Found problems: 3632
1964 AMC 12/AHSME, 14
A farmer bought $749$ sheeps. He sold $700$ of them for the price paid for the $749$ sheep. The remaining $49$ sheep were sold at the same price per head as the other $700$. Based on the cost, the percent gain on the entire transaction is:
${{ \textbf{(A)}\ 6.5 \qquad\textbf{(B)}\ 6.75 \qquad\textbf{(C)}\ 7 \qquad\textbf{(D)}\ 7.5 }\qquad\textbf{(E)}\ 8 } $
2010 AMC 10, 5
A month with 31 days has the same number of Mondays and Wednesdays. How many of the seven days of the week could be the first day of this month?
$ \textbf{(A)}\ 2\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}\ 5\qquad\textbf{(E)}\ 6$
1996 AMC 8, 16
$1-2-3+4+5-6-7+8+9-10-11+\cdots + 1992+1993-1994-1995+1996=$
$\text{(A)}\ -998 \qquad \text{(B)}\ -1 \qquad \text{(C)}\ 0 \qquad \text{(D)}\ 1 \qquad \text{(E)}\ 998$
2013 AMC 12/AHSME, 21
Consider \[A = \log (2013 + \log (2012 + \log (2011 + \log (\cdots + \log (3 + \log 2) \cdots )))).\] Which of the following intervals contains $ A $?
$ \textbf{(A)} \ (\log 2016, \log 2017) $
$ \textbf{(B)} \ (\log 2017, \log 2018) $
$ \textbf{(C)} \ (\log 2018, \log 2019) $
$ \textbf{(D)} \ (\log 2019, \log 2020) $
$ \textbf{(E)} \ (\log 2020, \log 2021) $
1995 AMC 12/AHSME, 26
In the figure, $\overline{AB}$ and $\overline{CD}$ are diameters of the circle with center $O$, $\overline{AB} \perp \overline{CD}$, and chord $\overline{DF}$ intersects $\overline{AB}$ at $E$. If $DE = 6$ and $EF = 2$, then the area of the circle is
[asy]
size(120); defaultpen(linewidth(0.7));
pair O=origin, A=(-5,0), B=(5,0), C=(0,5), D=(0,-5), F=5*dir(40), E=intersectionpoint(A--B, F--D);
draw(Circle(O, 5));
draw(A--B^^C--D--F);
dot(O^^A^^B^^C^^D^^E^^F);
markscalefactor=0.05;
draw(rightanglemark(B, O, D));
label("$A$", A, dir(O--A));
label("$B$", B, dir(O--B));
label("$C$", C, dir(O--C));
label("$D$", D, dir(O--D));
label("$F$", F, dir(O--F));
label("$O$", O, NW);
label("$E$", E, SE);[/asy]
$\textbf{(A)}\ 23\pi \qquad
\textbf{(B)}\ \dfrac{47}{2}\pi \qquad
\textbf{(C)}\ 24\pi \qquad
\textbf{(D)}\ \dfrac{49}{2}\pi \qquad
\textbf{(E)}\ 25\pi$
2011 AMC 12/AHSME, 13
Brian writes down four integers $w > x > y > z$ whose sum is $44$. The pairwise positive differences of these numbers are $1,3,4,5,6,$ and $9$. What is the sum of the possible values for $w$?
$ \textbf{(A)}\ 16 \qquad
\textbf{(B)}\ 31 \qquad
\textbf{(C)}\ 48 \qquad
\textbf{(D)}\ 62 \qquad
\textbf{(E)}\ 93 $
2022 AMC 10, 21
Let $P(x)$ be a polynomial with rational coefficients such that when $P(x)$ is divided by the polynomial $x^2 + x + 1$, the remainder is $x + 2$, and when $P(x)$ is divided by the polynomial $x^2 + 1$, the remainder is $2x + 1$. There is a unique polynomial of least degree with these two properties. What is the sum of the squares of the coefficients of that polynomial?
