Found problems: 229
MOAA Team Rounds, 2021.14
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]
2020 CMIMC Team, 11
Find the number of ordered triples of integers $(a,b,c)$, each between $1$ and $64$, such that
\[
a^2 + b^2 \equiv c^2\pmod{64}.
\]
2019 CMIMC, 11
Let $S$ be a subset of the natural numbers such that $0\in S$, and for all $n\in\mathbb N$, if $n$ is in $S$, then both $2n+1$ and $3n+2$ are in $S$. What is the smallest number of elements $S$ can have in the range $\{0,1,\ldots, 2019\}$?
2019 CMIMC, 8
A positive integer $n$ is [i]brgorable[/i] if it is possible to arrange the numbers $1, 1, 2, 2, ..., n, n$ such that between any two $k$'s there are exactly $k$ numbers (for example, $n=2$ is not brgorable, but $n = 3$ is as demonstrated by $3, 1, 2, 1, 3, 2$). How many brgorable numbers are less than 2019?
2022 CMIMC, 9
For natural numbers $n$, let $r(n)$ be the number formed by reversing the digits of $n$, and take $f(n)$ to be the maximum value of $\frac{r(k)}k$ across all $n$-digit positive integers $k$.
If we define $g(n)=\left\lfloor\frac1{10-f(n)}\right\rfloor$, what is the value of $g(20)$?
[i]Proposed by Adam Bertelli[/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]
2022 CMIMC, 12
Let $ABCD$ be a cyclic quadrilateral with $AB=3, BC=2, CD=6, DA=8,$ and circumcircle $\Gamma.$ The tangents to $\Gamma$ at $A$ and $C$ intersect at $P$ and the tangents to $\Gamma$ at $B$ and $D$ intersect at $Q.$ Suppose lines $PB$ and $PD$ intersect $\Gamma$ at points $W \neq B$ and $X \neq D,$ respectively. Similarly, suppose lines $QA$ and $QC$ intersect $\Gamma$ at points $Y \neq A$ and $Z \neq C,$ respectively. What is the value of $\frac{{WX}^2}{{YZ}^2}?$
[i]Proposed by Kyle Lee[/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.
2020 CMIMC Team, 14
Let $a_0=1$ and for all $n\ge 1$ let $a_n$ be the smaller root of the equation $$4^{-n}x^2-x+a_{n-1} = 0.$$ Given that $a_n$ approaches a value $L$ as $n$ goes to infinity, what is the value of $L$?
2019 CMIMC, 12
Call a convex quadrilateral [i]angle-Pythagorean[/i] if the degree measures of its angles are integers $w\leq x \leq y \leq z$ satisfying $$w^2+x^2+y^2=z^2.$$ Determine the maximum possible value of $x+y$ for an angle-Pythagorean quadrilateral.
2024 HMNT, 4
Albert writes down all of the multiples of $9$ between $9$ and $999,$ inclusive. Compute the sum of the digits he wrote.
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]
MOAA Team Rounds, 2019.5
Let $ABC$ be a triangle with $AB = AC = 10$ and $BC = 12$. Define $\ell_A$ as the line through $A$ perpendicular to $\overline{AB}$. Similarly, $\ell_B$ is the line through $B$ perpendicular to $\overline{BC}$ and $\ell_C$ is the line through $C$ perpendicular to $\overline{CA}$. These three lines $\ell_A, \ell_B, \ell_C$ form a triangle with perimeter $m/n$ for relatively prime positive integers $m$ and $n$. Find $m + n$.
2024 LMT Fall, 7
Let $A$, $F$, $L$, $M$, and $T$ be distinct digits such that $\overline{FALL} + \overline{LMT} = 2024$ and $F$, $L > 0$. Find the sum of all possible values of $\overline{FAT}$.
2021 MOAA, 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]
2024 CMIMC Team, 10
Square $ABCD$ has side length $2$. For each $0 \leq r \leq 2$, point $P_r$ is on side $\overline{AB}$ with $AP_r = r$, and square $\Sigma_r$ is constructed with diagonal $\overline{DP_r}$. Let region $\mathcal{R}$ be the set of all points that are in both $\Sigma_0$ and $\Sigma_2$, but not in $\Sigma_r$ for at least one value of $r$. Find the area of the convex hull of $\mathcal{R}$.
[i]Proposed by Justin Hsieh[/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.
2025 CMIMC Team, 8
Let $U$ be the set of all complex numbers $m$ such that the $4$ roots of $(x^2+2x+5)(x^2-2mx+25)=0$ are concyclic in the complex plane. One can show that when the points of $U$ are plotted on the complex plane, it is visualized as the finite union of some curves. Find the sum of the lengths of those curves (i.e. the perimeter of $U$).
2017 CMIMC Team, 4
Say an odd positive integer $n > 1$ is $\textit{twinning}$ if $p - 2 \mid n$ for every prime $p \mid n$. Find the number of twinning integers less than 250.
2019 CMIMC, 13
Points $A$, $B$, and $C$ lie in the plane such that $AB=13$, $BC=14$, and $CA=15$. A peculiar laser is fired from $A$ perpendicular to $\overline{BC}$. After bouncing off $BC$, it travels in a direction perpendicular to $CA$. When it hits $CA$, it travels in a direction perpendicular to $AB$, and after hitting $AB$ its new direction is perpendicular to $BC$ again. If this process is continued indefinitely, the laser path will eventually approach some finite polygonal shape $T_\infty$. What is the ratio of the perimeter of $T_\infty$ to the perimeter of $\triangle ABC$?
2020 CMIMC Team, 3
Let $ABC$ be a triangle with centroid $G$ and $BC = 3$. If $ABC$ is similar to $GAB$, compute the area of $ABC$.
2018 MOAA, 6
Consider an $m \times n$ grid of unit squares. Let $R$ be the total number of rectangles of any size, and let $S$ be the total number of squares of any size. Assume that the sides of the rectangles and squares are parallel to the sides of the $m \times n$ grid. If $\frac{R}{S} =\frac{759}{50}$ , then determine $mn$.
2021 MOAA, 6
Find the sum of all two-digit prime numbers whose digits are also both prime numbers.
[i]Proposed by Nathan Xiong[/i]
MOAA Team Rounds, 2019.7
Suppose $ABC$ is a triangle inscribed in circle $\omega$ . Let $A'$ be the point on $\omega$ so that $AA'$ is a diameter, and let $G$ be the centroid of $ABC$. Given that $AB = 13$, $BC = 14$, and $CA = 15$, let $x$ be the area of triangle $AGA'$ . If $x$ can be expressed in the form $m/n$ , where m and n are relatively prime positive integers, compute $100n + m$.
2019 CMIMC, 14
Consider the following function.
$\textbf{procedure }\textsc{M}(x)$
$\qquad\textbf{if }0\leq x\leq 1$
$\qquad\qquad\textbf{return }x$
$\qquad\textbf{return }\textsc{M}(x^2\bmod 2^{32})$
Let $f:\mathbb N\to\mathbb N$ be defined such that $f(x) = 0$ if $\textsc{M}(x)$ does not terminate, and otherwise $f(x)$ equals the number of calls made to $\textsc{M}$ during the running of $\textsc{M}(x)$, not including the initial call. For example, $f(1) = 0$ and $f(2^{31}) = 1$. Compute the number of ones in the binary expansion of
\[
f(0) + f(1) + f(2) + \cdots + f(2^{32} - 1).
\]