Found problems: 229
2023 CMIMC Team, 4
Suppose $a_1, a_2, a_3, \dots,$ is a sequence of real numbers such that $$a_n = \frac{a_{n-1}a_{n-2}}{3a_{n-2}-2a_{n-1}}$$ for all $n \ge 3$. If $a_1 = 1$ and $a_{10} = 10$, what is $a_{19}$?
[i]Proposed by Howard Halim[/i]
2021 MOAA, 11
Find the product of all possible real values for $k$ such that the system of equations
$$x^2+y^2= 80$$
$$x^2+y^2= k+2x-8y$$
has exactly one real solution $(x,y)$.
[i]Proposed by Nathan Xiong[/i]
2024 CMIMC Team, 8
Compute
\[\frac{(1-\tan10^\circ)(1-\tan 20^\circ)(1-\tan30^\circ)(1-\tan40^\circ)}{(1-\tan5^\circ)(1-\tan 15^\circ)(1-\tan25^\circ)(1-\tan35^\circ)}.\]
[i]Proposed by Connor Gordon[/i]
2024 CMIMC Team, 5
An ant is currently on a vertex of the top face on a 6-sided die. The ant wants to travel to the opposite vertex of the die (the vertex that is farthest from the start), and the ant can travel along edges of the die to other vertices that are on the top face of the die.
Every second, the ant picks a valid edge to move along, and the die randomly flips to an adjacent face. If the ant is on any of the bottom vertices after the flip, it is crushed and dies. What is the probability that the ant makes it to its target? (If the ant makes it to the target and the die rolls to crush it, it achieved its dreams before dying, so this counts.)
[i]Proposed by Lohith Tummala[/i]
MOAA Team Rounds, 2021.13
Bob has $30$ identical unit cubes. He can join two cubes together by gluing a face on one cube to a face on the other cube. He must join all the cubes together into one connected solid. Over all possible solids that Bob can build, what is the largest possible surface area of the solid?
[i]Proposed by Nathan Xiong[/i]
MOAA Team Rounds, 2021.12
Let $\triangle ABC$ have $AB=9$ and $AC=10$. A semicircle is inscribed in $\triangle ABC$ with its center on segment $BC$ such that it is tangent $AB$ at point $D$ and $AC$ at point $E$. If $AD=2DB$ and $r$ is the radius of the semicircle, $r^2$ can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m+n$.
[i]Proposed by Andy Xu[/i]
2022 CMIMC, 1
Let $A_1A_2A_3A_4$ and $B_1B_2B_3B_4$ be two squares such that the boundaries of $A_1A_2A_3A_4$ and $B_1B_2B_3B_4$ does not contain any line segment. Construct 16 line segments $A_iB_j$ for each possible $i,j \in \{1,2,3,4\}$. What is the maximum number of line segments that don't intersect the edges of $A_1A_2A_3A_4$ or $B_1B_2B_3B_4$? (intersection with a vertex is not counted).
[i]Proposed by Allen Zheng[/i]
2024 LMT Fall, 15
Amy has a six-sided die which always rolls values greater than or equal to the previous roll. She rolls the die repeatedly until she rolls a $6$. Find the expected value of the sum of all distinct values she has rolled when she finishes.
2023 CMIMC Team, 7
Compute the value of
$$\sin^2\left(\frac{\pi}{7}\right) + \sin^2\left(\frac{3\pi}{7}\right) + \sin^2\left(\frac{5\pi}{7}\right).$$
Your answer should not involve any trigonometric functions.
[i]Proposed by Howard Halim[/i]
2021 MOAA, 14
Evaluate
\[\left\lfloor\frac{1\times 5}{7}\right\rfloor + \left\lfloor\frac{2\times 5}{7}\right\rfloor + \left\lfloor\frac{3\times 5}{7}\right\rfloor+\cdots+\left\lfloor\frac{100\times 5}{7}\right\rfloor.\]
[i]Proposed by Nathan Xiong[/i]
2018 CMIMC Team, 2-1/2-2
Suppose that $a$ and $b$ are non-negative integers satisfying $a + b + ab + a^b = 42$. Find the sum of all possible values of $a + b$.
Let $T = TNYWR$. Suppose that a sequence $\{a_n\}$ is defined via $a_1 = 11, a_2 = T$, and $a_n = a_{n-1} + 2a_{n-2}$ for $n \ge 3$. Find $a_{19} + a_{20}$.
2020 CMIMC Team, Estimation
Choose a point $(x,y)$ in the square bounded by $(0,0), (0,1), (1,0)$ and $(1,1)$. Your score is the minimal distance from your point to any other team's submitted point. Your answer must be in the form $(0.abcd, 0.efgh)$ where $a, b, c, d, e, f, g, h$ are decimal digits.
2018 CMIMC Team, 4-1/4-2
Define an integer $n \ge 0$ to be \textit{two-far} if there exist integers $a$ and $b$ such that $a$, $b$, and $n + a + b$ are all powers of two. If $N$ is the number of two-far integers less than 2048, find the remainder when $N$ is divided by 100.
Let $T = TNYWR$. Let $CMU$ be a triangle with $CM=13$, $MU=14$, and $UC=15$. Rectangle $WEAN$ is inscribed in $\triangle CMU$ with points $W$ and $E$ on $\overline{MU}$, point $A$ on $\overline{CU}$, and point $N$ on $\overline{CM}$. If the area of $WEAN$ is $T$, what is its perimeter?
