Found problems: 161
A group of $5$ rappers wants to make a song together. They each make their own parts for the song and then arrange the $5$ parts. J Cole wants to be friends with both Drake and Kendrick, so he wants his part to be adjacent to both of theirs. Find the number of possible songs (distinct orders) that can be made.
Compute the sum of the three smallest positive integers $n$ for which $$\frac{1+2+3+\cdots+(2024n-1)+2024n}{1+2+3+\cdots+(4n-1)+4n}$$ is an integer.
The number $17^6$ when written out in base $10$ contains $8$ distinct digits from $1,2,\ldots,9,$ with no repeated digits or zeroes. Compute the missing nonzero digit.
Ben Y's favorite number $p$ is prime, and his second favorite number is some integer $n$. Given that $p$ divides $n$ and $n$ divides $3p+91$, find the maximum possible value of $n$.
Find the number of real numbers $0 \leq \alpha < 50$ such that $\alpha^2 + 2\{\alpha\}$ is an integer. (Here $\{\alpha\}$ denotes the fractional part of $\alpha$.)
Jacob rolls two fair six-sided dice. If the outcomes of these dice rolls are the same, he rolls a third fair six-sided die. Compute the probability that the sum of the outcomes of all the dice he rolls is even.
Compute the remainder when $$1002003004005006007008009$$ is divided by $13.$
One hundred points labeled $1$ to $100$ are arranged in a $10\times 10$ grid such that adjacent points are one unit apart. The labels are increasing left to right, top to bottom (so the first row has labels $1$ to $10,$ the second row has labels $11$ to $20,$ and so on).
Convex polygon $\mathcal{P}$ has the property that every point with a label divisible by $7$ is either on the boundary or in the interior of $\mathcal{P}.$ Compute the smallest possible area of $\mathcal{P}.$
Define $f(n)$ to be the sum of positive integers $k$ less than or equal to $n$ such that $\gcd(n, k)$ is prime. Find $f(2024)$.
Compute the unique five-digit positive integer $\underline{abcde}$ such that $a \neq 0, c \neq 0,$ and $$\underline{abcde}=(\underline{ab}+\underline{cde})^2.$$
Two distinct positive even integers sum to $8$. Find the larger of the two integers.
Find the number of ways to tile a $2 \times 2 \times 2 \times 2$ four dimensional hypercube with $2 \times 1 \times 1 \times 1$ blocks, with reflections and rotations of the large hypercube distinct.
A new meme is circling around social media known as the [i]DaDerek Convertible[/i]. The license plate number of the [i]DaDerek Convertible[/i] is such that the product of its nonzero digits times $5$ is equal to itself. Given that its license plate number has less than or equal to $3$ digits and that it has at least one nonzero digit, find the [i]DaDerek Convertible[/i]'s license plate number.
Regular hexagon $ABCDEF$ has side length $2.$ Circle $\omega$ lies inside the hexagon and is tangent to segments $\overline{AB}$ and $\overline{AF}.$ There exist two perpendicular lines tangent to $\omega$ that pass through $C$ and $E,$ respectively. Given that these two lines do not intersect on line $AD,$ compute the radius of $\omega.$
Right triangle $\triangle{DEF}$ with $\angle{D}=90^\circ$ and $\angle{F}=30^\circ$ is inscribed in equilateral triangle $\triangle{ABC}$ such that $D, E,$ and $F$ lie on segments $\overline{BC}, \overline{CA},$ and $\overline{AB},$ respectively. Given that $BD=7$ and $DC=4,$ compute $DE.$
Compute the number of ways to shade exactly $4$ distinct cells of a $4\times4$ grid such that no two shaded cells share one or more vertices.
Let $L$ be the answer to problem $9$. Find the solution to the equation $4x+\sqrt{L}=0$.
Let $M$ be the answer to problem $7$. Let $f(x)=x^4+4x^3+6x^2+1$. Find $f(M)$.
Let $T$ be the answer to problem $8$. Find the area of a square with side length $T$.
Compute the number of ordered pairs $(m,n)$ of [i]odd[/i] positive integers both less than $80$ such that $$\gcd(4^m+2^m+1, 4^n+2^n+1)>1.$$
A group of nine math team members like to play Survev.io. They noticed that the number of hours each of them played this week forms an arithmetic progression. The person who played the least played for $1$ hour, while the most played for $9.$ Find the total number of hours all nine group members spent playing Survev.io this week.
Let $ABCD$ be a trapezoid such that $AB \parallel CD, AD=13, BC=15, AB=20,$ and $CD=34.$ Point $X$ lies inside the trapezoid such that $\angle{XAB}=2\angle{XBA}$ and $\angle{XDC}=2\angle{XCD}.$ Compute $XD-XA.$
Let $a \star b=ab-2.$ Comute the remainder when $(((579\star569)\star559)\star\cdots\star19)\star9$ is divided by $100.$
Compute the smallest integer $n > 72$ that has the same set of prime divisors as $72.$
Compute the number of ways to shade in some subset of the $16$ cells in a $4 \times 4$ grid such that each of the $25$ vertices of the grid is a corner of at least one shaded cell.
A right rectangular prism of silly powder has dimensions $20 \times 24 \times 25.$ Jerry the wizard applies $10$ bouts of highdroxylation to the box, each of which increases one dimension of the silly powder by $1$ and decreases a different dimension of the silly powder by $1,$ with every possible choice of dimensions equally likely to be chosen and independent of all previous choices. Compute the expected volume of the silly powder after Jerry’s routine.
Positive integers $a, b,$ and $c$ have the property that $\text{lcm}(a,b), \text{lcm}(b,c),$ and $\text{lcm}(c,a)$ end in $4, 6,$ and $7,$ respectively, when written in base $10.$ Compute the minimum possible value of $a + b + c.$