Found problems: 3632
2007 AMC 10, 2
Define $ a@b \equal{} ab \minus{} b^{2}$ and $ a\#b \equal{} a \plus{} b \minus{} ab^{2}$. What is $ \frac {6@2}{6\#2}$?
$ \textbf{(A)}\ \minus{} \frac {1}{2}\qquad \textbf{(B)}\ \minus{} \frac {1}{4}\qquad \textbf{(C)}\ \frac {1}{8}\qquad \textbf{(D)}\ \frac {1}{4}\qquad \textbf{(E)}\ \frac {1}{2}$
2004 AMC 12/AHSME, 10
The sum of $ 49$ consecutive integers is $ 7^5$. What is their median?
$ \textbf{(A)}\ 7 \qquad \textbf{(B)}\ 7^2\qquad \textbf{(C)}\ 7^3\qquad \textbf{(D)}\ 7^4\qquad \textbf{(E)}\ 7^5$
1992 AMC 12/AHSME, 22
Ten points are selected on the positive x-axis, $X^{+}$, and fives points are selected on the positive y-axis, $Y^{+}$. The fifty segments connecting the ten selected points on $X^{+}$ to the five selected points on $Y^{+}$ are drawn. What is the maximum possible number of points of intersection of these fifty segments that could lie in the interior of the first quadrant?
$ \textbf{(A)}\ 250\qquad\textbf{(B)}\ 450\qquad\textbf{(C)}\ 500\qquad\textbf{(D)}\ 1250\qquad\textbf{(E)}\ 2500 $
1994 AMC 12/AHSME, 28
In the $xy$-plane, how many lines whose $x$-intercept is a positive prime number and whose $y$-intercept is a positive integer pass through the point $(4,3)$?
$ \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ 4$
2011 USAJMO, 4
A [i]word[/i] is defined as any finite string of letters. A word is a [i]palindrome[/i] if it reads the same backwards and forwards. Let a sequence of words $W_0, W_1, W_2,...$ be defined as follows: $W_0 = a, W_1 = b$, and for $n \ge 2$, $W_n$ is the word formed by writing $W_{n-2}$ followed by $W_{n-1}$. Prove that for any $n \ge 1$, the word formed by writing $W_1, W_2, W_3,..., W_n$ in succession is a palindrome.
2011 NIMO Summer Contest, 8
Triangle $ABC$ with $\measuredangle A = 90^\circ$ has incenter $I$. A circle passing through $A$ with center $I$ is drawn, intersecting $\overline{BC}$ at $E$ and $F$ such that $BE < BF$. If $\tfrac{BE}{EF} = \tfrac{2}{3}$, then $\tfrac{CF}{FE} = \tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
[i]Proposed by Lewis Chen
[/i]
2010 AIME Problems, 7
Let $ P(z) \equal{} z^3 \plus{} az^2 \plus{} bz \plus{} c$, where $ a$, $ b$, and $ c$ are real. There exists a complex number $ w$ such that the three roots of $ P(z)$ are $ w \plus{} 3i$, $ w \plus{} 9i$, and $ 2w \minus{} 4$, where $ i^2 \equal{} \minus{} 1$. Find $ |a \plus{} b \plus{} c|$.
2012 AMC 10, 8
What is the sum of all integer solutions to $1<(x-2)^2<25$?
$ \textbf{(A)}\ 10\qquad\textbf{(B)}\ 12\qquad\textbf{(C)}\ 15\qquad\textbf{(D)}\ 19\qquad\textbf{(E)}\ 25 $
2016 AMC 10, 8
Trickster Rabbit agrees with Foolish Fox to double Fox's money every time Fox crosses the bridge by Rabbit's house, as long as Fox pays $40$ coins in toll to Rabbit after each crossing. The payment is made after the doubling, Fox is excited about his good fortune until he discovers that all his money is gone after crossing the bridge three times. How many coins did Fox have at the beginning?
$\textbf{(A)}\ 20 \qquad\textbf{(B)}\ 30\qquad\textbf{(C)}\ 35\qquad\textbf{(D)}\ 40\qquad\textbf{(E)}\ 45$
2017 AMC 8, 23
Each day for four days, Linda traveled for one hour at a speed that resulted in her traveling one mile in an integer number of minutes. Each day after the first, her speed decreased so that the number of minutes to travel one mile increased by 5 minutes over the preceding day. Each of the four days, her distance traveled was also an integer number of miles. What was the total number of miles for the four trips?
