Found problems: 128
MOAA Team Rounds, 2021.20
Compute the sum of all integers $x$ for which there exists an integer $y$ such that
\[x^3+xy+y^3=503.\]
[i]Proposed by Nathan Xiong[/i]
2021 MOAA, 15
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]
MOAA Gunga Bowls, 2021.19
Let $S$ be the set of triples $(a,b,c)$ of non-negative integers with $a+b+c$ even. The value of the sum
\[\sum_{(a,b,c)\in S}\frac{1}{2^a3^b5^c}\]
can be expressed as $\frac{m}{n}$ for relative prime positive integers $m$ and $n$. Compute $m+n$.
[i]Proposed by Nathan Xiong[/i]
MOAA Individual Speed General Rounds, 2021.7
If positive real numbers $x,y,z$ satisfy the following system of equations, compute $x+y+z$.
$$xy+yz = 30$$
$$yz+zx = 36$$
$$zx+xy = 42$$
[i]Proposed by Nathan Xiong[/i]
2021 MOAA, 2
[asy]
size(5cm);
defaultpen(fontsize(6pt));
draw((0,0)--(4,0)--(4,4)--(0,4)--cycle);
draw((0,0)--(-4,0)--(-4,-4)--(0,-4)--cycle);
draw((1,-1)--(1,3)--(-3,3)--(-3,-1)--cycle);
draw((-1,1)--(-1,-3)--(3,-3)--(3,1)--cycle);
draw((-4,-4)--(0,-4)--(0,-3)--(3,-3)--(3,0)--(4,0)--(4,4)--(0,4)--(0,3)--(-3,3)--(-3,0)--(-4,0)--cycle, red+1.2);
label("1", (-3.5,0), S);
label("2", (-2,0), S);
label("1", (-0.5,0), S);
label("1", (3.5,0), S);
label("2", (2,0), S);
label("1", (0.5,0), S);
label("1", (0,3.5), E);
label("2", (0,2), E);
label("1", (0,0.5), E);
label("1", (0,-3.5), E);
label("2", (0,-2), E);
label("1", (0,-0.5), E);
[/asy]
Compute the area of the resulting shape, drawn in red above.
[i]Proposed by Nathan Xiong[/i]
MOAA Gunga Bowls, 2021.2
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 Gunga Bowls, 2021.7
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 Gunga Bowls, 2021.15
Let $a,b,c,d$ be the four roots of the polynomial
\[x^4+3x^3-x^2+x-2.\]
Given that $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}=\frac{1}{2}$ and $\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}+\frac{1}{d^2}=-\frac{3}{4}$, the value of
\[\frac{1}{a^3}+\frac{1}{b^3}+\frac{1}{c^3}+\frac{1}{d^3}\]
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, 10
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, 6
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, 21
King William is located at $(1, 1)$ on the coordinate plane. Every day, he chooses one of the eight lattice points closest to him and moves to one of them with equal probability. When he exits the region bounded by the $x, y$ axes and $x+y = 4$, he stops moving and remains there forever. Given that after an arbitrarily large amount of time he must exit the region, the probability he ends up on $x+y = 4$ can be expressed as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
[i]Proposed by Andrew Wen[/i]
MOAA Accuracy Rounds, 2021.10
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 Individual Speed General Rounds, 2021.1
What is $2021+20+21+2+0+2+1$?
[i]Proposed by Nathan Xiong[/i]
MOAA Gunga Bowls, 2021.14
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]
MOAA Team Rounds, 2021.19
Consider the $5$ by $5$ by $5$ equilateral triangular grid as shown:
[asy]
size(5cm);
real n = 5;
for (int i = 0; i < n; ++i) {
draw((0.5*i,0.866*i)--(n-0.5*i,0.866*i));
}
for (int i = 0; i < n; ++i) {
draw((n-i,0)--((n-i)/2,(n-i)*0.866));
}
for (int i = 0; i < n; ++i) {
draw((i,0)--((n+i)/2,(n-i)*0.866));
}
[/asy]
Ethan chooses two distinct upward-oriented equilateral triangles bounded by the gridlines. The probability that Ethan chooses two triangles that share exactly one vertex can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m+n$.
[i]Proposed by Andrew Wen[/i]
2021 MOAA, 3
For two real numbers $x$ and $y$, let $x\circ y=\frac{xy}{x+y}$. The value of
\[1 \circ (2 \circ (3 \circ (4 \circ 5)))\]
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, 3
What is the last digit of $2021^{2021}$?
[i]Proposed by Yifan Kang[/i]
2021 MOAA, 7
If positive real numbers $x,y,z$ satisfy the following system of equations, compute $x+y+z$.
$$xy+yz = 30$$
$$yz+zx = 36$$
$$zx+xy = 42$$
[i]Proposed by Nathan Xiong[/i]
2021 MOAA, 5
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, 22
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]
2021 MOAA, 18
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]
MOAA Gunga Bowls, 2021.3
What is the last digit of $2021^{2021}$?
[i]Proposed by Yifan Kang[/i]
MOAA Individual Speed General Rounds, 2021.10
Let $ABCD$ be a unit square in the plane. Points $X$ and $Y$ are chosen independently and uniformly at random on the perimeter of $ABCD$. If the expected value of the area of triangle $\triangle AXY$ 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, 4
Compute the number of two-digit numbers $\overline{ab}$ with nonzero digits $a$ and $b$ such that $a$ and $b$ are both factors of $\overline{ab}$.
[i]Proposed by Nathan Xiong[/i]
MOAA Individual Speed General Rounds, 2021.4
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]