Found problems: 3632
2012 AMC 12/AHSME, 24
Let $\{a_k\}^{2011}_{k=1}$ be the sequence of real numbers defined by $$a_1=0.201, \quad a_2=(0.2011)^{a_1},\quad a_3=(0.20101)^{a_2},\quad a_4=(0.201011)^{a_3},$$ and more generally \[ a_k = \begin{cases}(0.\underbrace{20101\cdots0101}_{k+2 \ \text{digits}})^{a_{k-1}}, &\text {if } k \text { is odd,} \\ (0.\underbrace{20101\cdots01011}_{k+2 \ \text{digits}})^{a_{k-1}}, &\text {if } k \text { is even.}\end{cases} \]
Rearranging the numbers in the sequence $\{a_k\}^{2011}_{k=1}$ in decreasing order produces a new sequence $\{b_k\}^{2011}_{k=1}$. What is the sum of all the integers $k$, $1\le k \le 2011$, such that $a_k = b_k$?
$ \textbf{(A)}\ 671\qquad\textbf{(B)}\ 1006\qquad\textbf{(C)}\ 1341\qquad\textbf{(D)}\ 2011\qquad\textbf{(E)}\ 2012 $
1997 AMC 8, 8
Walter gets up at 6:30 a.m., catches the school bus at 7:30 a.m., has 6 classes that last 50 minutes each, has 30 minutes for lunch, and has 2 hours additional time at school. He takes the bus home and arrives at 4:00 p.m. How many minutes has he spent on the bus?
$\textbf{(A)}\ 30 \qquad \textbf{(B)}\ 60 \qquad \textbf{(C)}\ 75 \qquad \textbf{(D)}\ 90 \qquad \textbf{(E)}\ 120$
2013 Princeton University Math Competition, 5
Suppose $w,x,y,z$ satisfy \begin{align*}w+x+y+z&=25,\\wx+wy+wz+xy+xz+yz&=2y+2z+193\end{align*} The largest possible value of $w$ can be expressed in lowest terms as $w_1/w_2$ for some integers $w_1,w_2>0$. Find $w_1+w_2$.
2012 Putnam, 1
Let $d_1,d_2,\dots,d_{12}$ be real numbers in the open interval $(1,12).$ Show that there exist distinct indices $i,j,k$ such that $d_i,d_j,d_k$ are the side lengths of an acute triangle.
1986 AMC 12/AHSME, 29
Two of the altitudes of the scalene triangle $ABC$ have length $4$ and $12$. If the length of the third altitude is also an integer, what is the biggest it can be?
$ \textbf{(A)}\ 4\qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\ 6\qquad\textbf{(D)}\ 7\qquad\textbf{(E)}\ \text{none of these} $
2013 AMC 10, 7
A student must choose a program of four courses from a menu of courses consisting of English, Algebra, Geometry, History, Art, and Latin. This program must contain English and at least one mathematics course. In how many ways can this program be chosen?
$ \textbf{(A)}\ 6\qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ 9\qquad\textbf{(D)}\ 12\qquad\textbf{(E)}\ 16$
1997 AIME Problems, 12
The function $f$ defined by $\displaystyle f(x)= \frac{ax+b}{cx+d}$. where $a,b,c$ and $d$ are nonzero real numbers, has the properties $f(19)=19, f(97)=97$ and $f(f(x))=x$ for all values except $\displaystyle \frac{-d}{c}$. Find the unique number that is not in the range of $f$.
2024 AIME, 2
A list of positive integers has the following properties:
- The sum of the items in the list is $30$.
- The unique mode of the list is $9$.
- The median of the list is a positive integer that does not appear in the list itself.
Find the sum of the squares of all the items in the list.
2006 USAMO, 4
Find all positive integers $n$ such that there are $k \geq 2$ positive rational numbers $a_1, a_2, \ldots, a_k$ satisfying $a_1 + a_2 + \ldots + a_k = a_1 \cdot a_2 \cdots a_k = n.$
1976 AMC 12/AHSME, 2
For how many real numbers $x$ is $\sqrt{-(x+1)^2}$ a real number?
