Found problems: 128
2021 MOAA, 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]
MOAA Gunga Bowls, 2021.9
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]
MOAA Team Rounds, 2021.3
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]
MOAA Team Rounds, 2021.17
Compute the remainder when $10^{2021}$ is divided by $10101$.
[i]Proposed by Nathan Xiong[/i]
MOAA Gunga Bowls, 2021.1
Evaluate $2\times 0+2\times 1+ 2+0\times 2 +1$.
[i]Proposed by Nathan Xiong[/i]
MOAA Accuracy Rounds, 2021.6
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]
MOAA Accuracy Rounds, 2021.3
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]
2021 MOAA, 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]
MOAA Team Rounds, 2021.16
Let $\triangle ABC$ have $\angle ABC=67^{\circ}$. Point $X$ is chosen such that $AB = XC$, $\angle{XAC}=32^\circ$, and $\angle{XCA}=35^\circ$. Compute $\angle{BAC}$ in degrees.
[i]Proposed by Raina Yang[/i]
MOAA Team Rounds, 2021.4
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, 5
Two right triangles are placed next to each other to form a quadrilateral as shown. What is the perimeter of the quadrilateral?
[asy]
size(4cm);
fill((-5,0)--(0,12)--(0,6)--(8,0)--cycle, gray+opacity(0.3));
draw((0,0)--(0,12)--(-5,0)--cycle);
draw((0,0)--(8,0)--(0,6));
label("5", (-2.5,0), S);
label("13", (-2.5,6), dir(140));
label("6", (0,3), E);
label("8", (4,0), S);
[/asy]
[i]Proposed by Nathan Xiong[/i]
2021 MOAA, 16
Let $1,7,19,\ldots$ be the sequence of numbers such that for all integers $n\ge 1$, the average of the first $n$ terms is equal to the $n$th perfect square. Compute the last three digits of the $2021$st term in the sequence.
[i]Proposed by Nathan Xiong[/i]
MOAA Team Rounds, 2021.10
For how many nonempty subsets $S \subseteq \{1, 2, \ldots , 10\}$ is the sum of all elements in $S$ even?
[i]Proposed by Andrew Wen[/i]
MOAA Gunga Bowls, 2021.16
Let $1,7,19,\ldots$ be the sequence of numbers such that for all integers $n\ge 1$, the average of the first $n$ terms is equal to the $n$th perfect square. Compute the last three digits of the $2021$st term in the sequence.
[i]Proposed by Nathan Xiong[/i]
MOAA Accuracy Rounds, 2021.4
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 Gunga Bowls, 2021.21
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, 2021.2
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, 8
Evaluate
\[2^{7}\times 3^{0}+2^{6}\times 3^{1}+2^{5}\times 3^{2}+\cdots+2^{0}\times 3^{7}.\]
[i]Proposed by Nathan Xiong[/i]
2021 MOAA, 18
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, 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, 8
Compute the number of triangles of different sizes which contain the gray triangle in the figure below.
[asy]
size(5cm);
real n = 4;
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));
}
filldraw((1.5,0.866)--(2,2*0.866)--(2.5,0.866)--cycle, gray);
[/asy]
[i]Proposed by Nathan Xiong[/i]
MOAA Team Rounds, 2021.5
Two right triangles are placed next to each other to form a quadrilateral as shown. What is the perimeter of the quadrilateral?
[asy]
size(4cm);
fill((-5,0)--(0,12)--(0,6)--(8,0)--cycle, gray+opacity(0.3));
draw((0,0)--(0,12)--(-5,0)--cycle);
draw((0,0)--(8,0)--(0,6));
label("5", (-2.5,0), S);
label("13", (-2.5,6), dir(140));
label("6", (0,3), E);
label("8", (4,0), S);
[/asy]
[i]Proposed by Nathan Xiong[/i]
2021 MOAA, 11
Let $ABCD$ be a rectangle with $AB=10$ and $BC=26$. Let $\omega_1$ be the circle with diameter $\overline{AB}$ and $\omega_2$ be the circle with diameter $\overline{CD}$. Suppose $\ell$ is a common internal tangent to $\omega_1$ and $\omega_2$ and that $\ell$ intersects $AD$ and $BC$ at $E$ and $F$ respectively. What is $EF$?
[asy]
size(10cm);
draw((0,0)--(26,0)--(26,10)--(0,10)--cycle);
draw((1,0)--(25,10));
draw(circle((0,5),5));
draw(circle((26,5),5));
dot((1,0));
dot((25,10));
label("$E$",(1,0),SE);
label("$F$",(25,10),NW);
label("$A$", (0,0), SW);
label("$B$", (0,10), NW);
label("$C$", (26,10), NE);
label("$D$", (26,0), SE);
dot((0,0));
dot((0,10));
dot((26,0));
dot((26,10));
[/asy]
[i]Proposed by Nathan Xiong[/i]
MOAA Team Rounds, 2021.8
Evaluate
\[2^{7}\times 3^{0}+2^{6}\times 3^{1}+2^{5}\times 3^{2}+\cdots+2^{0}\times 3^{7}.\]
[i]Proposed by Nathan Xiong[/i]
MOAA Gunga Bowls, 2021.11
Let $ABCD$ be a rectangle with $AB=10$ and $BC=26$. Let $\omega_1$ be the circle with diameter $\overline{AB}$ and $\omega_2$ be the circle with diameter $\overline{CD}$. Suppose $\ell$ is a common internal tangent to $\omega_1$ and $\omega_2$ and that $\ell$ intersects $AD$ and $BC$ at $E$ and $F$ respectively. What is $EF$?
[asy]
size(10cm);
draw((0,0)--(26,0)--(26,10)--(0,10)--cycle);
draw((1,0)--(25,10));
draw(circle((0,5),5));
draw(circle((26,5),5));
dot((1,0));
dot((25,10));
label("$E$",(1,0),SE);
label("$F$",(25,10),NW);
label("$A$", (0,0), SW);
label("$B$", (0,10), NW);
label("$C$", (26,10), NE);
label("$D$", (26,0), SE);
dot((0,0));
dot((0,10));
dot((26,0));
dot((26,10));
[/asy]
[i]Proposed by Nathan Xiong[/i]