Olympiad problems

2023 MOAA, 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, 3

Tags: MOAA 2023

At Andover, $35\%$ of students are lowerclassmen and the rest are upperclassmen. Given that $26\%$ of lowerclassmen and $6\%$ of upperclassmen take Latin, what percentage of all students take Latin? [i]Proposed by Anthony Yang[/i]

2023 MOAA, 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 Gunga Bowls, 2023.11

Tags: MOAA 2023

Let $s(n)$ denote the sum of the digits of $n$ and let $p(n)$ be the product of the digits of $n$. Find the smallest integer $k$ such that $s(k)+p(k)=49$ and $s(k+1)+p(k+1)=68$. [i]Proposed by Anthony Yang[/i]

MOAA Gunga Bowls, 2023.19

Tags: MOAA 2023

Compute the remainder when $\binom{205}{101}$ is divded by $101 \times 103$. [i]Proposed by Brandon Xu[/i]

MOAA Team Rounds, 2023.4

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]

MOAA Team Rounds, 2023.9

Tags: MOAA 2023

Let $ABCDEF$ be an equiangular hexagon. Let $P$ be the point that is a distance of 6 from $BC$, $DE$, and $FA$. If the distances from $P$ to $AB$, $CD$, and $EF$ are $8$, $11$, and $5$ respectively, find $(DE-AB)^2$. [i]Proposed by Andy Xu[/i]

MOAA Individual Speed General Rounds, 2023.1

Tags: MOAA 2023

Compute $\sqrt{202 \times 3 - 20 \times 23 + 2 \times 23 - 23}$. [i]Proposed by Andy Xu[/i]

2023 MOAA, 3

Tags: MOAA 2023

After the final exam, Mr. Liang asked each of his 17 students to guess the average final exam score. David, a very smart student, received a 100 and guessed the average would be 97. Each of the other 16 students guessed $30+\frac{n}{2}$ where $n$ was that student’s score. If the average of the final exam scores was the same as the average of the guesses, what was the average score on the final exam? [i]Proposed by Eric Wang[/i]

MOAA Gunga Bowls, 2023.10

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]

2023 MOAA, 11

Tags: MOAA 2023

Let the quadratic $P(x)=x^2+5x+1$. Two distinct real numbers $a,b$ satisfy \[P(a+b)=ab\] \[P(ab)=a+b\] Find the sum of all possible values of $a^2$. [i]Proposed by Harry Kim[/i]

2023 MOAA, 22

Tags: MOAA 2023

Harry the knight is positioned at the origin of the Cartesian plane. In a "knight hop", Harry can move from the point $(i,j)$ to a point with integer coordinates that is a distance of $\sqrt{5}$ away from $(i,j)$. What is the number of ways that Harry can return to the origin after 6 knight hops? [i]Proposed by Harry Kim[/i]

MOAA Accuracy Rounds, 2023.7

Tags: MOAA 2023

Pentagon $ANDD'Y$ has $AN \parallel DY$ and $AY \parallel D'N$ with $AN = D'Y$ and $AY = DN$. If the area of $ANDY$ is 20, the area of $AND'Y$ is 24, and the area of $ADD'$ is 26, the area of $ANDD'Y$ can be expressed in the form $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Find $m+n$. [i]Proposed by Andy Xu[/i]

MOAA Gunga Bowls, 2023.2

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, 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, 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]

2023 MOAA, 2

Tags: MOAA 2023

Let $ABCD$ be a square. Let $M$ be the midpoint of $BC$ and $N$ be the point on $AB$ such that $2AN=BN$. If the area of $\triangle DMN$ is 15, find the area of square $ABCD$. [i]Proposed by Harry Kim[/i]

2023 MOAA, 5

Tags: MOAA 2023

Andy creates a 3 sided dice with a side labeled $7$, a side labeled $17$, and a side labeled $27$. He then asks Anthony to roll the dice $3$ times. The probability that the product of Anthony's rolls is greater than $2023$ 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, 2023.6

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 Gunga Bowls, 2023.5

Tags: MOAA 2023

Andy creates a 3 sided dice with a side labeled $7$, a side labeled $17$, and a side labeled $27$. He then asks Anthony to roll the dice $3$ times. The probability that the product of Anthony's rolls is greater than $2023$ 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, 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]

2023 MOAA, 23

Tags: MOAA 2023

For every positive integer $n$ let $$f(n) = \frac{n^4+n^3+n^2-n+1}{n^6-1}$$ Given $$\sum_{n = 2}^{20} f(n) = \frac{a}{b}$$ for relatively prime positive integers $a$ and $b$, find the sum of the prime factors of $b$. [i]Proposed by Harry Kim[/i]

2023 MOAA, Tie

Tags: MOAA 2023

TB1. Two not necessarily distinct positive integers $a,b$ are randomly chosen from the set $\{1,2,\ldots, 20\}$. Find the expected value of the number of distinct prime factors of $ab$. [i]Proposed by Harry Kim[/i] TB2. Square $ABCD$ has side length $15$. Let $E$ and $F$ be points on $AD$ and $BC$ respectively such that $AE = 5$ and $BF = 5$. Find the area of intersection between triangles $\triangle{AFC}$ and $\triangle{BED}$. [i]Proposed by Andy Xu[/i] TB3. If $x$ and $y$ satisfy $$\frac{1}{x}+\frac{1}{y} = 2$$ $$\frac{x}{y}+\frac{y}{x} = 3$$ find $xy$. [i]Proposed by Harry Kim and Andy Xu[/i]

MOAA Gunga Bowls, 2023.3

Tags: MOAA 2023

At Andover, $35\%$ of students are lowerclassmen and the rest are upperclassmen. Given that $26\%$ of lowerclassmen and $6\%$ of upperclassmen take Latin, what percentage of all students take Latin? [i]Proposed by Anthony Yang[/i]

MOAA Individual Speed General Rounds, 2023.9

Tags: MOAA 2023

Let $ABCD$ be a trapezoid with $AB \parallel CD$ and $BC \perp CD$. There exists a point $P$ on $BC$ such that $\triangle{PAD}$ is equilateral. If $PB = 20$ and $PC = 23$, the area of $ABCD$ can be expressed in the form $\frac{a\sqrt{b}}{c}$ where $b$ is square-free and $a$ and $c$ are relatively prime. Find $a+b+c$. [i]Proposed by Andy Xu[/i]