Olympiad problems

2021 MOAA, 10

In $\triangle ABC$, let $X$ and $Y$ be points on segment $BC$ such that $AX=XB=20$ and $AY=YC=21$. Let $J$ be the $A$-excenter of triangle $\triangle AXY$. Given that $J$ lies on the circumcircle of $\triangle ABC$, the length of $BC$ can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m+n$. [i]Proposed by Andrew Wen[/i]

MOAA Accuracy Rounds, 2021.5

Tags: MOAA 2021 , Accuracy

If $x$, $y$, $z$ are nonnegative integers satisfying the equation below, then compute $x+y+z$. \[\left(\frac{16}{3}\right)^x\times \left(\frac{27}{25}\right)^y\times \left(\frac{5}{4}\right)^z=256.\] [i]Proposed by Jeffrey Shi[/i]

2021 MOAA, 8

Tags: MOAA 2021 , Accuracy

Will has a magic coin that can remember previous flips. If the coin has already turned up heads $m$ times and tails $n$ times, the probability that the next flip turns up heads is exactly $\frac{m+1}{m+n+2}$. Suppose that the coin starts at $0$ flips. The probability that after $10$ coin flips, heads and tails have both turned up exactly $5$ times can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m+n$. [i]Proposed by Nathan Xiong[/i]

MOAA Team Rounds, 2021.18

Tags: MOAA 2021 , team

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]

2021 MOAA, 3

Tags: MOAA 2021 , Accuracy

Arnav is placing three rectangles into a $3 \times 3$ grid of unit squares. He has a $1\times 3$ rectangle, a $1\times 2$ rectangle, and a $1\times 1$ rectangle. He must place the rectangles onto the grid such that the edges of the rectangles align with the gridlines of the grid. If he is allowed to rotate the rectangles, how many ways can he place the three rectangles into the grid, without overlap? [i]Proposed by William Yue[/i]

MOAA Gunga Bowls, 2021.10

Tags: MOAA 2021 , Gunga

We say that an ordered pair $(a,b)$ of positive integers with $a>b$ is square-ish if both $a+b$ and $a-b$ are perfect squares. For example, $(17,8)$ is square-ish because $17+8=25$ and $17-8=9$ are both perfect squares. How many square-ish pairs $(a,b)$ with $a+b<100$ are there? [i]Proposed by Nathan Xiong[/i]

2021 MOAA, 17

Tags: MOAA 2021 , Gunga

Isosceles trapezoid $ABCD$ has side lengths $AB = 6$ and $CD = 12$, while $AD = BC$. It is given that $O$, the circumcenter of $ABCD$, lies in the interior of the trapezoid. The extensions of lines $AD$ and $BC$ intersect at $T$. Given that $OT = 18$, the area of $ABCD$ can be expressed as $a + b\sqrt{c}$ where $a$, $b$, and $c$ are positive integers where $c$ is not divisible by the square of any prime. Compute $a+b+c$. [i]Proposed by Andrew Wen[/i]

MOAA Gunga Bowls, 2021.18

Tags: MOAA 2021 , Gunga

Find the largest positive integer $n$ such that the number $(2n)!$ ends with $10$ more zeroes than the number $n!$. [i]Proposed by Andy Xu[/i]

2021 MOAA, 8

Tags: MOAA 2021 , speed

Andrew chooses three (not necessarily distinct) integers $a$, $b$, and $c$ independently and uniformly at random from $\{1,2,3,4,5,6,7\}$. Let $p$ be the probability that $abc(a+b+c)$ is divisible by $4$. If $p$ can be written as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$, then compute $m+n$. [i]Proposed by Andrew Wen[/i]

2021 MOAA, 6

Tags: MOAA 2021 , Accuracy

Let $\triangle ABC$ be a triangle in a plane such that $AB=13$, $BC=14$, and $CA=15$. Let $D$ be a point in three-dimensional space such that $\angle{BDC}=\angle{CDA}=\angle{ADB}=90^\circ$. Let $d$ be the distance from $D$ to the plane containing $\triangle ABC$. The value $d^2$ can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m+n$. [i]Proposed by William Yue[/i]

2021 MOAA, 2

Tags: MOAA 2021 , Accuracy

On Andover's campus, Graves Hall is $60$ meters west of George Washington Hall, and George Washington Hall is $80$ meters north of Paresky Commons. Jessica wants to walk from Graves Hall to Paresky Commons. If she first walks straight from Graves Hall to George Washington Hall and then walks straight from George Washington Hall to Paresky Commons, it takes her $8$ minutes and $45$ seconds while walking at a constant speed. If she walks with the same speed directly from Graves Hall to Paresky Commons, how much time does she save, in seconds? [i]Proposed by Nathan Xiong[/i]

