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

2023 MOAA, 6
Tags: MOAA 2023
Call a set of integers [i]unpredictable[/i] if no four elements in the set form an arithmetic sequence. How many unordered [i]unpredictable[/i] sets of five distinct positive integers $\{a, b, c, d, e\}$ exist such that all elements are strictly less than $12$? [i]Proposed by Anthony Yang[/i]

2023 MOAA, 12
Tags: MOAA 2023
Let $N$ be the number of $105$-digit positive integers that contain the digit 1 an odd number of times. Find the remainder when $N$ is divided by $1000$. [i]Proposed by Harry Kim[/i]

MOAA Gunga Bowls, 2023.1
Tags: MOAA 2023
Find the last digit of $2023^{2023}$. [i]Proposed by Yifan Kang[/i]

MOAA Gunga Bowls, 2023.15
Tags: MOAA 2023
Triangle $ABC$ has $AB = 5$, $BC = 7$, $CA = 8$. Let $M$ be the midpoint of $BC$ and let points $P$ and $Q$ lie on $AB$ and $AC$ respectively such that $MP \perp AB$ and $MQ \perp AC$. If $H$ is the orthocenter of $\triangle{APQ}$ then the area of $\triangle{HPM}$ can be expressed in the form $\frac{a\sqrt{b}}{c}$ where $a$ and $c$ are relatively prime positive integers and $b$ is square-free. Find $a+b+c$. [i]Proposed by Harry Kim[/i]

MOAA Gunga Bowls, 2023.12
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]

MOAA Gunga Bowls, 2023.16
Tags: MOAA 2023
Compute the sum $$\frac{\varphi(50!)}{\varphi(49!)}+ \frac{\varphi(51!)}{\varphi(50!)} + \dots + \frac{\varphi(100!)}{\varphi(99!)}$$ where $\varphi(n)$ returns the number of positive integers less than $n$ that are relatively prime to $n$. [i]Proposed by Andy Xu[/i]

MOAA Accuracy Rounds, 2023.4
Tags: MOAA 2023
A two-digit number $\overline{ab}$ is [i]self-loving[/i] if $a$ and $b$ are relatively prime positive integers and $\overline{ab}$ is divisible by $a+b$. How many self-loving numbers are there? [i]Proposed by Anthony Yang and Andy Xu[/i]

MOAA Gunga Bowls, 2023.14
Tags: MOAA 2023
Let $N$ be the number of ordered triples of 3 positive integers $(a,b,c)$ such that $6a$, $10b$, and $15c$ are all perfect squares and $abc = 210^{210}$. Find the number of divisors of $N$. [i]Proposed by Andy Xu[/i]

2023 MOAA, 4
Tags: MOAA 2023
Andy has $4$ coins $c_1, c_2, c_3, c_4$ such that the probability that coin $c_i$ with $1 \leq i \leq 4$ lands tails is $\frac{1}{2^i}$. Andy flips each coin exactly once. The probability that only one coin lands on heads can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. [i]Proposed by Anthony Yang[/i]

2023 MOAA, 14
Tags: MOAA 2023
Let $N$ be the number of ordered triples of 3 positive integers $(a,b,c)$ such that $6a$, $10b$, and $15c$ are all perfect squares and $abc = 210^{210}$. Find the number of divisors of $N$. [i]Proposed by Andy Xu[/i]

MOAA Gunga Bowls, 2023.13
Tags: MOAA 2023
Let $\alpha$, $\beta$ and $\gamma$ be the roots of the polynomial $2023x^3-2023x^2-1$. Find $$\frac{1}{\alpha^3}+\frac{1}{\beta^3}+\frac{1}{\gamma^3}$$. [i]Proposed by Andy Xu[/i]

2023 MOAA, 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 Team Rounds, 2023.5
Tags: MOAA 2023
Angeline starts with a 6-digit number and she moves the last digit to the front. For example, if she originally had $100823$ she ends up with $310082$. Given that her new number is $4$ times her original number, find the smallest possible value of her original number. [i]Proposed by Angeline Zhao[/i]

