Found problems: 41
2017 CMIMC Individual Finals, 1
Jesse has ten squares, which are labeled $1, 2, \dots, 10$. In how many ways can he color each square either red, green, yellow, or blue such that for all $1 \le i < j \le 10$, if $i$ divides $j$, then the $i$-th and $j$-th squares have different colors?
2018 CMIMC Individual Finals, 1
Let $ABC$ be a triangle with $AB=9$, $BC=10$, $CA=11$, and orthocenter $H$. Suppose point $D$ is placed on $\overline{BC}$ such that $AH=HD$. Compute $AD$.
2016 CMIMC, 2
For each integer $n\geq 1$, let $S_n$ be the set of integers $k > n$ such that $k$ divides $30n-1$. How many elements of the set \[\mathcal{S} = \bigcup_{i\geq 1}S_i = S_1\cup S_2\cup S_3\cup\ldots\] are less than $2016$?
2017 CMIMC Individual Finals, 2
Let $x$ be a real number between $0$ and $\tfrac{\pi}2$ such that \[\dfrac{\sin^4(x)}{42}+\dfrac{\cos^4(x)}{75} = \dfrac{1}{117}.\] Find $\tan(x)$.
2017 CMIMC Individual Finals, 2
Define
\[f(h,t) =
\begin{cases}
8h & h = t \\
(h-t)^2 & h \neq t.
\end{cases}\]
Cody plays a game with a fair coin, where he begins by flipping it once. At each turn in the game, if he has flipped $h$ heads and $t$ tails and $h + t < 6$, he can choose either to stop and receive $f(h,t)$ dollars or he can flip the coin again; if $h + t = 6$ then the game ends and he receives $f(h,t)$ dollars. If Cody plays to maximize expectancy, how much money, in dollars, can he expect to win from this game?
2018 CMIMC Individual Finals, 1
Alex has one-pound red bricks and two-pound blue bricks, and has 360 total pounds of brick. He observes that it is impossible to rearrange the bricks into piles that all weigh three pounds, but he can put them in piles that each weigh five pounds. Finally, when he tries to put them into piles that all have three bricks, he has one left over. If Alex has $r$ red bricks, find the number of values $r$ could take on.
2018 CMIMC Individual Finals, 1
The [i]distance[/i] between two vertices in a connected graph is defined to be the length of the shortest path between them. How many graphs with the vertex set $\{0,1,2,\dots,6\}$ satisfy the following property: there are $3$ vertices of distance $1$ away from vertex $0$, $2$ of distance $2$ away, and $1$ of distance $3$ away?
2016 CMIMC, 2
Identical spherical tennis balls of radius 1 are placed inside a cylindrical container of radius 2 and height 19. Compute the maximum number of tennis balls that can fit entirely inside this container.
2018 CMIMC Individual Finals, 1
For all real numbers $r$, denote by $\{r\}$ the fractional part of $r$, i.e. the unique real number $s\in[0,1)$ such that $r-s$ is an integer. How many real numbers $x\in[1,2)$ satisfy the equation $\left\{x^{2018}\right\} = \left\{x^{2017}\right\}?$
2018 CMIMC Individual Finals, 3
Let $a$ be a complex number, and set $\alpha$, $\beta$, and $\gamma$ to be the roots of the polynomial $x^3 - x^2 + ax - 1$. Suppose \[(\alpha^3+1)(\beta^3+1)(\gamma^3+1) = 2018.\] Compute the product of all possible values of $a$.
2018 CMIMC Individual Finals, 2
Compute the sum of the digits of \[\prod_{n=0}^{2018}\left(10^{2\cdot 3^n} - 10^{3^n} + 1\right)\left(10^{2\cdot 3^n} + 10^{3^n} + 1\right).\]
2018 CMIMC Individual Finals, 2
How many integer values of $k$, with $1 \leq k \leq 70$, are such that $x^{k}-1 \equiv 0 \pmod{71}$ has at least $\sqrt{k}$ solutions?
2016 CMIMC, 1
A $\emph{planar}$ graph is a connected graph that can be drawn on a sphere without edge crossings. Such a drawing will divide the sphere into a number of faces. Let $G$ be a planar graph with $11$ vertices of degree $2$, $5$ vertices of degree $3$, and $1$ vertex of degree $7$. Find the number of faces into which $G$ divides the sphere.
2016 CMIMC, 3
Suppose $x$ and $y$ are real numbers which satisfy the system of equations \[x^2-3y^2=\frac{17}x\qquad\text{and}\qquad 3x^2-y^2=\frac{23}y.\] Then $x^2+y^2$ can be written in the form $\sqrt[m]{n}$, where $m$ and $n$ are positive integers and $m$ is as small as possible. Find $m+n$.
2017 CMIMC Individual Finals, 3
Let $n=2017$ and $x_1,\dots,x_n$ be boolean variables. An \emph{$7$-CNF clause} is an expression of the form $\phi_1(x_{i_1})+\dots+\phi_7(x_{i_7})$, where $\phi_1,\dots,\phi_7$ are each either the function $f(x)=x$ or $f(x)=1-x$, and $i_1,i_2,\dots,i_7\in\{1,2,\dots,n\}$. For example, $x_1+(1-x_1)+(1-x_3)+x_2+x_4+(1-x_3)+x_{12}$ is a $7$-CNF clause. What's the smallest number $k$ for which there exist $7$-CNF clauses $f_1,\dots,f_k$ such that \[f(x_1,\dots,x_n):=f_1(x_1,\dots,x_n)\cdots f_k(x_1,\dots,x_n)\] is $0$ for all values of $(x_1,\dots,x_n)\in\{0,1\}^n$?
2017 CMIMC Individual Finals, 1
Let $\tau(n)$ denote the number of positive integer divisors of $n$. For example, $\tau(4) = 3$. Find the sum of all positive integers $n$ such that $2 \tau(n) = n$.