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.

AND:
OR:
NO:

Found problems: 128

MOAA Team Rounds, 2021.7
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 Team Rounds, 2021.2
Tags: MOAA 2021 , team
Four students Alice, Bob, Charlie, and Diana want to arrange themselves in a line such that Alice is at either end of the line, i.e., she is not in between two students. In how many ways can the students do this? [i]Proposed by Nathan Xiong[/i]

MOAA Gunga Bowls, 2021.22
Tags: MOAA 2021 , Gunga
Let $p$ and $q$ be positive integers such that $p$ is a prime, $p$ divides $q-1$, and $p+q$ divides $p^2+2020q^2$. Find the sum of the possible values of $p$. [i]Proposed by Andy Xu[/i]

MOAA Gunga Bowls, 2021.6
Tags: MOAA 2021 , Gunga
Determine the number of triangles, of any size and shape, in the following figure: [asy] size(4cm); draw(2*dir(0)--dir(120)--dir(240)--cycle); draw(dir(60)--2*dir(180)--dir(300)--cycle); [/asy] [i]Proposed by William Yue[/i]

2021 MOAA, 20
Tags: MOAA 2021 , Gunga
In the interior of square $ABCD$ with side length $1$, a point $P$ is chosen such that the lines $\ell_1, \ell_2$ through $P$ parallel to $AC$ and $BD$, respectively, divide the square into four distinct regions, the smallest of which has area $\mathcal{R}$. The area of the region of all points $P$ for which $\mathcal{R} \geq \tfrac{1}{6}$ can be expressed as $\frac{a-b\sqrt{c}}{d}$ where $\gcd(a,b,d)=1$ and $c$ is not divisible by the square of any prime. Compute $a+b+c+d$. [i]Proposed by Andrew Wen[/i]

MOAA Gunga Bowls, 2021.17
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, 2021.4
Tags: MOAA 2021 , Gunga
How many of the following capital English letters look the same when rotated $180^\circ$ about their center? [center]A B C D E F G H I J K L M N O P Q R S T U V W X Y Z[/center] [i]Proposed by William Yue[/i]

2021 MOAA, 14
Tags: MOAA 2021 , Gunga
Sinclair starts with the number $1$. Every minute, he either squares his number or adds $1$ to his number, both with equal probability. What is the expected number of minutes until his number is divisible by $3$? [i]Proposed by Nathan Xiong[/i]

2021 MOAA, 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, 1
Tags: MOAA 2021 , Gunga
Evaluate $2\times 0+2\times 1+ 2+0\times 2 +1$. [i]Proposed by Nathan Xiong[/i]

MOAA Individual Speed General Rounds, 2021.5
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 Gunga Bowls, 2021.5
Tags: MOAA 2021 , Gunga
Joshua rolls two dice and records the product of the numbers face up. The probability that this product is composite 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, 17
Tags: MOAA 2021 , team
Compute the remainder when $10^{2021}$ is divided by $10101$. [i]Proposed by Nathan Xiong[/i]

MOAA Individual Speed General Rounds, 2021.3
Tags: MOAA 2021 , speed
Find the number of ordered pairs $(x,y)$, where $x$ and $y$ are both integers between $1$ and $9$, inclusive, such that the product $x\times y$ ends in the digit $5$. [i]Proposed by Andrew Wen[/i]

2021 MOAA, 12
Tags: MOAA 2021 , team
Let $\triangle ABC$ have $AB=9$ and $AC=10$. A semicircle is inscribed in $\triangle ABC$ with its center on segment $BC$ such that it is tangent $AB$ at point $D$ and $AC$ at point $E$. If $AD=2DB$ and $r$ is the radius of the semicircle, $r^2$ can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m+n$. [i]Proposed by Andy Xu[/i]

2021 MOAA, 9
Tags: MOAA 2021 , Gunga
William is biking from his home to his school and back, using the same route. When he travels to school, there is an initial $20^\circ$ incline for $0.5$ kilometers, a flat area for $2$ kilometers, and a $20^\circ$ decline for $1$ kilometer. If William travels at $8$ kilometers per hour during uphill $20^\circ$ sections, $16$ kilometers per hours during flat sections, and $20$ kilometers per hour during downhill $20^\circ$ sections, find the closest integer to the number of minutes it take William to get to school and back. [i]Proposed by William Yue[/i]

2021 MOAA, 3
Tags: MOAA 2021 , speed
Find the number of ordered pairs $(x,y)$, where $x$ and $y$ are both integers between $1$ and $9$, inclusive, such that the product $x\times y$ ends in the digit $5$. [i]Proposed by Andrew Wen[/i]

2021 MOAA, 9
Tags: MOAA 2021 , speed
Triangle $\triangle ABC$ has $\angle{A}=90^\circ$ with $BC=12$. Square $BCDE$ is drawn such that $A$ is in its interior. The line through $A$ tangent to the circumcircle of $\triangle ABC$ intersects $CD$ and $BE$ at $P$ and $Q$, respectively. If $PA=4\cdot QA$, and the area of $\triangle ABC$ can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$, then compute $m+n$. [i]Proposed by Andy Xu[/i]

2021 MOAA, 7
Tags: MOAA 2021 , Gunga
Andover has a special weather forecast this week. On Monday, there is a $\frac{1}{2}$ chance of rain. On Tuesday, there is a $\frac{1}{3}$ chance of rain. This pattern continues all the way to Sunday, when there is a $\frac{1}{8}$ chance of rain. The probability that it doesn't rain in Andover all week 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, 2021.15
Tags: MOAA 2021 , team
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]

2021 MOAA, 4
Tags: MOAA 2021 , speed , algebra , inequalities
Let $a$, $b$, and $c$ be real numbers such that $0\le a,b,c\le 5$ and $2a + b + c = 10$. Over all possible values of $a$, $b$, and $c$, determine the maximum possible value of $a + 2b + 3c$. [i]Proposed by Andrew Wen[/i]

2021 MOAA, 6
Tags: MOAA 2021 , Gunga
Determine the number of triangles, of any size and shape, in the following figure: [asy] size(4cm); draw(2*dir(0)--dir(120)--dir(240)--cycle); draw(dir(60)--2*dir(180)--dir(300)--cycle); [/asy] [i]Proposed by William Yue[/i]

2021 MOAA, 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, 13
Tags: MOAA 2021 , team
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]

MOAA Gunga Bowls, 2021.24
Tags: MOAA 2021 , Gunga
Freddy the Frog is situated at 1 on an infinitely long number line. On day $n$, where $n\ge 1$, Freddy can choose to hop 1 step to the right, stay where he is, or hop $k$ steps to the left, where $k$ is an integer at most $n+1$. After day 5, how many sequences of moves are there such that Freddy has landed on at least one negative number? [i]Proposed by Andy Xu[/i]