Found problems: 3632
2016 AMC 12/AHSME, 4
The mean, median, and mode of the $7$ data values $60, 100, x, 40, 50, 200, 90$ are all equal to $x$. What is the value of $x$?
$\textbf{(A)}\ 50 \qquad\textbf{(B)}\ 60 \qquad\textbf{(C)}\ 75 \qquad\textbf{(D)}\ 90 \qquad\textbf{(E)}\ 100$
2009 AMC 12/AHSME, 13
Triangle $ ABC$ has $ AB\equal{}13$ and $ AC\equal{}15$, and the altitude to $ \overline{BC}$ has length $ 12$. What is the sum of the two possible values of $ BC$?
$ \textbf{(A)}\ 15\qquad
\textbf{(B)}\ 16\qquad
\textbf{(C)}\ 17\qquad
\textbf{(D)}\ 18\qquad
\textbf{(E)}\ 19$
2009 AMC 10, 15
When a bucket is two-thirds full of water, the bucket and water weigh $ a$ kilograms. When the bucket is one-half full of water the total weight is $ b$ kilograms. In terms of $ a$ and $ b$, what is the total weight in kilograms when the bucket is full of water?
$ \textbf{(A)}\ \frac23a\plus{}\frac13b\qquad
\textbf{(B)}\ \frac32a\minus{}\frac12b\qquad
\textbf{(C)}\ \frac32a\plus{}b$
$ \textbf{(D)}\ \frac32a\plus{}2b\qquad
\textbf{(E)}\ 3a\minus{}2b$
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} $
2001 AMC 12/AHSME, 6
A telephone number has the form $ ABC \minus{} DEF \minus{} GHIJ$, where each letter represents a different digit. The digits in each part of the numbers are in decreasing order; that is, $ A > B > C$, $ D > E > F$, and $ G > H > I > J$. Furthermore, $ D$, $ E$, and $ F$ are consecutive even digits; $ G$, $ H$, $ I$, and $ J$ are consecutive odd digits; and $ A \plus{} B \plus{} C \equal{} 9$. Find $ A$.
$ \textbf{(A)} \ 4 \qquad \textbf{(B)} \ 5 \qquad \textbf{(C)} \ 6 \qquad \textbf{(D)} \ 7 \qquad \textbf{(E)} \ 8$
2017 AMC 12/AHSME, 17
There are 24 different complex numbers $z$ such that $z^{24} = 1$. For how many of these is $z^6$ a real number?
$\textbf{(A) }1\qquad\textbf{(B) }3\qquad\textbf{(C) }6\qquad\textbf{(D) }12\qquad\textbf{(E) }24$
2014 AMC 12/AHSME, 15
A five-digit palindrome is a positive integer with respective digits $abcba$, where $a$ is non-zero. Let $S$ be the sum of all five-digit palindromes. What is the sum of the digits of $S$?
$\textbf{(A) }9\qquad
\textbf{(B) }18\qquad
\textbf{(C) }27\qquad
\textbf{(D) }36\qquad
\textbf{(E) }45\qquad$
2017 AIME Problems, 11
Five towns are connected by a system of roads. There is exactly one road connecting each pair of towns. Find the number of ways there are to make all the roads one-way in such a way that it is still possible to get from any town to any other town using the roads (possibly passing through other towns on the way).
2022 AMC 12/AHSME, 6
Consider the following $100$ sets of $10$ elements each:
\begin{align*}
&\{1,2,3,\cdots,10\}, \\
&\{11,12,13,\cdots,20\},\\
&\{21,22,23,\cdots,30\},\\
&\vdots\\
&\{991,992,993,\cdots,1000\}.
\end{align*}
How many of these sets contain exactly two multiples of $7$?
$\textbf{(A)} 40\qquad\textbf{(B)} 42\qquad\textbf{(C)} 43\qquad\textbf{(D)} 49\qquad\textbf{(E)} 50$
2023 AMC 12/AHSME, 2
The weight of $\frac 13$ of a large pizza together with $3 \frac 12$ cups of orange slices is the same as the weight of $\frac 34$ of a large pizza together with $\frac 12$ cup of orange slices. A cup of orange slices weighs $\frac 14$ of a pound. What is the weight, in pounds, of a large pizza?
$\textbf{(A)}~1\frac45\qquad\textbf{(B)}~2\qquad\textbf{(C)}~2\frac25\qquad\textbf{(D)}~3\qquad\textbf{(E)}~3\frac35$
2010 USAMO, 1
Let $AXYZB$ be a convex pentagon inscribed in a semicircle of diameter $AB$. Denote by $P$, $Q$, $R$, $S$ the feet of the perpendiculars from $Y$ onto lines $AX$, $BX$, $AZ$, $BZ$, respectively. Prove that the acute angle formed by lines $PQ$ and $RS$ is half the size of $\angle XOZ$, where $O$ is the midpoint of segment $AB$.
2015 AMC 10, 7
How many terms are there in the arithmetic sequence $13, 16, 19, \dots, 70,73$?
$ \textbf{(A) }20\qquad\textbf{(B) }21\qquad\textbf{(C) }24\qquad\textbf{(D) }60\qquad\textbf{(E) }61 $
1961 AMC 12/AHSME, 2
An automobile travels $a/6$ feet in $r$ seconds. If this rate is maintained for $3$ minutes, how many yards does it travel in $3$ minutes?
