Found problems: 3632
2009 AMC 10, 25
For $ k>0$, let $ I_k\equal{}10\ldots 064$, where there are $ k$ zeros between the $ 1$ and the $ 6$. Let $ N(k)$ be the number of factors of $ 2$ in the prime factorization of $ I_k$. What is the maximum value of $ N(k)$?
$ \textbf{(A)}\ 6\qquad \textbf{(B)}\ 7\qquad \textbf{(C)}\ 8\qquad \textbf{(D)}\ 9\qquad \textbf{(E)}\ 10$
2014 AIME Problems, 10
Let $z$ be a complex number with $|z| = 2014$. Let $P$ be the polygon in the complex plane whose vertices are $z$ and every $w$ such that $\tfrac{1}{z+w} = \tfrac{1}{z} + \tfrac{1}{w}$. Then the area enclosed by $P$ can be written in the form $n\sqrt{3},$ where $n$ is an integer. Find the remainder when $n$ is divided by $1000$.
2009 Harvard-MIT Mathematics Tournament, 8
If $a, b, x$ and $y$ are real numbers such that $ax + by = 3,$ $ax^2+by^2=7,$ $ax^3+bx^3=16$, and $ax^4+by^4=42,$ find $ax^5+by^5$.
2017 AMC 12/AHSME, 21
A set $S$ is constructed as follows. To begin, $S=\{0,10\}$. Repeatedly, as long as possible, if $x$ is an integer root of some polynomial $a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$ for some $n\geq 1$, all of whose coefficients $a_i$ are elements of $S$, then $x$ is put into $S$. When no more elements can be added to $S$, how many elements does $S$ have?
$\textbf{(A) } 4 \qquad \textbf{(B) } 5 \qquad \textbf{(C) } 7 \qquad \textbf{(D) } 9 \qquad \textbf{(E) } 11$
2006 AMC 10, 18
A license plate in a certain state consists of 4 digits, not necessarily distinct, and 2 letters, also not necessarily distinct. These six characters may appear in any order, except that the two letters must appear next to each other. How many distinct license plates are possible?
$ \textbf{(A) } 10^4\cdot 26^2 \qquad \textbf{(B) } 10^3\cdot 26^3 \qquad \textbf{(C) } 5\cdot 10^4\cdot 26^2 \qquad \textbf{(D) } 10^2\cdot 26^4\\
\textbf{(E) } 5\cdot 10^3\cdot 26^3$
2012 AMC 8, 15
The smallest number greater than 2 that leaves a remainder of 2 when divided by 3, 4, 5, or 6 lies between what numbers?
$\textbf{(A)}\hspace{.05in}40\text{ and }50 \qquad \textbf{(B)}\hspace{.05in}51\text{ and }55 \qquad \textbf{(C)}\hspace{.05in}56\text{ and }60 \qquad \textbf{(D)}\hspace{.05in} \text{61 and 65}\qquad \textbf{(E)}\hspace{.05in} \text{66 and 99}$
2016 AMC 10, 16
A triangle with vertices $A(0, 2)$, $B(-3, 2)$, and $C(-3, 0)$ is reflected about the $x$-axis, then the image $\triangle A'B'C'$ is rotated counterclockwise about the origin by $90^{\circ}$ to produce $\triangle A''B''C''$. Which of the following transformations will return $\triangle A''B''C''$ to $\triangle ABC$?
$\textbf{(A)}$ counterclockwise rotation about the origin by $90^{\circ}$.
$\textbf{(B)}$ clockwise rotation about the origin by $90^{\circ}$.
$\textbf{(C)}$ reflection about the $x$-axis
$\textbf{(D)}$ reflection about the line $y = x$
$\textbf{(E)}$ reflection about the $y$-axis.
2012 AMC 8, 7
Isabella must take four 100-point tests in her math class. Her goal is to achieve an average grade of 95 on the tests. Her first two test scores were 97 and 91. After seeing her score on the third test, she realized she can still reach her goal. What is the lowest possible score she could have made on the third test?
$\textbf{(A)}\hspace{.05in}90 \qquad \textbf{(B)}\hspace{.05in}92 \qquad \textbf{(C)}\hspace{.05in}95 \qquad \textbf{(D)}\hspace{.05in}96 \qquad \textbf{(E)}\hspace{.05in}97 $
2010 AMC 10, 4
For a real number $ x$, define $ \heartsuit (x)$ to be the average of $ x$ and $ x^2$. What is $ \heartsuit(1) \plus{} \heartsuit(2) \plus{}\heartsuit(3)$?
