Found problems: 3632
2024 AMC 10, 1
What is the value of $9901\cdot101-99\cdot10101?$
$\textbf{(A) }2\qquad\textbf{(B) }20\qquad\textbf{(C) }21\qquad\textbf{(D) }200\qquad\textbf{(E) }2020$
1999 Niels Henrik Abels Math Contest (Norwegian Math Olympiad) Round 2, 4
Two semicircles are tangent to middle circle, and both semicircles and middle circle are tangent to the horizontal line as shown. If $ PQ \equal{} QR \equal{} RS \equal{} 24,$ then find the length of radius $ r$.
[img]http://i250.photobucket.com/albums/gg265/geometry101/NielsHenrikAbel1999Number3.jpg[/img]
2010 AMC 10, 21
A palindrome between $ 1000$ and $ 10,000$ is chosen at random. What is the probability that it is divisible by $ 7?$
$ \textbf{(A)}\ \dfrac{1}{10} \qquad \textbf{(B)}\ \dfrac{1}{9} \qquad \textbf{(C)}\ \dfrac{1}{7} \qquad \textbf{(D)}\ \dfrac{1}{6}\qquad \textbf{(E)}\ \dfrac{1}{5}$
1985 ITAMO, 9
In a circle, parallel chords of lengths 2, 3, and 4 determine central angles of $\alpha$, $\beta$, and $\alpha + \beta$ radians, respectively, where $\alpha + \beta < \pi$. If $\cos \alpha$, which is a positive rational number, is expressed as a fraction in lowest terms, what is the sum of its numerator and denominator?
2013 AIME Problems, 9
A $7 \times 1$ board is completely covered by $m \times 1$ tiles without overlap; each tile may cover any number of consecutive squares, and each tile lies completely on the board. Each tile is either red, blue, or green. Let $N$ be the number of tilings of the $7 \times 1$ board in which all three colors are used at least once. For example, a $1 \times 1$ red tile followed by a $2 \times 1$ green tile, a $1 \times 1$ green tile, a $2 \times 1$ blue tile, and a $1 \times 1$ green tile is a valid tiling. Note that if the $2 \times 1$ blue tile is replaced by two $1 \times 1$ blue tiles, this results in a different tiling. Find the remainder when $N$ is divided by $1000$.
2011 AMC 12/AHSME, 22
Let $T_1$ be a triangle with sides $2011, 2012,$ and $2013$. For $n \ge 1$, if $T_n=\triangle ABC$ and $D,E,$ and $F$ are the points of tangency of the incircle of $\triangle ABC$ to the sides $AB,BC$ and $AC$, respectively, then $T_{n+1}$ is a triangle with side lengths $AD,BE,$ and $CF$, if it exists. What is the perimeter of the last triangle in the sequence $(T_n)$?
$ \textbf{(A)}\ \frac{1509}{8} \qquad
\textbf{(B)}\ \frac{1509}{32} \qquad
\textbf{(C)}\ \frac{1509}{64} \qquad
\textbf{(D)}\ \frac{1509}{128} \qquad
\textbf{(E)}\ \frac{1509}{256} $
1997 AMC 8, 22
A two-inch cube $(2\times 2\times 2)$ of silver weighs 3 pounds and is worth \$200. How much is a three-inch cube of silver worth?
$\textbf{(A)}\ 300\text{ dollars} \qquad \textbf{(B)}\ 375\text{ dollars} \qquad \textbf{(C)}\ 450\text{ dollars} \qquad \textbf{(D)}\ 560\text{ dollars} \qquad \textbf{(E)}\ 675\text{ dollars}$
2006 AMC 12/AHSME, 18
An object in the plane moves from one lattice point to another. At each step, the object may move one unit to the right, one unit to the left, one unit up, or one unit down. If the object starts at the origin and takes a ten-step path, how many different points could be the final point?
$ \textbf{(A) } 120 \qquad \textbf{(B) } 121 \qquad \textbf{(C) } 221 \qquad \textbf{(D) } 230 \qquad \textbf{(E) } 231$
1989 AMC 12/AHSME, 28
Find the sum of the roots of $\tan^2x-9\tan x+1=0$ that are between $x=0$ and $x=2\pi$ radians.
