Found problems: 229
2024 HMNT, 1
The integers from $1$ to $9$ are arranged in a $3\times3$ grid. The rows and columns of the grid correspond to $6$ three-digit numbers, reading rows from left to right, and columns from top to bottom. Compute the least possible value of the largest of the $6$ numbers.
2023 CMIMC Team, 8
NASA is launching a spaceship at the south pole, but a sudden earthquake shock caused the spaceship to be launched at an angle of $\theta$ from vertical ($0 < \theta < 90^\circ$). The spaceship crashed back to Earth, and NASA found the debris floating in the ocean in the northern hemisphere. NASA engineers concluded that $\tan \theta > M$, where $M$ is maximal. Find $M$.
Assume that the Earth is a sphere, and the trajectory of the spaceship (in the reference frame of Earth) is an ellipse with the center of the Earth one of the foci.
[i]Proposed by Kevin You[/i]
2021 MOAA, 15
Consider the polynomial
\[P(x)=x^3+3x^2+6x+10.\]
Let its three roots be $a$, $b$, $c$. Define $Q(x)$ to be the monic cubic polynomial with roots $ab$, $bc$, $ca$. Compute $|Q(1)|$.
[i]Proposed by Nathan Xiong[/i]
2024 HMNT, 10
For each positive integer $n,$ let $f(n)$ be either the unique integer $r \in \{0,1, \ldots, n-1\}$ such that $n$ divides $15r-1,$ or $0$ if such $r$ does not exist. Compute $$f(16)+f(17)+f(18)+\cdots+f(300).$$
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, 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, 2021.15
Consider the polynomial
\[P(x)=x^3+3x^2+6x+10.\]
Let its three roots be $a$, $b$, $c$. Define $Q(x)$ to be the monic cubic polynomial with roots $ab$, $bc$, $ca$. Compute $|Q(1)|$.
[i]Proposed by Nathan Xiong[/i]
MOAA Team Rounds, 2021.3
For two real numbers $x$ and $y$, let $x\circ y=\frac{xy}{x+y}$. The value of
\[1 \circ (2 \circ (3 \circ (4 \circ 5)))\]
can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m+n$.
[i]Proposed by Nathan Xiong[/i]
2024 LMT Fall, 1
A positive integer $n$ is called "foursic'' if there exists a placement of $0$ in the digits of $n$ such that the resulting number a multiple of $4.$ For example, $14$ is foursic because $104$ is a multiple of $4.$ Find the number of two-digit foursic numbers.
2024 LMT Fall, 4
A rhombus has vertices at $(0,0)$, $(6, 8)$, $(16, 8)$, and $(10, 0)$. A line with slope $m$ passes through the point $(3, 1)$ and splits the rhombus into $2$ regions of equal area. Find $m$.
2025 CMIMC Team, 1
I define a "good day" as a day when both the day and the month evenly divide the concatenation of the two. For example, today (March $15$) is a good day since $3$ and $15$ both divide $315.$ However, March $9$ is not a good day since $9$ does not divide $39.$ How many good days are in March, April, and May combined?
MOAA Team Rounds, 2021.10
For how many nonempty subsets $S \subseteq \{1, 2, \ldots , 10\}$ is the sum of all elements in $S$ even?
[i]Proposed by Andrew Wen[/i]
2024 HMNT, 2
Compute the sum of all positive integers $x$ such that $(x-17)\sqrt{x-1}+(x-1)\sqrt{x+15}$ is an integer.
2023 CMIMC Team, 13
Suppose that the sequence of real numbers $a_1,a_2,\ldots$ satisfies $a_1 = - \sqrt{1}, a_2 = \sqrt{2}$, and for all $k > 1$,
\[ \frac{a_{k+1}+a_{k-1}}{a_k} = \frac{\sqrt{3} + \sqrt{1}}{\sqrt{2}}. \]
Find $a_{2023}$.
[i]Proposed by Kevin You[/i]
2019 CMIMC, 4
Let $\triangle A_1B_1C_1$ be an equilateral triangle of area $60$. Chloe constructs a new triangle $\triangle A_2B_2C_2$ as follows. First, she flips a coin. If it comes up heads, she constructs point $A_2$ such that $B_1$ is the midpoint of $\overline{A_2C_1}$. If it comes up tails, she instead constructs $A_2$ such that $C_1$ is the midpoint of $\overline{A_2B_1}$. She performs analogous operations on $B_2$ and $C_2$. What is the expected value of the area of $\triangle A_2B_2C_2$?
