Found problems: 5
2012 AMC 12/AHSME, 13
Paula the painter and her two helpers each paint at constant, but different, rates. They always start at $\text{8:00 AM}$, and all three always take the same amount of time to eat lunch. On Monday the three of them painted $50\%$ of a house, quitting at $\text{4:00 PM}$. On Tuesday, when Paula wasn't there, the two helpers painted only $24\%$ of the house and quit at $\text{2:12 PM}$. On Wednesday Paula worked by herself and finished the house by working until $\text{7:12 PM}$. How long, in minutes, was each day's lunch break?
$ \textbf{(A)}\ 30
\qquad\textbf{(B)}\ 36
\qquad\textbf{(C)}\ 42
\qquad\textbf{(D)}\ 48
\qquad\textbf{(E)}\ 60
$
2012 AMC 10, 19
Paula the painter and her two helpers each paint at constant, but different, rates. They always start at $\text{8:00 AM}$, and all three always take the same amount of time to eat lunch. On Monday the three of them painted $50\%$ of a house, quitting at $\text{4:00 PM}$. On Tuesday, when Paula wasn't there, the two helpers painted only $24\%$ of the house and quit at $\text{2:12 PM}$. On Wednesday Paula worked by herself and finished the house by working until $\text{7:12 PM}$. How long, in minutes, was each day's lunch break?
$ \textbf{(A)}\ 30
\qquad\textbf{(B)}\ 36
\qquad\textbf{(C)}\ 42
\qquad\textbf{(D)}\ 48
\qquad\textbf{(E)}\ 60
$
2018 AMC 12/AHSME, 17
Farmer Pythagoras has a field in the shape of a right triangle. The right triangle's legs have lengths 3 and 4 units. In the corner where those sides meet at a right angle, he leaves a small unplanted square $S$ so that from the air it looks like the right angle symbol. The rest of the field is planted. The shortest distance from $S$ to the hypotenuse is 2 units. What fraction of the field is planted?
[asy]
draw((0,0)--(4,0)--(0,3)--(0,0));
draw((0,0)--(0.3,0)--(0.3,0.3)--(0,0.3)--(0,0));
fill(origin--(0.3,0)--(0.3,0.3)--(0,0.3)--cycle, gray);
label("$4$", (2,0), N);
label("$3$", (0,1.5), E);
label("$2$", (.8,1), E);
label("$S$", (0,0), NE);
draw((0.3,0.3)--(1.4,1.9), dashed);
[/asy]
$\textbf{(A) } \frac{25}{27} \qquad \textbf{(B) } \frac{26}{27} \qquad \textbf{(C) } \frac{73}{75} \qquad \textbf{(D) } \frac{145}{147} \qquad \textbf{(E) } \frac{74}{75} $
2018 AMC 10, 23
Farmer Pythagoras has a field in the shape of a right triangle. The right triangle's legs have lengths 3 and 4 units. In the corner where those sides meet at a right angle, he leaves a small unplanted square $S$ so that from the air it looks like the right angle symbol. The rest of the field is planted. The shortest distance from $S$ to the hypotenuse is 2 units. What fraction of the field is planted?
[asy]
draw((0,0)--(4,0)--(0,3)--(0,0));
draw((0,0)--(0.3,0)--(0.3,0.3)--(0,0.3)--(0,0));
fill(origin--(0.3,0)--(0.3,0.3)--(0,0.3)--cycle, gray);
label("$4$", (2,0), N);
label("$3$", (0,1.5), E);
label("$2$", (.8,1), E);
label("$S$", (0,0), NE);
draw((0.3,0.3)--(1.4,1.9), dashed);
[/asy]
$\textbf{(A) } \frac{25}{27} \qquad \textbf{(B) } \frac{26}{27} \qquad \textbf{(C) } \frac{73}{75} \qquad \textbf{(D) } \frac{145}{147} \qquad \textbf{(E) } \frac{74}{75} $
2003 AMC 12-AHSME, 20
How many $ 15$-letter arrangements of $ 5$ A's, $ 5$ B's, and $ 5$ C's have no A's in the first $ 5$ letters, no B's in the next $ 5$ letters, and no C's in the last $ 5$ letters?
$ \textbf{(A)}\ \sum_{k\equal{}0}^5\binom{5}{k}^3 \qquad
\textbf{(B)}\ 3^5\cdot 2^5 \qquad
\textbf{(C)}\ 2^{15} \qquad
\textbf{(D)}\ \frac{15!}{(5!)^3} \qquad
\textbf{(E)}\ 3^{15}$