$ \textbf{(A)}\ \frac{\pi}{2} \qquad\textbf{(B)}\ \pi \qquad\textbf{(C)}\ \frac{3\pi}{2} \qquad\textbf{(D)}\ 3\pi \qquad\textbf{(E)}\ 4\pi $
1998 AMC 8, 15
Problems $15, 16$, and $17$ all refer to the following:
In the very center of the Irenic Sea lie the beautiful Nisos Isles. In $1998$ the number of people on these islands is only 200, but the population triples every $25$ years. Queen Irene has decreed that there must be at least $1.5$ square miles for every person living in the Isles. The total area of the Nisos Isles is $24,900$ square miles.
15. Estimate the population of Nisos in the year $2050$.
$ \text{(A)}\ 600\qquad\text{(B)}\ 800\qquad\text{(C)}\ 1000\qquad\text{(D)}\ 2000\qquad\text{(E)}\ 3000 $
2017 AMC 12/AHSME, 5
The data set $[6, 19, 33, 33, 39, 41, 41, 43, 51, 57]$ has median $Q_2=40$, first quartile $Q_1=33$, and third quartile $Q_3=43$. An outlier in a data set is a value that is more than $1.5$ times the interquartile range below the first quartile ($Q_1$) or more than $1.5$ times the interquartile range above the third quartile ($Q_3$), where the interquartile range is defined as $Q_3-Q_1$. How many outliers does this data set have?
$\textbf{(A)}\ 0\qquad\textbf{(B)}\ 1\qquad\textbf{(C)}\ 2\qquad\textbf{(D)}\ 3\qquad\textbf{(E)}\ 4$
1978 AMC 12/AHSME, 29
Sides $AB,~ BC, ~CD$ and $DA$, respectively, of convex quadrilateral $ABCD$ are extended past $B,~ C ,~ D$ and $A$ to points $B',~C',~ D'$ and $A'$. Also, $AB = BB' = 6,~ BC = CC' = 7, ~CD = DD' = 8$ and $DA = AA' = 9$; and the area of $ABCD$ is 10. The area of $A 'B 'C'D'$ is
$\textbf{(A) }20\qquad\textbf{(B) }40\qquad\textbf{(C) }45\qquad\textbf{(D) }50\qquad \textbf{(E) }60$
2008 AMC 10, 13
For each positive integer $ n$, the mean of the first $ n$ terms of a sequence is $ n$. What is the $ 2008$th term of the sequence?
$ \textbf{(A)}\ 2008 \qquad
\textbf{(B)}\ 4015 \qquad
\textbf{(C)}\ 4016 \qquad
\textbf{(D)}\ 4,030,056 \qquad
\textbf{(E)}\ 4,032,064$
2009 AMC 12/AHSME, 8
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$
1971 AMC 12/AHSME, 10
Each of a group of $50$ girls is blonde or brunette and is blue eyed of brown eyed. If $14$ are blue-eyed blondes, $31$ are brunettes, and $18$ are brown-eyed, then the number of brown-eyed brunettes is
$\textbf{(A) }5\qquad\textbf{(B) }7\qquad\textbf{(C) }9\qquad\textbf{(D) }11\qquad \textbf{(E) }13$
1961 AMC 12/AHSME, 33
The number of solutions of $2^{2x}-3^{2y}=55$, in which $x$ and $y$ are integers, is:
${{ \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1\qquad\textbf{(C)}\ 2\qquad\textbf{(D)}\ 3}\qquad\textbf{(E)}\ \text{More than three, but finite} } $
1983 AMC 12/AHSME, 14
The units digit of $3^{1001}7^{1002}13^{1003}$ is
$ \textbf{(A)}\ 1\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 5\qquad\textbf{(D)}\ 7\qquad\textbf{(E)}\ 9 $
1977 AMC 12/AHSME, 22
If $f(x)$ is a real valued function of the real variable $x$, and $f(x)$ is not identically zero, and for all $a$ and $b$ \[f(a+b)+f(a-b)=2f(a)+2f(b),\] then for all $x$ and $y$
$\textbf{(A) }f(0)=1\qquad\textbf{(B) }f(-x)=-f(x)\qquad$
$\textbf{(C) }f(-x)=f(x)\qquad\textbf{(D) }f(x+y)=f(x)+f(y)\qquad$
$\textbf{(E) }\text{there is a positive real number }T\text{ such that }f(x+T)=f(x)$
2016 AMC 12/AHSME, 22
How many ordered triples $(x, y, z)$ of positive integers satisfy $\text{lcm}(x, y) = 72$, $\text{lcm}(x, z)= 600$, and $\text{lcm}(y, z) = 900$?