2021 MOAA, 11

Tags: MOAA 2021 , team

Find the product of all possible real values for $k$ such that the system of equations $$x^2+y^2= 80$$ $$x^2+y^2= k+2x-8y$$ has exactly one real solution $(x,y)$. [i]Proposed by Nathan Xiong[/i]

2021 MOAA, 4

Tags: MOAA 2021 , team

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]

2021 MOAA, 7

Tags: MOAA 2021 , team

Compute the number of ordered pairs $(a,b)$ of positive integers satisfying $a^b=2^{100}$. [i]Proposed by Nathan Xiong[/i]

MOAA Accuracy Rounds, 2021.1

Tags: MOAA 2021 , Accuracy

Evaluate \[2\times (2\times (2\times (2\times (2\times (2\times 2-2)-2)-2)-2)-2)-2.\] [i]Proposed by Nathan Xiong[/i]

2021 MOAA, 5

Tags: MOAA 2021 , speed

There are 12 students in Mr. DoBa's math class. On the final exam, the average score of the top 3 students was 8 more than the average score of the other students, and the average score of the entire class was 85. Compute the average score of the top $3$ students. [i]Proposed by Yifan Kang[/i]

MOAA Team Rounds, 2021.11

Tags: MOAA 2021 , team

Find the product of all possible real values for $k$ such that the system of equations $$x^2+y^2= 80$$ $$x^2+y^2= k+2x-8y$$ has exactly one real solution $(x,y)$. [i]Proposed by Nathan Xiong[/i]

MOAA Individual Speed General Rounds, 2021.6

Tags: MOAA 2021 , speed

Suppose $(a,b)$ is an ordered pair of integers such that the three numbers $a$, $b$, and $ab$ form an arithmetic progression, in that order. Find the sum of all possible values of $a$. [i]Proposed by Nathan Xiong[/i]

2021 MOAA, 12

Tags: MOAA 2021 , Gunga

Andy wishes to open an electronic lock with a keypad containing all digits from $0$ to $9$. He knows that the password registered in the system is $2469$. Unfortunately, he is also aware that exactly two different buttons (but he does not know which ones) $\underline{a}$ and $\underline{b}$ on the keypad are broken $-$ when $\underline{a}$ is pressed the digit $b$ is registered in the system, and when $\underline{b}$ is pressed the digit $a$ is registered in the system. Find the least number of attempts Andy needs to surely be able to open the lock. [i]Proposed by Andrew Wen[/i]

MOAA Team Rounds, 2021.9

Tags: MOAA 2021 , team

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, 1

Tags: MOAA 2021 , team

The value of \[\frac{1}{20}-\frac{1}{21}+\frac{1}{20\times 21}\] can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m+n$. [i]Proposed by Nathan Xiong[/i]

MOAA Accuracy Rounds, 2021.8

Tags: MOAA 2021 , Accuracy

Will has a magic coin that can remember previous flips. If the coin has already turned up heads $m$ times and tails $n$ times, the probability that the next flip turns up heads is exactly $\frac{m+1}{m+n+2}$. Suppose that the coin starts at $0$ flips. The probability that after $10$ coin flips, heads and tails have both turned up exactly $5$ times can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m+n$. [i]Proposed by Nathan Xiong[/i]

2021 MOAA, 2

Tags: MOAA 2021 , Gunga

Add one pair of brackets to the expression \[1+2\times 3+4\times 5+6\] so that the resulting expression has a valid mathematical value, e.g., $1+2\times (3 + 4\times 5)+6=53$. What is the largest possible value that one can make? [i]Proposed by Nathan Xiong[/i]

MOAA Individual Speed General Rounds, 2021.8

Tags: MOAA 2021 , speed

Andrew chooses three (not necessarily distinct) integers $a$, $b$, and $c$ independently and uniformly at random from $\{1,2,3,4,5,6,7\}$. Let $p$ be the probability that $abc(a+b+c)$ is divisible by $4$. If $p$ can be written as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$, then compute $m+n$. [i]Proposed by Andrew Wen[/i]

MOAA Accuracy Rounds, 2021.7

Tags: MOAA 2021 , Accuracy

Jeffrey rolls fair three six-sided dice and records their results. The probability that the mean of these three numbers is greater than the median of these three numbers can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m+n$. [i]Proposed by Nathan Xiong[/i]