MOAA Team Rounds, 2021.5
Two right triangles are placed next to each other to form a quadrilateral as shown. What is the perimeter of the quadrilateral?
[asy]
size(4cm);
fill((-5,0)--(0,12)--(0,6)--(8,0)--cycle, gray+opacity(0.3));
draw((0,0)--(0,12)--(-5,0)--cycle);
draw((0,0)--(8,0)--(0,6));
label("5", (-2.5,0), S);
label("13", (-2.5,6), dir(140));
label("6", (0,3), E);
label("8", (4,0), S);
[/asy]
[i]Proposed by Nathan Xiong[/i]
2016 CMIMC, 3
We have 7 buckets labelled 0-6. Initially bucket 0 is empty, while bucket $n$ (for each $1 \leq n \leq 6$) contains the list $[1,2, \ldots, n]$. Consider the following program: choose a subset $S$ of $[1,2,\ldots,6]$ uniformly at random, and replace the contents of bucket $|S|$ with $S$. Let $\tfrac{p}{q}$ be the probability that bucket 5 still contains $[1,2, \ldots, 5]$ after two executions of this program, where $p,q$ are positive coprime integers. Find $p$.
2017 CMIMC Team, 3
Suppose Pat and Rick are playing a game in which they take turns writing numbers from $\{1, 2, \dots, 97\}$ on a blackboard. In each round, Pat writes a number, then Rick writes a number; Rick wins if the sum of all the numbers written on the blackboard after $n$ rounds is divisible by 100. Find the minimum positive value of $n$ for which Rick has a winning strategy.
2018 MOAA, 9
Quadrilateral $ABCD$ with $AC = 800$ is inscribed in a circle, and $E, W, X, Y, Z$ are the midpoints of segments $BD$, $AB$, $BC$, $CD$, $DA$, respectively. If the circumcenters of $EW Z$ and $EXY$ are $O_1$ and $O_2$, respectively, determine $O_1O_2$.
MOAA Team Rounds, 2021.6
Find the sum of all two-digit prime numbers whose digits are also both prime numbers.
[i]Proposed by Nathan Xiong[/i]
2017 CMIMC Team, 2
Suppose $x$, $y$, and $z$ are nonzero complex numbers such that $(x+y+z)(x^2+y^2+z^2)=x^3+y^3+z^3$. Compute \[(x+y+z)\left(\dfrac1x+\dfrac1y+\dfrac1z\right).\]
MOAA Team Rounds, 2021.9
Mr. DoBa has a bag of markers. There are 2 blue, 3 red, 4 green, and 5 yellow markers. Mr. DoBa randomly takes out two markers from the bag. The probability that these two markers are different colors can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m+n$.
[i]Proposed by Raina Yang[/i]
2018 CMIMC Team, 7-1/7-2
Let $ABCD$ be a unit square, and suppose that $E$ and $F$ are on $\overline{AD}$ and $\overline{AB}$ such that $AE = AF = \tfrac23$. Let $\overline{CE}$ and $\overline{DF}$ intersect at $G$. If the area of $\triangle CFG$ can be expressed as simplified fraction $\frac{p}{q}$, find $p + q$.
Let $T = TNYWR$. A total of $2T$ students go on a road trip. They take two cars, each of which seats $T$ people. Call two students \textit{friendly} if they sat together in the same car going to the trip and in the same car going back home. What is the smallest possible number of friendly pairs of students on the trip?
2019 CMIMC, 1
David recently bought a large supply of letter tiles. One day he arrives back to his dorm to find that some of the tiles have been arranged to read $\textsc{Central Michigan University}$. What is the smallest number of tiles David must remove and/or replace so that he can rearrange them to read $\textsc{Carnegie Mellon University}$?
MOAA Team Rounds, 2021.16
Let $\triangle ABC$ have $\angle ABC=67^{\circ}$. Point $X$ is chosen such that $AB = XC$, $\angle{XAC}=32^\circ$, and $\angle{XCA}=35^\circ$. Compute $\angle{BAC}$ in degrees.
[i]Proposed by Raina Yang[/i]
2023 CMIMC Team, 10
Consider the set of all permutations, $\mathcal{P}$, of $\{1,2,\ldots,2022\}$. For permutation $P\in \mathcal{P}$, let $P_1$ denote the first element in $P$. Let $\text{sgn}(P)$ denote the sign of the permutation. Compute the following number modulo 1000: $$\displaystyle\sum_{P\in\mathcal{P}}\dfrac{P_1\cdot\text{sgn}(P)^{P_1}}{2020!}.$$
(The [i]sign[/i] of a permutation $P$ is $(-1)^k$, where $k$ is the minimum number of two-element swaps needed to reach that permutation).
[i]Proposed by Nairit Sarkar[/i]
2025 Harvard-MIT Mathematics Tournament, 9
Let $\mathbb{Z}$ be the set of integers. Determine, with proof, all primes $p$ for which there exists a function $f:\mathbb{Z}\to\mathbb{Z}$ such that for any integer $x,$
$\quad \bullet \ f(x+p)=f(x)\text{ and}$
$\quad \bullet \ p \text{ divides } f(x+f(x))-x.$