$\textbf{(A) } 10 \qquad \textbf{(B) } 13 \qquad \textbf{(C) } 19 \qquad \textbf{(D) } 20 \qquad \textbf{(E) } 23$
1971 AMC 12/AHSME, 21
If $\log_2(\log_3(\log_4 x))=\log_3(\log_4(\log_2 y))=\log_4(\log_2(\log_3 z))=0$, then the sum $x+y+z$ is equal to
$\textbf{(A) }50\qquad\textbf{(B) }58\qquad\textbf{(C) }89\qquad\textbf{(D) }111\qquad \textbf{(E) }1296$
1977 AMC 12/AHSME, 13
If $a_1,a_2,a_3,\dots$ is a sequence of positive numbers such that $a_{n+2}=a_na_{n+1}$ for all positive integers $n$, then the sequence $a_1,a_2,a_3,\dots$ is a geometric progression
$\textbf{(A) }\text{for all positive values of }a_1\text{ and }a_2\qquad$
$\textbf{(B) }\text{if and only if }a_1=a_2\qquad$
$\textbf{(C) }\text{if and only if }a_1=1\qquad$
$\textbf{(D) }\text{if and only if }a_2=1\qquad $
$\textbf{(E) }\text{if and only if }a_1=a_2=1$
1961 AMC 12/AHSME, 1
When simplified, $(-\frac{1}{125})^{-2/3}$ becomes:
${{ \textbf{(A)}\ \frac{1}{25} \qquad\textbf{(B)}\ -\frac{1}{25} \qquad\textbf{(C)}\ 25\qquad\textbf{(D)}\ -25}\qquad\textbf{(E)}\ 25\sqrt{-1}} $
2001 AMC 10, 13
A telephone number has the form $ ABC \minus{} DEF \minus{} GHIJ$, where each letter represents a different digit. The digits in each part of the numbers are in decreasing order; that is, $ A > B > C$, $ D > E > F$, and $ G > H > I > J$. Furthermore, $ D$, $ E$, and $ F$ are consecutive even digits; $ G$, $ H$, $ I$, and $ J$ are consecutive odd digits; and $ A \plus{} B \plus{} C \equal{} 9$. Find $ A$.
$ \textbf{(A)} \ 4 \qquad \textbf{(B)} \ 5 \qquad \textbf{(C)} \ 6 \qquad \textbf{(D)} \ 7 \qquad \textbf{(E)} \ 8$
2018 AMC 12/AHSME, 22
Consider polynomials $P(x)$ of degree at most $3$, each of whose coefficients is an element of $\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}$. How many such polynomials satisfy $P(-1) = -9$?
$\textbf{(A) } 110 \qquad \textbf{(B) } 143 \qquad \textbf{(C) } 165 \qquad \textbf{(D) } 220 \qquad \textbf{(E) } 286 $
2012 AIME Problems, 12
Let $\triangle ABC$ be a right triangle with right angle at $C$. Let $D$ and $E$ be points on $\overline{AB}$ with $D$ between $A$ and $E$ such that $\overline{CD}$ and $\overline{CE}$ trisect $\angle C$. If $\frac{DE}{BE} = \frac{8}{15}$, then $\tan B$ can be written as $\frac{m\sqrt{p}}{n}$, where $m$ and $n$ are relatively prime positive integers, and $p$ is a positive integer not divisible by the square of any prime. Find $m+n+p$.
2017 AMC 12/AHSME, 13
Driving at a constant speed, Sharon usually takes $180$ minutes to drive from her house to her mother's house. One day Sharon begins the drive at her usual speed, but after driving $\frac{1}{3}$ of the way, she hits a bad snowstorm and reduces her speed by $20$ miles per hour. This time the trip takes her a total of $276$ minutes. How many miles is it from Sharon's house to her mother's house?
$\textbf{(A)}\ 132\qquad\textbf{(B)}\ 135\qquad\textbf{(C)}\ 138\qquad\textbf{(D)}\ 141\qquad\textbf{(E)}\ 144$
1999 AIME Problems, 2
Consider the parallelogram with vertices $(10,45),$ $(10,114),$ $(28,153),$ and $(28,84).$ A line through the origin cuts this figure into two congruent polygons. The slope of the line is $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$
2003 AIME Problems, 8
In an increasing sequence of four positive integers, the first three terms form an arithmetic progression, the last three terms form a geometric progression, and the first and fourth terms differ by 30. Find the sum of the four terms.