$ \textbf{(A)}\ 3 \qquad \textbf{(B)}\ 6 \qquad \textbf{(C)}\ 10 \qquad \textbf{(D)}\ 12 \qquad \textbf{(E)}\ 20$
2007 AIME Problems, 11
Two long cylindrical tubes of the same length but different diameters lie parallel to each other on a flat surface. The larger tube has radius $72$ and rolls along the surface toward the smaller tube, which has radius $24$. It rolls over the smaller tube and continues rolling along the flat surface until it comes to rest on the same point of its circumference as it started, having made one complete revolution. If the smaller tube never moves, and the rolling occurs with no slipping, the larger tube ends up a distance $x$ from where it starts. The distance $x$ can be expressed in the form $a\pi+b\sqrt{c},$ where $a,$ $b,$ and $c$ are integers and $c$ is not divisible by the square of any prime. Find $a+b+c.$
2024 AMC 8 -, 8
On Monday Taye has \$2. Everyday he either gains \$3 or doubles the amount of money he had on the previous day. How many different dollar amounts could Taye have on Thursday, 3 days later?
$\textbf{(A) } 3\qquad\textbf{(B) } 4\qquad\textbf{(C) } 5\qquad\textbf{(D) } 6\qquad\textbf{(E) } 7$
2022 AMC 10, 10
Daniel finds a rectangular index card and measures its diagonal to be 8 centimeters. Daniel then cuts out equal squares of side 1 cm at two opposite corners of the index card and measures the distance between the two closest vertices of these squares to be $4\sqrt{2}$ centimeters, as shown below. What is the area of the original index card?
[asy]
unitsize(0.6 cm);
pair A, B, C, D, E, F, G, H;
real x, y;
x = 9;
y = 5;
A = (0,y);
B = (x - 1,y);
C = (x - 1,y - 1);
D = (x,y - 1);
E = (x,0);
F = (1,0);
G = (1,1);
H = (0,1);
draw(A--B--C--D--E--F--G--H--cycle);
draw(interp(C,G,0.03)--interp(C,G,0.97), dashed, Arrows(6));
draw(interp(A,E,0.03)--interp(A,E,0.97), dashed, Arrows(6));
label("$1$", (B + C)/2, W);
label("$1$", (C + D)/2, S);
label("$8$", interp(A,E,0.3), NE);
label("$4 \sqrt{2}$", interp(G,C,0.2), SE);
[/asy]
$\textbf{(A) }14\qquad\textbf{(B) }10\sqrt{2}\qquad\textbf{(C) }16\qquad\textbf{(D) }12\sqrt{2}\qquad\textbf{(E) }18$
1951 AMC 12/AHSME, 39
A stone is dropped into a well and the report of the stone striking the bottom is heard $ 7.7$ seconds after it is dropped. Assume that the stone falls $ 16t^2$ feet in $ t$ seconds and that the velocity of sound is $ 1120$ feet per second. The depth of the well is:
$ \textbf{(A)}\ 784 \text{ ft.} \qquad\textbf{(B)}\ 342 \text{ ft.} \qquad\textbf{(C)}\ 1568 \text{ ft.} \qquad\textbf{(D)}\ 156.8 \text{ ft.} \qquad\textbf{(E)}\ \text{none of these}$
1996 AMC 8, 9
If $5$ times a number is $2$, then $100$ times the reciprocal of the number is
$\text{(A)}\ 2.5 \qquad \text{(B)}\ 40 \qquad \text{(C)}\ 50 \qquad \text{(D)}\ 250 \qquad \text{(E)}\ 500$
2024 AMC 12/AHSME, 13
There are real numbers $x,y,h$ and $k$ that satisfy the system of equations $$x^2 + y^2 - 6x - 8y = h$$ $$x^2 + y^2 - 10x + 4y = k$$
What is the minimum possible value of $h+k$?
$
\textbf{(A) }-54 \qquad
\textbf{(B) }-46 \qquad
\textbf{(C) }-34 \qquad
\textbf{(D) }-16 \qquad
\textbf{(E) }16 \qquad
$
2024 AMC 8 -, 4
When Yunji added all the integers from $1$ to $9$, she mistakenly left out a number. Her incorrect sum turned out to be a square number. Which number did Yunji leave out?