$\textbf{(A) }\text{none}\qquad\textbf{(B) }\text{one}\qquad\textbf{(C) }\text{two}\qquad\textbf{(D) }\text{a finite number greater than two}\qquad \textbf{(E) }\text{infinitely many}$
2010 AMC 10, 7
Crystal has a running course marked out for her daily run. She starts this run by heading due north for one mile. She then runs northeast for one mile, then southeast for one mile. The last portion of her run takes her on a straight line back to where she started. How far, in miles is this last portion of her run?
$ \textbf{(A)}\ 1 \qquad
\textbf{(B)}\ \sqrt2 \qquad
\textbf{(C)}\ \sqrt3 \qquad
\textbf{(D)}\ 2 \qquad
\textbf{(E)}\ 2\sqrt2$
2023 AIME, 4
The sum of all positive integers $m$ for which $\tfrac{13!}{m}$ is a perfect square can be written as $2^{a}3^{b}5^{c}7^{d}11^{e}13^{f}$, where $a, b, c, d, e,$ and $f$ are positive integers. Find $a+b+c+d+e+f$.
2014 AMC 10, 1
1. Leah has 13 coins, all of which are pennies and nickels. If she had one more nickel than she has now, then she would have the same number of pennies and nickels. In cents, how much are Leah's coins worth?
${ \textbf{(A)}\ \ 33\qquad\textbf{(B)}\ 35\qquad\textbf{(C)}\ 37\qquad\textbf{(D)}}\ 39\qquad\textbf{(E)}\ 41 $
1967 AMC 12/AHSME, 11
If the perimeter of rectangle $ABCD$ is $20$ inches, the least value of diagonal $\overline{AC}$, in inches, is:
$\textbf{(A)}\ 0\qquad
\textbf{(B)}\ \sqrt{50}\qquad
\textbf{(C)}\ 10\qquad
\textbf{(D)}\ \sqrt{200}\qquad
\textbf{(E)}\ \text{none of these}$
2022 AMC 12/AHSME, 16
Suppose $x$ and $y$ are positive real numbers such that
$x^y=2^{64}$ and $(\log_2{x})^{\log_2{y}}=2^{7}$.
What is the greatest possible value of $\log_2{y}$?
$\textbf{(A)}3~\textbf{(B)}4~\textbf{(C)}3+\sqrt{2}~\textbf{(D)}4+\sqrt{3}~\textbf{(E)}7$
1970 AMC 12/AHSME, 21
On an auto trip, the distance read from the instrument panel was $450$ miles. With snow tires on for the return trip over the same route, the reading was $440$ miles. Find, to the nearest hundredth of an inch, the increase in radius of the wheels if the original radius was $15$ inches.
$\textbf{(A) }.33\qquad\textbf{(B) }.34\qquad\textbf{(C) }.35\qquad\textbf{(D) }.38\qquad \textbf{(E) }.66$
1977 AMC 12/AHSME, 5
The set of all points $P$ such that the sum of the (undirected) distances from $P$ to two fixed points $A$ and $B$ equals the distance between $A$ and $B$ is
$\textbf{(A) }\text{the line segment from }A\text{ to }B\qquad$
$\textbf{(B) }\text{the line passing through }A\text{ and }B\qquad$
$\textbf{(C) }\text{the perpendicular bisector of the line segment from }A\text{ to }B\qquad$
$\textbf{(D) }\text{an elllipse having positive area}\qquad$
$\textbf{(E) }\text{a parabola}$
2003 USAMO, 5
Let $ a$, $ b$, $ c$ be positive real numbers. Prove that
\[ \dfrac{(2a \plus{} b \plus{} c)^2}{2a^2 \plus{} (b \plus{} c)^2} \plus{} \dfrac{(2b \plus{} c \plus{} a)^2}{2b^2 \plus{} (c \plus{} a)^2} \plus{} \dfrac{(2c \plus{} a \plus{} b)^2}{2c^2 \plus{} (a \plus{} b)^2} \le 8.