1972 AMC 12/AHSME, 26
[asy]
real t=pi/8;real u=7*pi/12;real v=13*pi/12;
real ct=cos(t);real st=sin(t);real cu=cos(u);real su=sin(u);
draw(unitcircle);
draw((ct,st)--(-ct,st)--(cos(v),sin(v)));
draw((cu,su)--(cu,st));
label("A",(-ct,st),W);label("B",(ct,st),E);
label("M",(cu,su),N);label("P",(cu,st),S);
label("C",(cos(v),sin(v)),W);
//Credit to Zimbalono for the diagram[/asy]
In the circle above, $M$ is the midpoint of arc $CAB$ and segment $MP$ is perpendicular to chord $AB$ at $P$. If the measure of chord $AC$ is $x$ and that of segment $AP$ is $(x+1)$, then segment $PB$ has measure equal to
$\textbf{(A) }3x+2\qquad\textbf{(B) }3x+1\qquad\textbf{(C) }2x+3\qquad\textbf{(D) }2x+2\qquad \textbf{(E) }2x+1$
1960 AMC 12/AHSME, 36
Let $s_1, s_2, s_3$ be the respective sums of $n$, $2n$, $3n$ terms of the same arithmetic progression with $a$ as the first term and $d$ as the common difference. Let $R=s_3-s_2-s_1$. Then $R$ is dependent on:
$ \textbf{(A)}\ a \text{ } \text{and} \text{ } d\qquad\textbf{(B)}\ d \text{ } \text{and} \text{ } n\qquad\textbf{(C)}\ a \text{ } \text{and} \text{ } n\qquad\textbf{(D)}\ a, d, \text{ } \text{and} \text{ } n\qquad$
$\textbf{(E)}\ \text{neither} \text{ } a \text{ } \text{nor} \text{ } d \text{ } \text{nor} \text{ } n $
1970 AMC 12/AHSME, 8
If $a=\log_8225$ and $b=\log_215$, then
$\textbf{(A) }a=\frac{1}{2}b\qquad\textbf{(B) }a=\frac{2b}{3}\qquad\textbf{(C) }a=b\qquad\textbf{(D) }b=\frac{1}{2}a\qquad \textbf{(E) }a=\frac{3b}{2}$
2014 AMC 8, 8
Eleven members of the Middle School Math Club each paid the same amount for a guest speaker to talk about problem solving at their math club meeting. They paid their guest speaker $ \$ \underline{1}$ $ \underline{A}$ $ \underline{2}$. What is the missing digit $A$ of this $3$-digit number?
$\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }3\qquad \textbf{(E) }4$
2012 Balkan MO Shortlist, N3
Let $\mathbb{Z}^+$ be the set of positive integers. Find all functions $f:\mathbb{Z}^+ \rightarrow\mathbb{Z}^+$ such that the following conditions both hold:
(i) $f(n!)=f(n)!$ for every positive integer $n$,
(ii) $m-n$ divides $f(m)-f(n)$ whenever $m$ and $n$ are different positive integers.
2012 AMC 10, 13
It takes Clea $60$ seconds to walk down an escalator when it is not operating and only $24$ seconds to walk down the escalator when it is operating. How many seconds does it take Clea to ride down the operating escalator when she just stands on it?
$ \textbf{(A)}\ 36\qquad\textbf{(B)}\ 40\qquad\textbf{(C)}\ 42\qquad\textbf{(D)}\ 48\qquad\textbf{(E)}\ 52 $
2012 AMC 12/AHSME, 12
How many sequences of zeros and ones of length 20 have all the zeros consecutive, or all the ones consecutive, or both?
$ \textbf{(A)}\ 190\qquad\textbf{(B)}\ 192\qquad\textbf{(C)}\ 211\qquad\textbf{(D)}\ 380\qquad\textbf{(E)}\ 382$
2024 AMC 12/AHSME, 16
A set of $12$ tokens ---- $3$ red, $2$ white, $1$ blue, and $6$ black ---- is to be distributed at random to $3$ game players, $4$ tokens per player. The probability that some player gets all the red tokens, another gets all the white tokens, and the remaining player gets the blue token can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$?
$
\textbf{(A) }387 \qquad
\textbf{(B) }388 \qquad
\textbf{(C) }389 \qquad
\textbf{(D) }390 \qquad
\textbf{(E) }391 \qquad
$
2021 AMC 10 Spring, 9
The point $P(a,b)$ in the $xy$-plane is first rotated counterclockwise by $90^{\circ}$ around the point $(1,5)$ and then reflected about the line $y=-x$. The image of $P$ after these two transformations is at $(-6,3)$. What is $b-a$?
$\textbf{(A) }1 \qquad \textbf{(B) }3 \qquad \textbf{(C) }5 \qquad \textbf{(D) }7 \qquad \textbf{(E) }9$