$\textbf{(A) } 5\qquad\textbf{(B) } 6\qquad\textbf{(C) } 7\qquad\textbf{(D) } 8\qquad\textbf{(E) } 9$
1993 AMC 12/AHSME, 7
The symbol $R_k$ stands for an integer whose base-ten representation is a sequence of $k$ ones. For example, $R_3=111$, $R_5=11111$, etc. When $R_{24}$ is divided by $R_4$, the quotient $Q=\frac{R_{24}}{R_4}$ is an integer whose base-ten representation is a sequence containing only ones and zeros. The number of zeros in $Q$ is
$ \textbf{(A)}\ 10 \qquad\textbf{(B)}\ 11 \qquad\textbf{(C)}\ 12 \qquad\textbf{(D)}\ 13 \qquad\textbf{(E)}\ 15 $
2013 AMC 10, 21
Two non-decreasing sequences of nonnegative integers have different first terms. Each sequence has the property that each term beginning with the third is the sum of the previous two terms, and the seventh term of each sequence is $N$. What is the smallest possible value of $N$?
${ \textbf{(A)}\ 55\qquad\textbf{(B)}\ 89\qquad\textbf{(C)}\ 104\qquad\textbf{(D}}\ 144\qquad\textbf{(E)}\ 273 $
2012 AMC 12/AHSME, 6
The sums of three whole numbers taken in pairs are $12$, $17$, and $19$. What is the middle number?
$ \textbf{(A)}\ 4\qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\ 6\qquad\textbf{(D)}\ 7\qquad\textbf{(E)}\ 8 $
2007 National Olympiad First Round, 19
If $x_1=5, x_2=401$, and
\[
x_n=x_{n-2}-\frac 1{x_{n-1}}
\]
for every $3\leq n \leq m$, what is the largest value of $m$?
$
\textbf{(A)}\ 406
\qquad\textbf{(B)}\ 2005
\qquad\textbf{(C)}\ 2006
\qquad\textbf{(D)}\ 2007
\qquad\textbf{(E)}\ \text{None of the above}
$
2019 AMC 12/AHSME, 7
Melanie computes the mean $\mu$, the median $M$, and the modes of the $365$ values that are the dates in the months of $2019$. Thus her data consist of $12$ $1\text{s}$, $12$ $2\text{s}$, . . . , $12$ $28\text{s}$, $11$ $29\text{s}$, $11$ $30\text{s}$, and $7$ $31\text{s}$. Let $d$ be the median of the modes. Which of the following statements is true?
$\textbf{(A) } \mu < d < M \qquad\textbf{(B) } M < d < \mu \qquad\textbf{(C) } d = M =\mu \qquad\textbf{(D) } d < M < \mu \qquad\textbf{(E) } d < \mu < M$
1983 USAMO, 4
Six segments $S_1, S_2, S_3, S_4, S_5,$ and $S_6$ are given in a plane. These are congruent to the edges $AB, AC, AD, BC, BD,$ and $CD$, respectively, of a tetrahedron $ABCD$. Show how to construct a segment congruent to the altitude of the tetrahedron from vertex $A$ with straight-edge and compasses.
1961 AMC 12/AHSME, 6
When simplified, $\log{8} \div \log{\frac{1}{8}}$ becomes:
${{{ \textbf{(A)}\ 6\log{2} \qquad\textbf{(B)}\ \log{2} \qquad\textbf{(C)}\ 1 \qquad\textbf{(D)}\ 0}\qquad\textbf{(E)}\ -1}} $
1998 AMC 8, 13
What is the ratio of the area of the shaded square to the area of the large square? (The figure is drawn to scale)
[asy]
draw((0,0)--(0,4)--(4,4)--(4,0)--cycle);
draw((0,0)--(4,4));
draw((0,4)--(3,1)--(3,3));
draw((1,1)--(2,0)--(4,2));
fill((1,1)--(2,0)--(3,1)--(2,2)--cycle,black);[/asy]
$ \text{(A)}\ \frac{1}{6}\qquad\text{(B)}\ \frac{1}{7}\qquad\text{(C)}\ \frac{1}{8}\qquad\text{(D)}\ \frac{1}{12}\qquad\text{(E)}\ \frac{1}{16} $
1971 AMC 12/AHSME, 20
The sum of the squares of the roots of the equation $x^2+2hx=3$ is $10$. The absolute value of $h$ is equal to
$\textbf{(A) }-1\qquad\textbf{(B) }\textstyle\frac{1}{2}\qquad\textbf{(C) }\textstyle\frac{3}{2}\qquad\textbf{(D) }2\qquad \textbf{(E) }\text{None of these}$