Found problems: 3632
2007 AIME Problems, 7
Given a real number $x,$ let $\lfloor x \rfloor$ denote the greatest integer less than or equal to $x.$ For a certain integer $k,$ there are exactly $70$ positive integers $n_{1}, n_{2}, \ldots, n_{70}$ such that $k=\lfloor\sqrt[3]{n_{1}}\rfloor = \lfloor\sqrt[3]{n_{2}}\rfloor = \cdots = \lfloor\sqrt[3]{n_{70}}\rfloor$ and $k$ divides $n_{i}$ for all $i$ such that $1 \leq i \leq 70.$
Find the maximum value of $\frac{n_{i}}{k}$ for $1\leq i \leq 70.$
2008 AMC 12/AHSME, 24
Let $ A_0\equal{}(0,0)$. Distinct points $ A_1,A_2,\ldots$ lie on the $ x$-axis, and distinct points $ B_1,B_2,\ldots$ lie on the graph of $ y\equal{}\sqrt{x}$. For every positive integer $ n$, $ A_{n\minus{}1}B_nA_n$ is an equilateral triangle. What is the least $ n$ for which the length $ A_0A_n\ge100$?
$ \textbf{(A)}\ 13\qquad
\textbf{(B)}\ 15\qquad
\textbf{(C)}\ 17\qquad
\textbf{(D)}\ 19\qquad
\textbf{(E)}\ 21$
2011 AMC 12/AHSME, 5
Last summer $30\%$ of the birds living on Town Lake were geese, $25\%$ were swans, $10\%$ were herons, and $35\%$ were ducks. What percent of the birds that were not swans were geese?
$ \textbf{(A)}\ 20 \qquad
\textbf{(B)}\ 30 \qquad
\textbf{(C)}\ 40 \qquad
\textbf{(D)}\ 50 \qquad
\textbf{(E)}\ 60
$
2012 AMC 12/AHSME, 2
A circle of radius $5$ is inscribed in a rectangle as shown. The ratio of the the length of the rectangle to its width is $2\ :\ 1$. What is the area of the rectangle?
[asy]
draw((0,0)--(0,10)--(20,10)--(20,0)--cycle);
draw(circle((10,5),5));
[/asy]
$ \textbf{(A)}\ 50\qquad\textbf{(B)}\ 100\qquad\textbf{(C)}\ 125\qquad\textbf{(D)}\ 150\qquad\textbf{(E)}\ 200 $
2025 USAMO, 4
Let $H$ be the orthocenter of acute triangle $ABC$, let $F$ be the foot of the altitude from $C$ to $AB$, and let $P$ be the reflection of $H$ across $BC$. Suppose that the circumcircle of triangle $AFP$ intersects line $BC$ at two distinct points $X$ and $Y$. Prove that $C$ is the midpoint of $XY$.
2009 AMC 12/AHSME, 25
The first two terms of a sequence are $ a_1 \equal{} 1$ and $ a_2 \equal{} \frac {1}{\sqrt3}$. For $ n\ge1$,
\[ a_{n \plus{} 2} \equal{} \frac {a_n \plus{} a_{n \plus{} 1}}{1 \minus{} a_na_{n \plus{} 1}}.
\]What is $ |a_{2009}|$?
$ \textbf{(A)}\ 0\qquad \textbf{(B)}\ 2 \minus{} \sqrt3\qquad \textbf{(C)}\ \frac {1}{\sqrt3}\qquad \textbf{(D)}\ 1\qquad \textbf{(E)}\ 2 \plus{} \sqrt3$
2011 AMC 10, 16
Which of the following is equal to $\sqrt{9-6\sqrt{2}}+\sqrt{9+6\sqrt{2}}$?
$\textbf{(A)}\,3\sqrt2 \qquad\textbf{(B)}\,2\sqrt6 \qquad\textbf{(C)}\,\frac{7\sqrt2}{2} \qquad\textbf{(D)}\,3\sqrt3 \qquad\textbf{(E)}\,6$
2000 AMC 10, 12
Figures $ 0$, $ 1$, $ 2$, and $ 3$ consist of $ 1$, $ 5$, $ 13$, and $ 25$ nonoverlapping squares, respectively. If the pattern were continued, how many nonoverlapping squares would there be in figure $ 100$?