$\textbf{(A) }10\qquad\textbf{(B) }15\qquad\textbf{(C) }25\qquad\textbf{(D) }50\qquad\textbf{(E) }82$
2015 AMC 10, 13
The line $12x+5y=60$ forms a triangle with the coordinate axes. What is the sum of the lengths of the altitudes of this triangle?
$\textbf{(A) } 20
\qquad\textbf{(B) } \dfrac{360}{17}
\qquad\textbf{(C) } \dfrac{107}{5}
\qquad\textbf{(D) } \dfrac{43}{2}
\qquad\textbf{(E) } \dfrac{281}{13}
$
2024 AMC 10, 12
A group of $100$ students from different countries meet at a mathematics competition. Each student speaks the same number of languages, and, for every pair of students $A$ and $B$, student $A$ speaks some language that student $B$ does not speak, and student $B$ speaks some language that student $A$ does not speak. What is the least possible total number of languages spoken by all the students?
$
\textbf{(A) }9 \qquad
\textbf{(B) }10 \qquad
\textbf{(C) }12 \qquad
\textbf{(D) }51 \qquad
\textbf{(E) }100 \qquad
$
2014 AMC 10, 3
Bridget bakes $48$ loaves of bread for her bakery. She sells half of them in the morning for $\$2.50$ each. In the afternoon she sells two thirds of what she has left, and because they are not fresh, she charges only half price. In the late afternoon she sells the remaining loaves at a dollar each. Each loaf costs $\$0.75$ for her to make. In dollars, what is her profit for the day?
${ \textbf{(A)}\ 24\qquad\textbf{(B)}\ 36\qquad\textbf{(C)}\ 44\qquad\textbf{(D)}}\ 48\qquad\textbf{(E)}\ 52$
2014 USAMO, 2
Let $\mathbb{Z}$ be the set of integers. Find all functions $f : \mathbb{Z} \rightarrow \mathbb{Z}$ such that \[xf(2f(y)-x)+y^2f(2x-f(y))=\frac{f(x)^2}{x}+f(yf(y))\] for all $x, y \in \mathbb{Z}$ with $x \neq 0$.
2019 AIME Problems, 3
Find the number of $7$-tuples of positive integers $(a,b,c,d,e,f,g)$ that satisfy the following systems of equations:
\begin{align*}
abc&=70,\\
cde&=71,\\
efg&=72.
\end{align*}
2021 USAMO, 6
Let $ABCDEF$ be a convex hexagon satisfying $\overline{AB} \parallel \overline{DE}$, $\overline{BC} \parallel \overline{EF}$, $\overline{CD} \parallel \overline{FA}$, and
\[
AB \cdot DE = BC \cdot EF = CD \cdot FA.
\]
Let $X$, $Y$, and $Z$ be the midpoints of $\overline{AD}$, $\overline{BE}$, and $\overline{CF}$. Prove that the circumcenter of $\triangle ACE$, the circumcenter of $\triangle BDF$, and the orthocenter of $\triangle XYZ$ are collinear.
2019 AMC 12/AHSME, 22
Define a sequence recursively by $x_0=5$ and
\[x_{n+1}=\frac{x_n^2+5x_n+4}{x_n+6}\]
for all nonnegative integers $n.$ Let $m$ be the least positive integer such that
\[x_m\leq 4+\frac{1}{2^{20}}.\] In which of the following intervals does $m$ lie?
$\textbf{(A) } [9,26] \qquad\textbf{(B) } [27,80] \qquad\textbf{(C) } [81,242]\qquad\textbf{(D) } [243,728] \qquad\textbf{(E) } [729,\infty]$
2021 AIME Problems, 3
Find the number of positive integers less than $1000$ that can be expressed as the difference of two integral powers of $2.$
2013 AMC 10, 10
A basketball team's players were successful on $50\%$ of their two-point shots and $40\%$ of their three-point shots, which resulted in $54$ points. They attempted $50\%$ more two-point shots than three-point shots. How many three-point shots did they attempt?
