Found problems: 3632
2021 AMC 10 Fall, 11
Emily sees a ship traveling at a constant speed along a straight section of a river. She walks parallel to the riverbank at a uniform rate faster tha the ship. She counts $210$ equal steps walking from the back of the ship to the front. Walking in the opposite direction, she counts $42$ steps of the same size from the front of the ship to the back. In terms of Emily's equal steps, what is the length of the ship?
$\textbf{(A) }70\qquad\textbf{(B) }84\qquad\textbf{(C) }98\qquad\textbf{(D) }105\qquad\textbf{(E) }126$
2012 AMC 8, 13
Jamar bought some pencils costing more than a penny each at the school bookstore and paid $\$1.43$. Sharona bought some of the same pencils and paid $\$1.87$. How many more pencils did Sharona buy than Jamar?
$\textbf{(A)}\hspace{.05in}2 \qquad \textbf{(B)}\hspace{.05in}3 \qquad \textbf{(C)}\hspace{.05in}4 \qquad \textbf{(D)}\hspace{.05in}5 \qquad \textbf{(E)}\hspace{.05in}6 $
2014 AMC 8, 7
There are four more girls than boys in Ms. Raub's class of $28$ students. What is the ratio of number of girls to the number of boys in her class?
$\textbf{(A) }3 : 4\qquad\textbf{(B) }4 : 3\qquad\textbf{(C) }3 : 2\qquad\textbf{(D) }7 : 4\qquad \textbf{(E) }2 : 1$
2009 AMC 12/AHSME, 16
A circle with center $ C$ is tangent to the positive $ x$ and $ y$-axes and externally tangent to the circle centered at $ (3,0)$ with radius $ 1$. What is the sum of all possible radii of the circle with center $ C$?
$ \textbf{(A)}\ 3 \qquad
\textbf{(B)}\ 4 \qquad
\textbf{(C)}\ 6 \qquad
\textbf{(D)}\ 8 \qquad
\textbf{(E)}\ 9$
2014 AMC 8, 21
The $7$-digit numbers $\underline{7}$ $ \underline{4}$ $ \underline{A}$ $ \underline{5}$ $ \underline{2}$ $ \underline{B}$ $ \underline{1}$ and $\underline{3}$ $ \underline{2}$ $ \underline{6}$ $ \underline{A}$ $ \underline{B}$ $ \underline{4}$ $ \underline{C}$ are each multiples of $3$. Which of the following could be the value of $C$?
$\textbf{(A) }1\qquad\textbf{(B) }2\qquad\textbf{(C) }3\qquad\textbf{(D) }5\qquad \textbf{(E) }8$
2013 AMC 12/AHSME, 17
Let $a,b,$ and $c$ be real numbers such that \begin{align*}
a+b+c &= 2, \text{ and} \\
a^2+b^2+c^2&= 12
\end{align*}
What is the difference between the maximum and minimum possible values of $c$?
$ \textbf{(A)}\ 2\qquad\textbf{(B)}\ \frac{10}{3}\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}\ \frac{16}{3}\qquad\textbf{(E)}\ \frac{20}{3} $
2021 AMC 10 Fall, 20
In a particular game, each of $4$ players rolls a standard $6{ }$-sided die. The winner is the player who rolls the highest number. If there is a tie for the highest roll, those involved in the tie will roll again and this process will continue until one player wins. Hugo is one of the players in this game. What is the probability that Hugo's first roll was a $5,$ given that he won the game?
$(\textbf{A})\: \frac{61}{216}\qquad(\textbf{B}) \: \frac{367}{1296}\qquad(\textbf{C}) \: \frac{41}{144}\qquad(\textbf{D}) \: \frac{185}{648}\qquad(\textbf{E}) \: \frac{11}{36}$
2014 AMC 10, 25
The number $5^{867}$ is between $2^{2013}$ and $2^{2014}$. How many pairs of integers $(m,n)$ are there such that $1\leq m\leq 2012$ and \[5^n<2^m<2^{m+2}<5^{n+1}?\]
$\textbf{(A) }278\qquad
\textbf{(B) }279\qquad
\textbf{(C) }280\qquad
\textbf{(D) }281\qquad
\textbf{(E) }282\qquad$
2014 South East Mathematical Olympiad, 3
In an obtuse triangle $ABC$ $(AB>AC)$,$O$ is the circumcentre and $D,E,F$ are the midpoints of $BC,CA,AB$ respectively.Median $AD$ intersects $OF$ and $OE$ at $M$ and $N$ respectively.$BM$ meets $CN$ at point $P$.Prove that $OP\perp AP$
PEN M Problems, 32
In an increasing infinite sequence of positive integers, every term starting from the $2002$-th term divides the sum of all preceding terms. Prove that every term starting from some term is equal to the sum of all preceding terms.