[asy]
unitsize(8);
draw((0,0)--(1,0)--(1,1)--(0,1)--cycle);
draw((9,0)--(10,0)--(10,3)--(9,3)--cycle);
draw((8,1)--(11,1)--(11,2)--(8,2)--cycle);
draw((19,0)--(20,0)--(20,5)--(19,5)--cycle);
draw((18,1)--(21,1)--(21,4)--(18,4)--cycle);
draw((17,2)--(22,2)--(22,3)--(17,3)--cycle);
draw((32,0)--(33,0)--(33,7)--(32,7)--cycle);
draw((29,3)--(36,3)--(36,4)--(29,4)--cycle);
draw((31,1)--(34,1)--(34,6)--(31,6)--cycle);
draw((30,2)--(35,2)--(35,5)--(30,5)--cycle);
label("Figure",(0.5,-1),S);
label("$0$",(0.5,-2.5),S);
label("Figure",(9.5,-1),S);
label("$1$",(9.5,-2.5),S);
label("Figure",(19.5,-1),S);
label("$2$",(19.5,-2.5),S);
label("Figure",(32.5,-1),S);
label("$3$",(32.5,-2.5),S);[/asy]$ \textbf{(A)}\ 10401 \qquad \textbf{(B)}\ 19801 \qquad \textbf{(C)}\ 20201 \qquad \textbf{(D)}\ 39801 \qquad \textbf{(E)}\ 40801$
2013 AIME Problems, 4
In the array of $13$ squares shown below, $8$ squares are colored red, and the remaining $5$ squares are colored blue. If one of all possible such colorings is chosen at random, the probability that the chosen colored array appears the same when rotated $90^{\circ}$ around the central square is $\tfrac{1}{n}$, where $n$ is a positive integer. Find $n$.
[asy]
draw((0,0)--(1,0)--(1,1)--(0,1)--(0,0));
draw((2,0)--(2,2)--(3,2)--(3,0)--(3,1)--(2,1)--(4,1)--(4,0)--(2,0));
draw((1,2)--(1,4)--(0,4)--(0,2)--(0,3)--(1,3)--(-1,3)--(-1,2)--(1,2));
draw((-1,1)--(-3,1)--(-3,0)--(-1,0)--(-2,0)--(-2,1)--(-2,-1)--(-1,-1)--(-1,1));
draw((0,-1)--(0,-3)--(1,-3)--(1,-1)--(1,-2)--(0,-2)--(2,-2)--(2,-1)--(0,-1));
size(100);
[/asy]
2011 AMC 10, 1
A cell phone plan costs $\$20$ each month, plus $5$¢ per text message sent, plus 10¢ for each minute used over $30$ hours. In January Michelle sent $100$ text messages and talked for $30.5$ hours. How much did she have to pay?
$ \textbf{(A)}\$ 24.00\qquad\textbf{(B)}\$ 24.50\qquad\textbf{(C)}\$25.50\qquad\textbf{(D)}\$28.00\qquad\textbf{(E)}\$30.00 $
2022 AMC 12/AHSME, 21
Let $S$ be the set of circles in the coordinate plane that are tangent to each of the three circles with equations $x^{2}+y^{2}=4$, $x^{2}+y^{2}=64$, and $(x-5)^{2}+y^{2}=3$. What is the sum of the areas of all circles in $S$?
$\textbf{(A)}~48\pi\qquad\textbf{(B)}~68\pi\qquad\textbf{(C)}~96\pi\qquad\textbf{(D)}~102\pi\qquad\textbf{(E)}~136\pi\qquad$
2024 AMC 12/AHSME, 11
Let $x_{n} = \sin^2(n^\circ)$. What is the mean of $x_{1}, x_{2}, x_{3}, \cdots, x_{90}$?
$
\textbf{(A) }\frac{11}{45} \qquad
\textbf{(B) }\frac{22}{45} \qquad
\textbf{(C) }\frac{89}{180} \qquad
\textbf{(D) }\frac{1}{2} \qquad
\textbf{(E) }\frac{91}{180} \qquad
$
2023 USAMO, 5
Let $n\geq3$ be an integer. We say that an arrangement of the numbers $1$, $2$, $\dots$, $n^2$ in a $n \times n$ table is [i]row-valid[/i] if the numbers in each row can be permuted to form an arithmetic progression, and [i]column-valid[/i] if the numbers in each column can be permuted to form an arithmetic progression. For what values of $n$ is it possible to transform any row-valid arrangement into a column-valid arrangement by permuting the numbers in each row?
2014 Contests, 2
Roy's cat eats $\frac{1}{3}$ of a can of cat food every morning and $\frac{1}{4}$ of a can of cat food every evening. Before feeding his cat on Monday morning, Roy opened a box containing $6$ cans of cat food. On what day of the week did the cat finish eating all the cat food in the box?
