This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 119

MOAA Accuracy Rounds, 2023.6
Tags: MOAA 2023
Let $b$ be a positive integer such that 2032 has 3 digits when expressed in base $b$. Define the function $S_k(n)$ as the sum of the digits of the base $k$ representation of $n$. Given that $S_b(2032)+S_{b^2}(2032) = 14$, find $b$. [i]Proposed by Anthony Yang[/i]

MOAA Team Rounds, 2023.8
Tags: MOAA 2023
Two consecutive positive integers $n$ and $n+1$ have the property that they both have $6$ divisors but a different number of distinct prime factors. Find the sum of the possible values of $n$. [i]Proposed by Harry Kim[/i]

MOAA Individual Speed General Rounds, 2023.10
Tags: MOAA 2023
If $x,y,z$ satisfy the system of equations \[xy+yz+zx=23\] \[\frac{y}{x+y}+\frac{z}{y+z}+\frac{x}{z+x}=-1\] \[\frac{z^2x}{x+y}+\frac{x^2y}{y+z}+\frac{y^2z}{z+x}=202\] Find the value of $x^2+y^2+z^2$. [i]Proposed by Harry Kim[/i]

MOAA Team Rounds, 2023.2
Tags: MOAA 2023
Let $ABCD$ be a square with side length $6$. Let $E$ be a point on the perimeter of $ABCD$ such that the area of $\triangle{AEB}$ is $\frac{1}{6}$ the area of $ABCD$. Find the maximum possible value of $CE^2$. [i]Proposed by Anthony Yang[/i]

2023 MOAA, 21
Tags: MOAA 2023
In obtuse triangle $ABC$ where $\angle B > 90^\circ$ let $H$ and $O$ be its orthocenter and circumcenter respectively. Let $D$ be the foot of the altitude from $A$ to $HC$ and $E$ be the foot of the altitude from $B$ to $AC$ such that $O,E,D$ lie on a line. If $OC=8$ and $OE=4$, find the area of triangle $HAB$. [i]Proposed by Harry Kim[/i]

2023 MOAA, 2
Tags: MOAA 2023
Harry wants to put $5$ identical blue books, $3$ identical red books, and $1$ white book on his bookshelf. If no two adjacent books may be the same color, how many distinct arrangements can Harry make? [i]Proposed by Anthony Yang[/i]

2023 MOAA, 5
Tags: MOAA 2023
Let $P(x)$ be a nonzero quadratic polynomial such that $P(1) = P(2) = 0$. Given that $P(3)^2 = P(4)+P(5)$, find $P(6)$. [i]Proposed by Andy Xu[/i]

2023 MOAA, 6
Tags: MOAA 2023
Andy chooses not necessarily distinct digits $G$, $U$, $N$, and $A$ such that the $5$ digit number $GUNGA$ is divisible by $44$. Find the least possible value of $G+U+N+G+A$. [i]Proposed by Andy Xu[/i]

MOAA Team Rounds, 2023.7
Tags: MOAA 2023
In a cube, let $M$ be the midpoint of one of the segments. Choose two vertices of the cube, $A$ and $B$. What is the number of distinct possible triangles $\triangle AMB$ up to congruency? [i]Proposed by Harry Kim[/i]

2023 MOAA, 1
Tags: MOAA 2023
Find the last digit of $2023^{2023}$. [i]Proposed by Yifan Kang[/i]

MOAA Individual Speed General Rounds, 2023.7
Tags: MOAA 2023
Andy flips a strange coin for which the probability of flipping heads is $\frac{1}{2^k+1}$, where $k$ is the number of heads that appeared previously. If Andy flips the coin repeatedly until he gets heads 10 times, what is the expected number of total flips he performs? [i]Proposed by Harry Kim[/i]

MOAA Accuracy Rounds, 2023.3
Tags: MOAA 2023
Ms. Raina's math class has 6 students, including the troublemakers Andy and Harry. For a group project, Ms. Raina randomly divides the students into three groups containing 1, 2, and 3 people. The probability that Andy and Harry unfortunately end up in the same group can be expressed in the form $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$. [i]Proposed by Andy Xu[/i]

2023 MOAA, 2
Tags: MOAA 2023
In the coordinate plane, the line passing through points $(2023,0)$ and $(-2021,2024)$ also passes through $(1,c)$ for a constant $c$. Find $c$. [i]Proposed by Andy Xu[/i]

2023 MOAA, 12
Tags: MOAA 2023
Andy is planning to flip a fair coin 10 times. Among the 10 flips, Valencia randomly chooses one flip to exchange Andy's fair coin with her special coin which lands on heads with a probability of $\frac{1}{4}$. If the coin is exchanged in a certain flip, then that flip, along with all following flips will be performed with the special coin. The expected number of heads Andy flips can be expressed as $\frac{m}{n}$ where $m$ and $n$ are positive integers. Find $m+n$. [i]Proposed by Andy Xu[/i]

2023 MOAA, 10
Tags: MOAA 2023
Let $S$ be the set of lattice points $(a,b)$ in the coordinate plane such that $1\le a\le 30$ and $1\le b\le 30$. What is the maximum number of lattice points in $S$ such that no four points form a square of side length 2? [i]Proposed by Harry Kim[/i]

MOAA Accuracy Rounds, 2023.5
Tags: MOAA 2023
Let $k$ be a constant such that exactly three real values of $x$ satisfy $$x-|x^2-4x+3| = k$$ The sum of all possible values of $k$ can be expressed in the form $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers, find $m+n$. [i]Proposed by Andy Xu[/i]

MOAA Gunga Bowls, 2023.9
Tags: MOAA 2023
Real numbers $x$ and $y$ satisfy $$xy+\frac{x}{y} = 3$$ $$\frac{1}{x^2y^2}+\frac{y^2}{x^2} = 4$$ If $x^2$ can be expressed in the form $\frac{a+\sqrt{b}}{c}$ for integers $a$, $b$, and $c$. Find $a+b+c$. [i]Proposed by Andy Xu[/i]

MOAA Gunga Bowls, 2023.18
Tags: MOAA 2023
Triangle $\triangle{ABC}$ is isosceles with $AB = AC$. Let the incircle of $\triangle{ABC}$ intersect $BC$ and $AC$ at $D$ and $E$ respectively. Let $F \neq A$ be the point such that $DF = DA$ and $EF = EA$. If $AF = 8$ and the circumradius of $\triangle{AED}$ is $5$, find the area of $\triangle{ABC}$. [i]Proposed by Anthony Yang and Andy Xu[/i]

2023 MOAA, 10
Tags: MOAA 2023
A number is called [i]winning[/i] if it can be expressed in the form $\frac{a}{20}+\frac{b}{23}$ where $a$ and $b$ are positive integers. How many [i]winning[/i] numbers are less than 1? [i]Proposed by Andy Xu[/i]