\]
1978 AMC 12/AHSME, 12
In $\triangle ADE$, $\measuredangle ADE=140^\circ$, points $B$ and $C$ lie on sides $AD$ and $AE$, respectively, and points $A,~B,~C,~D,~E$ are distinct.* If lengths $AB,~BC,~CD,$ and $DE$ are all equal, then the measure of $\measuredangle EAD$ is
$\textbf{(A) }5^\circ\qquad\textbf{(B) }6^\circ\qquad\textbf{(C) }7.5^\circ\qquad\textbf{(D) }8^\circ\qquad \textbf{(E) }10^\circ$
[size=50]* The specification that points $A,B,C,D,E$ be distinct was not included in the original statement of the problem. If $B=D$, then $C=E$ and $\measuredangle EAD=20^\circ$.[/size]
2009 AMC 12/AHSME, 22
A regular octahedron has side length $ 1$. A plane parallel to two of its opposite faces cuts the octahedron into the two congruent solids. The polygon formed by the intersection of the plane and the octahedron has area $ \frac {a\sqrt {b}}{c}$, where $ a$, $ b$, and $ c$ are positive integers, $ a$ and $ c$ are relatively prime, and $ b$ is not divisible by the square of any prime. What is $ a \plus{} b \plus{} c$?
$ \textbf{(A)}\ 10\qquad \textbf{(B)}\ 11\qquad \textbf{(C)}\ 12\qquad \textbf{(D)}\ 13\qquad \textbf{(E)}\ 14$
2014 AIME Problems, 3
A rectangle has sides of length $a$ and $36$. A hinge is installed at each vertex of the rectangle and at the midpoint of each side of length $36$. The sides of length $a$ can be pressed toward each other keeping those two sides parallel so the rectangle becomes a convex hexagon as shown. When the figure is a hexagon with the sides of length $a$ parallel and separated by a distance of $24,$ the hexagon has the same area as the original rectangle. Find $a^2$.
[asy]
pair A,B,C,D,E,F,R,S,T,X,Y,Z;
dotfactor = 2;
unitsize(.1cm);
A = (0,0);
B = (0,18);
C = (0,36);
// don't look here
D = (12*2.236, 36);
E = (12*2.236, 18);
F = (12*2.236, 0);
draw(A--B--C--D--E--F--cycle);
dot(" ",A,NW);
dot(" ",B,NW);
dot(" ",C,NW);
dot(" ",D,NW);
dot(" ",E,NW);
dot(" ",F,NW);
//don't look here
R = (12*2.236 +22,0);
S = (12*2.236 + 22 - 13.4164,12);
T = (12*2.236 + 22,24);
X = (12*4.472+ 22,24);
Y = (12*4.472+ 22 + 13.4164,12);
Z = (12*4.472+ 22,0);
draw(R--S--T--X--Y--Z--cycle);
dot(" ",R,NW);
dot(" ",S,NW);
dot(" ",T,NW);
dot(" ",X,NW);
dot(" ",Y,NW);
dot(" ",Z,NW);
// sqrt180 = 13.4164
// sqrt5 = 2.236
[/asy]
2017 AMC 12/AHSME, 4
Samia set off on her bicycle to visit her friend, traveling at an average speed of 17 kilometers per hour. When she had gone half the distance to her friend's house, a tire went flat, and she walked the rest of the way at 5 kilometers per hour. In all it took her 44 minutes to reach her friend's house. In kilometers rounded to the nearest tenth, how far did Samia walk?
$\textbf{(A)}\ 2.0 \qquad \textbf{(B)}\ 2.2\qquad \textbf{(C)}\ 2.8 \qquad \textbf{(D)}\ 3.4 \qquad \textbf{(E)}\ 4.4$
1992 AIME Problems, 15
Define a positive integer $ n$ to be a factorial tail if there is some positive integer $ m$ such that the decimal representation of $ m!$ ends with exactly $ n$ zeroes. How many positive integers less than $ 1992$ are not factorial tails?
2020 AMC 12/AHSME, 19
There exists a unique strictly increasing sequence of nonnegative integers $a_1 < a_2 < \dots < a_k$ such that \[\frac{2^{289}+1}{2^{17}+1} = 2^{a_1} + 2^{a_2} + \dots + 2^{a_k}.\] What is $k?$
$\textbf{(A) } 117 \qquad \textbf{(B) } 136 \qquad \textbf{(C) } 137 \qquad \textbf{(D) } 273 \qquad \textbf{(E) } 306$
1967 AMC 12/AHSME, 1
The three-digit number $2a3$ is added to the number $326$ to give the three-digit number $5b9$. If $5b9$ is divisible by 9, then $a+b$ equals
$ \text{(A)}\ 2\qquad\text{(B)}\ 4\qquad\text{(C)}\ 6\qquad\text{(D)}\ 8\qquad\text{(E)}\ 9$