${ \textbf{(A)}\ \text{Tuesday}\qquad\textbf{(B)}\ \text{Wednesday}\qquad\textbf{(C)}\ \text{Thursday}\qquad\textbf{(D)}}\ \text{Friday}\qquad\textbf{(E)}\ \text{Saturday}$
1970 AMC 12/AHSME, 25
For every real number $x$, let $[x]$ be the greatest integer less than or equal to $x$. If the postal rate for first class mail is six cents for every ounce or portion thereof, then the cost in cents of first-class postage on a letter weighing $W$ ounces is always
$\textbf{(A) }6W\qquad\textbf{(B) }6[W]\qquad\textbf{(C) }6([W]-1)\qquad\textbf{(D) }6([W]+1)\qquad \textbf{(E) }-6[-W]$
2021 AMC 10 Fall, 14
How many ordered pairs $(x,y)$ of real numbers satisfy the following system of equations?
\begin{align*}
x^2+3y&=9\\
(|x|+|y|-4)^2&=1\\
\end{align*}
$\textbf{(A)}\: 1\qquad\textbf{(B)} \: 2\qquad\textbf{(C)} \: 3\qquad\textbf{(D)} \: 5\qquad\textbf{(E)} \: 7$
1998 AMC 12/AHSME, 23
The graphs of $ x^2 \plus{} y^2 \equal{} 4 \plus{} 12x \plus{} 6y$ and $ x^2 \plus{} y^2 \equal{} k \plus{} 4x \plus{} 12y$ intersect when $ k$ satisfies $ a \leq k \leq b$, and for no other values of $ k$. Find $ b \minus{} a$.
$ \textbf{(A)}\ 5\qquad
\textbf{(B)}\ 68\qquad
\textbf{(C)}\ 104\qquad
\textbf{(D)}\ 140\qquad
\textbf{(E)}\ 144$
1986 AMC 12/AHSME, 13
A parabola $y = ax^{2} + bx + c$ has vertex $(4,2)$. If $(2,0)$ is on the parabola, then $abc$ equals
$ \textbf{(A)}\ -12\qquad\textbf{(B)}\ -6\qquad\textbf{(C)}\ 0\qquad\textbf{(D)}\ 6\qquad\textbf{(E)}\ 12$
1958 AMC 12/AHSME, 25
If $ \log_{k}{x}\cdot \log_{5}{k} \equal{} 3$, then $ x$ equals:
$ \textbf{(A)}\ k^6\qquad
\textbf{(B)}\ 5k^3\qquad
\textbf{(C)}\ k^3\qquad
\textbf{(D)}\ 243\qquad
\textbf{(E)}\ 125$
2013 AMC 10, 15
A wire is cut into two pieces, one of length $a$ and the other of length $b$. The piece of length $a$ is bent to form an equilateral triangle, and the piece of length $b$ is bent to form a regular hexagon. The triangle and the hexagon have equal area. What is $\frac{a}{b}$?
${ \textbf{(A)}\ 1 \qquad\textbf{(B)}\ \frac{\sqrt{6}}{2} \qquad\textbf{(C)}\ \sqrt{3}\qquad\textbf{(D}}\ 2 \qquad\textbf{(E)}\ \frac{3\sqrt{2}}{2} $
1987 AMC 12/AHSME, 23
If $p$ is a prime and both roots of $x^2+px-444p=0$ are integers, then
$ \textbf{(A)}\ 1<p\le 11 \qquad\textbf{(B)}\ 11<p \le 21 \qquad\textbf{(C)}\ 21< p \le 31 \\ \qquad\textbf{(D)}\ 31< p \le 41 \qquad\textbf{(E)}\ 41< p \le 51 $
2000 AIME Problems, 9
The system of equations
\begin{eqnarray*}\log_{10}(2000xy) - (\log_{10}x)(\log_{10}y) & = & 4 \\
\log_{10}(2yz) - (\log_{10}y)(\log_{10}z) & = & 1 \\
\log_{10}(zx) - (\log_{10}z)(\log_{10}x) & = & 0 \\
\end{eqnarray*}
has two solutions $ (x_{1},y_{1},z_{1})$ and $ (x_{2},y_{2},z_{2}).$ Find $ y_{1} + y_{2}.$
2012 AMC 8, 18
What is the smallest positive integer that is neither prime nor square and that has no prime factor less than 50?
$\textbf{(A)}\hspace{.05in}3127 \qquad \textbf{(B)}\hspace{.05in}3133 \qquad \textbf{(C)}\hspace{.05in}3137 \qquad \textbf{(D)}\hspace{.05in}3139 \qquad \textbf{(E)}\hspace{.05in}3149 $
2013 AIME Problems, 11
Let $A = \left\{ 1,2,3,4,5,6,7 \right\}$ and let $N$ be the number of functions $f$ from set $A$ to set $A$ such that $f(f(x))$ is a constant function. Find the remainder when $N$ is divided by $1000$.
2010 AMC 12/AHSME, 11
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}$