${{ \textbf{(A)}\ \frac{a}{1080r}\qquad\textbf{(B)}\ \frac{30r}{a}\qquad\textbf{(C)}\ \frac{30a}{r}\qquad\textbf{(D)}\ \frac{10r}{a} }\qquad\textbf{(E)}\ \frac{10a}{r} } $
2019 AMC 12/AHSME, 13
A red ball and a green ball are randomly and independently tossed into bins numbered with positive integers so that for each ball, the probability that it is tossed into bin $k$ is $2^{-k}$ for $k=1,2,3,\ldots.$ What is the probability that the red ball is tossed into a higher-numbered bin than the green ball?
$\textbf{(A) } \frac{1}{4} \qquad\textbf{(B) } \frac{2}{7} \qquad\textbf{(C) } \frac{1}{3} \qquad\textbf{(D) } \frac{3}{8} \qquad\textbf{(E) } \frac{3}{7}$
2014 AMC 8, 6
Six rectangles each with a common base width of $2$ have lengths of $1, 4, 9, 16, 25,$ and $36$. What is the sum of the areas of the six rectangles?
$\textbf{(A) }91\qquad\textbf{(B) }93\qquad\textbf{(C) }162\qquad\textbf{(D) }182\qquad \textbf{(E) }202$
2016 AMC 12/AHSME, 10
Five friends sat in a movie theater in a row containing $5$ seats, numbered $1$ to $5$ from left to right. (The directions "left" and "right" are from the point of view of the people as they sit in the seats.) During the movie Ada went to the lobby to get some popcorn. When she returned, she found that Bea had moved two seats to the right, Ceci had moved one seat to the left, and Dee and Edie had switched seats, leaving an end seat for Ada. In which seat had Ada been sitting before she got up?
$\textbf{(A) }1 \qquad \textbf{(B) } 2 \qquad \textbf{(C) } 3 \qquad \textbf{(D) } 4\qquad \textbf{(E) } 5$
2017 AMC 10, 6
Ms. Carroll promised that anyone who got all the multiple choice questions right on the upcoming exam would receive an A on the exam. Which of these statements necessarily follows logically?
$\textbf{(A)}$ If Lewis did not receive an A, then he got all of the multiple choice questions wrong. \\
$\textbf{(B)}$ If Lewis did not receive an A, then he got at least one of the multiple choice questions wrong. \\
$\textbf{(C)}$ If Lewis got at least one of the multiple choice questions wrong, then he did not receive an A. \\
$\textbf{(D)}$ If Lewis received an A, then he got all of the multiple choice questions right. \\
$\textbf{(E)}$ If Lewis received an A, then he got at least one of the multiple choice questions right.
2014 AIME Problems, 12
Let $A=\{1,2,3,4\}$, and $f$ and $g$ be randomly chosen (not necessarily distinct) functions from $A$ to $A$. The probability that the range of $f$ and the range of $g$ are disjoint is $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m$.
2021 AMC 12/AHSME Fall, 16
Let $a, b,$ and $c$ be positive integers such that $a+b+c=23$ and \[\gcd(a,b)+\gcd(b,c)+\gcd(c,a)=9.\] What is the sum of all possible distinct values of $a^{2}+b^{2}+c^{2}$?
$\textbf{(A)} ~259\qquad\textbf{(B)} ~438\qquad\textbf{(C)} ~516\qquad\textbf{(D)} ~625\qquad\textbf{(E)} ~687$
Proposed by [b]djmathman[/b]
2008 AMC 10, 15
How many right triangles have integer leg lengths $ a$ and $ b$ and a hypotenuse of length $ b\plus{}1$, where $ b<100$?
$ \textbf{(A)}\ 6 \qquad
\textbf{(B)}\ 7 \qquad
\textbf{(C)}\ 8 \qquad
\textbf{(D)}\ 9 \qquad
\textbf{(E)}\ 10$
2022 AMC 12/AHSME, 2
The sum of three numbers is $96$. The first number is $6$ times the third number, and the third number is $40$ less than the second number. What is the absolute value of the difference between the first and second numbers?
$\textbf{(A) } 1 \qquad \textbf{(B) } 2 \qquad \textbf{(C) } 3 \qquad \textbf{(D) } 4 \qquad \textbf{(E) } 5$
2020 AIME Problems, 11
For integers $a$, $b$, $c$, and $d$, let $f(x) = x^2 + ax + b$ and $g(x) = x^2 + cx + d$. Find the number of ordered triples $(a,b,c)$ of integers with absolute values not exceeding $10$ for which there is an integer $d$ such that $g(f(2)) = g(f(4)) = 0$.
2005 AIME Problems, 10
Triangle $ABC$ lies in the Cartesian Plane and has an area of 70. The coordinates of $B$ and $C$ are $(12,19)$ and $(23,20)$, respectively, and the coordinates of $A$ are $(p,q)$. The line containing the median to side $BC$ has slope $-5$. Find the largest possible value of $p+q$.
2006 AMC 10, 19
How many non-similar triangle have angles whose degree measures are distinct positive integers in arithmetic progression?
$ \textbf{(A) } 0 \qquad \textbf{(B) } 1 \qquad \textbf{(C) } 59 \qquad \textbf{(D) } 89 \qquad \textbf{(E) } 178$
2015 AMC 10, 15
The town of Hamlet has $3$ people for each horse, $4$ sheep for each cow, and $3$ ducks for each person. Which of the following could not possibly be the total number of people, horses, sheep, cows, and ducks in Hamlet?
$\textbf{(A) } 41
\qquad\textbf{(B) } 47
\qquad\textbf{(C) } 59
\qquad\textbf{(D) } 61
\qquad\textbf{(E) } 66
$