$ \textbf{(A) }10\qquad\textbf{(B) }15\qquad\textbf{(C) }20\qquad\textbf{(D) }25\qquad\textbf{(E) }30 $
2018 AMC 12/AHSME, 11
A closed box with a square base is to be wrapped with a square sheet of wrapping paper. The box is centered on the wrapping paper with the vertices of the base lying on the midlines of the square sheet of paper, as shown in the figure on the left. The four corners of the wrapping paper are to be folded up over the sides and brought together to meet at the center of the top of the box, point $A$ in the figure on the right. The box has base length $w$ and height $h$. What is the area of the sheet of wrapping paper?
[asy]size(270pt);
defaultpen(fontsize(10pt));
filldraw(((3,3)--(-3,3)--(-3,-3)--(3,-3)--cycle),lightgrey);
dot((-3,3));
label("$A$",(-3,3),NW);
draw((1,3)--(-3,-1),dashed+linewidth(.5));
draw((-1,3)--(3,-1),dashed+linewidth(.5));
draw((-1,-3)--(3,1),dashed+linewidth(.5));
draw((1,-3)--(-3,1),dashed+linewidth(.5));
draw((0,2)--(2,0)--(0,-2)--(-2,0)--cycle,linewidth(.5));
draw((0,3)--(0,-3),linetype("2.5 2.5")+linewidth(.5));
draw((3,0)--(-3,0),linetype("2.5 2.5")+linewidth(.5));
label('$w$',(-1,-1),SW);
label('$w$',(1,-1),SE);
draw((4.5,0)--(6.5,2)--(8.5,0)--(6.5,-2)--cycle);
draw((4.5,0)--(8.5,0));
draw((6.5,2)--(6.5,-2));
label("$A$",(6.5,0),NW);
dot((6.5,0));
[/asy]
$\textbf{(A) } 2(w+h)^2 \qquad \textbf{(B) } \frac{(w+h)^2}2 \qquad \textbf{(C) } 2w^2+4wh \qquad \textbf{(D) } 2w^2 \qquad \textbf{(E) } w^2h $
2022 AMC 10, 11
Ted mistakenly wrote $2^m \cdot \sqrt{\frac{1}{4096}}$ as $2\cdot \sqrt[m]{\frac{1}{4096}}$. What is the sum of all real numbers $m$ for which these two expressions have the same value?
$\textbf{(A) }5\qquad\textbf{(B) }6\qquad\textbf{(C) }7\qquad\textbf{(D) }8\qquad\textbf{(E) }9$
2023 AMC 10, 4
Jackson's paintbrush makes a narrow strip that is $6.5$ mm wide. Jackson has enough paint to make a strip of 25 meters. How much can he paint, in $\text{cm}^2$?
$\textbf{(A) }162{,}500\qquad\textbf{(B) }162.5\qquad\textbf{(C) }1{,}625\qquad\textbf{(D) }1{,}625{,}000\qquad\textbf{(E) }16{,}250$
1971 AMC 12/AHSME, 33
If $P$ is the product of $n$ quantities in Geometric Progression, $S$ their sum, and $S'$ the sum of their reciprocals, then $P$ in terms of $S$, $S'$, and $n$ is
$\textbf{(A) }(SS')^{\frac{1}{2}n}\qquad\textbf{(B) }(S/S')^{\frac{1}{2}n}\qquad\textbf{(C) }(SS')^{n-2}\qquad\textbf{(D) }(S/S')^n\qquad \textbf{(E) }(S/S')^{\frac{1}{2}(n-1)}$
1989 AIME Problems, 5
When a certain biased coin is flipped five times, the probability of getting heads exactly once is not equal to $0$ and is the same as that of getting heads exactly twice. Let $\frac ij$, in lowest terms, be the probability that the coin comes up heads in exactly $3$ out of $5$ flips. Find $i+j$.
2017 AMC 10, 1
What is the value of $2(2(2(2(2(2+1)+1)+1)+1)+1)+1$?
$\textbf{(A) } 70 \qquad
\textbf{(B) } 97 \qquad
\textbf{(C) } 127 \qquad
\textbf{(D) } 159 \qquad
\textbf{(E) } 729 $