$\textbf{(A) } 15 \qquad\textbf{(B) } 16 \qquad\textbf{(C) } 24 \qquad\textbf{(D) } 27 \qquad\textbf{(E) } 64$
2021 AMC 12/AHSME Fall, 25
Let $m\ge 5$ be an odd integer, and let $D(m)$ denote the number of quadruples $\big(a_1, a_2, a_3, a_4\big)$ of distinct integers with $1\le a_i \le m$ for all $i$ such that $m$ divides $a_1+a_2+a_3+a_4$. There is a polynomial
$$q(x) = c_3x^3+c_2x^2+c_1x+c_0$$such that $D(m) = q(m)$ for all odd integers $m\ge 5$. What is $c_1?$
$(\textbf{A})\: {-}6\qquad(\textbf{B}) \: {-}1\qquad(\textbf{C}) \: 4\qquad(\textbf{D}) \: 6\qquad(\textbf{E}) \: 11$
2022 AMC 10, 17
How many three-digit positive integers $\underline{a}$ $\underline{b}$ $\underline{c}$ are there whose nonzero digits $a$, $b$, and $c$ satisfy
$$0.\overline{\underline{a}~\underline{b}~\underline{c}} = \frac{1}{3} (0.\overline{a} + 0.\overline{b} + 0.\overline{c})?$$
(The bar indicates repetition, thus $0.\overline{\underline{a}~\underline{b}~\underline{c}}$ in the infinite repeating decimal $0.\underline{a}~\underline{b}~\underline{c}~\underline{a}~\underline{b}~\underline{c}~\cdots$)
$\textbf{(A) }9\qquad\textbf{(B) }10\qquad\textbf{(C) }11\qquad\textbf{(D) }13\qquad\textbf{(E) }14$
2008 AMC 10, 3
Assume that $ x$ is a positive real number. Which is equivalent to $ \sqrt[3]{x\sqrt{x}}$?
$ \textbf{(A)}\ x^{1/6} \qquad
\textbf{(B)}\ x^{1/4} \qquad
\textbf{(C)}\ x^{3/8} \qquad
\textbf{(D)}\ x^{1/2} \qquad
\textbf{(E)}\ x$
1960 AMC 12/AHSME, 23
The radius $R$ of a cylindrical box is $8$ inches, the height $H$ is $3$ inches. The volume $V = \pi R^2H$ is to be increased by the same fixed positive amount when $R$ is increased by $x$ inches as when $H$ is increased by $x$ inches. This condition is satisfied by:
$ \textbf{(A)}\ \text{no real value of} \text{ } x\qquad$
$\textbf{(B)}\ \text{one integral value of} \text{ } x\qquad$
$\textbf{(C)}\ \text{one rational, but not integral, value of} \text{ } x\qquad$
$\textbf{(D)}\ \text{one irrational value of} \text{ } x\qquad$
$\textbf{(E)}\ \text{two real values of} \text{ } x $
1976 AMC 12/AHSME, 9
In triangle $ABC$, $D$ is the midpoint of $AB$; $E$ is the midpoint of $DB$; and $F$ is the midpoint of $BC$. If the area of $\triangle ABC$ is $96$, then the area of $\triangle AEF$ is
$\textbf{(A) }16\qquad\textbf{(B) }24\qquad\textbf{(C) }32\qquad\textbf{(D) }36\qquad \textbf{(E) }48$
1988 AMC 12/AHSME, 15
If $a$ and $b$ are integers such that $x^2 - x - 1$ is a factor of $ax^3 + bx^2 + 1$, then $b$ is
$ \textbf{(A)}\ -2\qquad\textbf{(B)}\ -1\qquad\textbf{(C)}\ 0\qquad\textbf{(D)}\ 1\qquad\textbf{(E)}\ 2 $