2023 CMIMC Team, 1
On a plane, two equilateral triangles (of side length $1$) share a side, and a circle is drawn with the common side as a diameter. Find the area of the set of all points that lie inside exactly one of these shapes.
[i]Proposed by Howard Halim[/i]
MOAA Team Rounds, 2021.4
Compute the number of ordered triples $(x,y,z)$ of integers satisfying
\[x^2+y^2+z^2=9.\]
[i]Proposed by Nathan Xiong[/i]
2023 CMIMC Team, 2
Real numbers $x$ and $y$ satisfy
\begin{align*}
x^2 + y^2 &= 2023 \\
(x-2)(y-2) &= 3.
\end{align*}
Find the largest possible value of $|x-y|$.
[i]Proposed by Howard Halim[/i]
MOAA Team Rounds, 2018.4
Michael and Andrew are playing the game Bust, which is played as follows: Michael chooses a positive integer less than or equal to $99$, and writes it on the board. Andrew then makes a move, which consists of him choosing a positive integer less than or equal to $ 8$ and increasing the integer on the board by the integer he chose. Play then alternates in this manner, with each person making exactly one move, until the integer on the board becomes greater than or equal to $100$. The person who made the last move loses. Let S be the sum of all numbers for which Michael could choose initially and win with both people playing optimally. Find S.
2024 LMT Fall, 3
Jason starts in a cell of the grid below. Every second he moves to an adjacent cell (i.e., two cells that share a side) that he has not visited yet. Find the maximum possible number of cells that Jason can visit.
[asy]
size(3cm);
draw((1,0)--(4,0));
draw((0,1)--(5,1));
draw((0,2)--(5,2));
draw((0,3)--(5,3));
draw((0,4)--(5,4));
draw((1,5)--(4,5));
draw((0,1)--(0,4));
draw((1,0)--(1,5));
draw((2,0)--(2,5));
draw((3,0)--(3,5));
draw((4,0)--(4,5));
draw((5,1)--(5,4));
[/asy]
2022 CMIMC, 4
Let $\triangle ABC$ be equilateral with integer side length. Point $X$ lies on $\overline{BC}$ strictly between $B$ and $C$ such that $BX<CX$. Let $C'$ denote the reflection of $C$ over the midpoint of $\overline{AX}$. If $BC'=30$, find the sum of all possible side lengths of $\triangle ABC$.
[i]Proposed by Connor Gordon[/i]
2019 MOAA, 6
Let $f(x, y) = \left\lfloor \frac{5x}{2y} \right\rfloor + \left\lceil \frac{5y}{2x} \right\rceil$. Suppose $x, y$ are chosen independently uniformly at random from the interval $(0, 1]$. Let $p$ be the probability that $f(x, y) < 6$. If $p$ can be expressed in the form $m/n$ for relatively prime positive integers $m$ and $n$, compute $m + n$.
(Note: $\lfloor x\rfloor $ is defined as the greatest integer less than or equal to $x$ and $\lceil x \rceil$ is defined as the least integer greater than or equal to$ x$.)
2019 MOAA, 2
The lengths of the two legs of a right triangle are the two distinct roots of the quadratic $x^2 - 36x + 70$. What is the length of the triangle’s hypotenuse?
2021 MOAA, 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]
2021 MOAA, 18
Let $\triangle ABC$ be a triangle with side length $BC= 4\sqrt{6}$. Denote $\omega$ as the circumcircle of $\triangle{ABC}$. Point $D$ lies on $\omega$ such that $AD$ is the diameter of $\omega$. Let $N$ be the midpoint of arc $BC$ that contains $A$. $H$ is the intersection of the altitudes in $\triangle{ABC}$ and it is given that $HN = HD= 6$. If the area of $\triangle{ABC}$ can be expressed as $\frac{a\sqrt{b}}{c}$, where $a,b,c$ are positive integers with $a$ and $c$ relatively prime and $b$ not divisible by the square of any prime, compute $a+b+c$.
[i]Proposed by Andy Xu[/i]