of positive integers $(a, b)$ such that $a^2 + 80a + 2023 = b^2$

Find the number of intersections of $y = \frac{x}{2023}$ and $y=\{x ([x]^3 + [x]^2 + [x])\}$ for $x > 0$ where $[x]$ is the floor of $x$ (the greatest integer $\le$ x) and $\{x\}$ is the fractional part of $x$ ($\{x\}$ = $x-[x]$).

Find all possible integer roots of polynomial $$Q(x)= x^4+kx^3-82x^2-216x+576$$ where $k$ is a real number and for any fixed $k$, one of the four roots is the product of the other three roots.