Found problems: 128
2021 MOAA, 9
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]
MOAA Accuracy Rounds, 2021.9
Let $S$ be the set of ordered pairs $(x,y)$ of positive integers for which $x+y\le 20$. Evaluate
\[\sum_{(x, y) \in S} (-1)^{x+y}xy.\]
[i]Proposed by Andrew Wen[/i]
2021 MOAA, 6
Find the sum of all two-digit prime numbers whose digits are also both prime numbers.
[i]Proposed by Nathan Xiong[/i]
2021 MOAA, 9
Let $S$ be the set of ordered pairs $(x,y)$ of positive integers for which $x+y\le 20$. Evaluate
\[\sum_{(x, y) \in S} (-1)^{x+y}xy.\]
[i]Proposed by Andrew Wen[/i]
MOAA Team Rounds, 2021.1
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 Gunga Bowls, 2021.13
Determine the greatest power of $2$ that is a factor of $3^{15}+3^{11}+3^{6}+1$.
[i]Proposed by Nathan Xiong[/i]
MOAA Team Rounds, 2021.12
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]
MOAA Team Rounds, 2021.13
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 Individual Speed General Rounds, 2021.9
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, 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]
2021 MOAA, 14
Evaluate
\[\left\lfloor\frac{1\times 5}{7}\right\rfloor + \left\lfloor\frac{2\times 5}{7}\right\rfloor + \left\lfloor\frac{3\times 5}{7}\right\rfloor+\cdots+\left\lfloor\frac{100\times 5}{7}\right\rfloor.\]
[i]Proposed by Nathan Xiong[/i]
2021 MOAA, 2
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]
2021 MOAA, 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 Gunga Bowls, 2021.8
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 Gunga Bowls, 2021.20
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, 2021.23
Let $P$ be a point chosen on the interior of side $\overline{BC}$ of triangle $\triangle ABC$ with side lengths $\overline{AB} = 10, \overline{BC} = 10, \overline{AC} = 12$. If $X$ and $Y$ are the feet of the perpendiculars from $P$ to the sides $AB$ and $AC$, then the minimum possible value of $PX^2 + PY^2$ 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 Team Rounds, 2021.14
Evaluate
\[\left\lfloor\frac{1\times 5}{7}\right\rfloor + \left\lfloor\frac{2\times 5}{7}\right\rfloor + \left\lfloor\frac{3\times 5}{7}\right\rfloor+\cdots+\left\lfloor\frac{100\times 5}{7}\right\rfloor.\]
[i]Proposed by Nathan Xiong[/i]
2021 MOAA, 24
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]
MOAA Team Rounds, 2021.6
Find the sum of all two-digit prime numbers whose digits are also both prime numbers.
[i]Proposed by Nathan Xiong[/i]
MOAA Individual Speed General Rounds, 2021.2
[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]
2021 MOAA, 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, 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]
2021 MOAA, 23
Let $P$ be a point chosen on the interior of side $\overline{BC}$ of triangle $\triangle ABC$ with side lengths $\overline{AB} = 10, \overline{BC} = 10, \overline{AC} = 12$. If $X$ and $Y$ are the feet of the perpendiculars from $P$ to the sides $AB$ and $AC$, then the minimum possible value of $PX^2 + PY^2$ 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]
2021 MOAA, 7
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]
2021 MOAA, 1
What is $2021+20+21+2+0+2+1$?
[i]Proposed by Nathan Xiong[/i]