MOAA Accuracy Rounds, 2023.8
Tags: MOAA 2023
Harry wants to label the points of a regular octagon with numbers $1,2,\ldots ,8$ and label the edges with $1,2,\ldots, 8$. There are special rules he must follow: If an edge is numbered even, then the sum of the numbers of its endpoints must also be even. If an edge is numbered odd, then the sum of the numbers of its endpoints must also be odd. Two octagon labelings are equivalent if they can be made equal up to rotation, but not up to reflection. If $N$ is the number of possible octagon labelings, find the remainder when $N$ is divided by $100$. [i]Proposed by Harry Kim[/i]

MOAA Accuracy Rounds, 2023.9
Tags: MOAA 2023
Let $\triangle{ABC}$ be a triangle with $AB = 10$ and $AC = 11$. Let $I$ be the center of the inscribed circle of $\triangle{ABC}$. If $M$ is the midpoint of $AI$ such that $BM = BC$ and $CM = 7$, then $BC$ can be expressed in the form $\frac{\sqrt{a}-b}{c}$ where $a$, $b$, and $c$ are positive integers. Find $a+b+c$. [color=#00f]Note that this problem is null because a diagram is impossible.[/color] [i]Proposed by Andy Xu[/i]

2023 MOAA, 3
Tags: MOAA 2023
Andy and Harry are trying to make an O for the MOAA logo. Andy starts with a circular piece of leather with radius 3 feet and cuts out a circle with radius 2 feet from the middle. Harry starts with a square piece of leather with side length 3 feet and cuts out a square with side length 2 feet from the middle. In square feet, what is the positive difference in area between Andy and Harry's final product to the nearest integer? [i]Proposed by Andy Xu[/i]

MOAA Individual Speed General Rounds, 2023.6
Tags: MOAA 2023
Define the function $f(x) = \lfloor x \rfloor + \lfloor \sqrt{x} \rfloor + \lfloor \sqrt{\sqrt{x}} \rfloor$ for all positive real numbers $x$. How many integers from $1$ to $2023$ inclusive are in the range of $f(x)$? Note that $\lfloor x\rfloor$ is known as the $\textit{floor}$ function, which returns the greatest integer less than or equal to $x$. [i]Proposed by Harry Kim[/i]

2023 MOAA, 4
Tags: MOAA 2023
A number is called \textit{super odd} if it is an odd number divisible by the square of an odd prime. For example, $2023$ is a \textit{super odd} number because it is odd and divisible by $17^2$. Find the sum of all \textit{super odd} numbers from $1$ to $100$ inclusive. [i]Proposed by Andy Xu[/i]

2023 MOAA, 13
Tags: MOAA 2023
If real numbers $x$, $y$, and $z$ satisfy $x^2-yz = 1$ and $y^2-xz = 4$ such that $|x+y+z|$ is minimized, then $z^2-xy$ can be expressed in the form $\sqrt{a}-b$ where $a$ and $b$ are positive integers. Find $a+b$. [i]Proposed by Andy Xu[/i]

2023 MOAA, 4
Tags: MOAA 2023
A two-digit number $\overline{ab}$ is [i]self-loving[/i] if $a$ and $b$ are relatively prime positive integers and $\overline{ab}$ is divisible by $a+b$. How many self-loving numbers are there? [i]Proposed by Anthony Yang and Andy Xu[/i]

2023 MOAA, 1
Tags: MOAA 2023
Compute $\sqrt{202 \times 3 - 20 \times 23 + 2 \times 23 - 23}$. [i]Proposed by Andy Xu[/i]

MOAA Individual Speed General Rounds, 2023.3
Tags: MOAA 2023
Andy and Harry are trying to make an O for the MOAA logo. Andy starts with a circular piece of leather with radius 3 feet and cuts out a circle with radius 2 feet from the middle. Harry starts with a square piece of leather with side length 3 feet and cuts out a square with side length 2 feet from the middle. In square feet, what is the positive difference in area between Andy and Harry's final product to the nearest integer? [i]Proposed by Andy Xu[/i]

MOAA Individual Speed General Rounds, 2023.5
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, 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 Team Rounds, 2023.1
Tags: MOAA 2023
Find the last two digits of $2023+202^3+20^{23}$. [i]Proposed by Anthony Yang[/i]