2008 AMC 10, 17
An equilateral triangle has side length $ 6$. What is the area of the region containing all points that are outside the triangle and not more than $ 3$ units from a point of the triangle?
$ \textbf{(A)}\ 36\plus{}24\sqrt{3} \qquad
\textbf{(B)}\ 54\plus{}9\pi \qquad
\textbf{(C)}\ 54\plus{}18\sqrt{3}\plus{}6\pi \qquad
\textbf{(D)}\ \left(2\sqrt{3}\plus{}3\right)^2\pi \\
\textbf{(E)}\ 9\left(\sqrt{3}\plus{}1\right)^2\pi$
1959 AMC 12/AHSME, 25
The symbol $|a|$ means $+a$ if $a$ is greater than or equal to zero, and $-a$ if $a$ is less than or equal to zero; the symbol $<$ means "less than"; the symbol $>$ means "greater than."
The set of values $x$ satisfying the inequality $|3-x|<4$ consists of all $x$ such that:
$ \textbf{(A)}\ x^2<49 \qquad\textbf{(B)}\ x^2>1 \qquad\textbf{(C)}\ 1<x^2<49\qquad\textbf{(D)}\ -1<x<7\qquad\textbf{(E)}\ -7<x<1 $
2013 AMC 10, 12
In $\triangle ABC$, $AB=AC=28$ and $BC=20$. Points $D,E,$ and $F$ are on sides $\overline{AB}$, $\overline{BC}$, and $\overline{AC}$, respectively, such that $\overline{DE}$ and $\overline{EF}$ are parallel to $\overline{AC}$ and $\overline{AB}$, respectively. What is the perimeter of parallelogram $ADEF$?
[asy]
size(180);
pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps);
real r=5/7;
pair A=(10,sqrt(28^2-100)),B=origin,C=(20,0),D=(A.x*r,A.y*r);
pair bottom=(C.x+(D.x-A.x),C.y+(D.y-A.y));
pair E=extension(D,bottom,B,C);
pair top=(E.x+D.x,E.y+D.y);
pair F=extension(E,top,A,C);
draw(A--B--C--cycle^^D--E--F);
dot(A^^B^^C^^D^^E^^F);
label("$A$",A,NW);
label("$B$",B,SW);
label("$C$",C,SE);
label("$D$",D,W);
label("$E$",E,S);
label("$F$",F,dir(0));
[/asy]
$\textbf{(A) }48\qquad
\textbf{(B) }52\qquad
\textbf{(C) }56\qquad
\textbf{(D) }60\qquad
\textbf{(E) }72\qquad$
2008 AMC 12/AHSME, 6
Heather compares the price of a new computer at two different stores. Store A offers $ 15\%$ off the sticker price followed by a $ \$90$ rebate, and store B offers $ 25\%$ off the same sticker price with no rebate. Heather saves $ \$15$ by buying the computer at store A instead of store B. What is the sticker price of the computer, in dollars?
$ \textbf{(A)}\ 750 \qquad \textbf{(B)}\ 900 \qquad \textbf{(C)}\ 1000 \qquad \textbf{(D)}\ 1050 \qquad \textbf{(E)}\ 1500$
2022 USAJMO, 6
Let $a_0, b_0, c_0$ be complex numbers, and define \begin{align*}a_{n+1} &= a_n^2 + 2b_nc_n \\ b_{n+1} &= b_n^2 + 2c_na_n \\ c_{n+1} &= c_n^2 + 2a_nb_n\end{align*}for all nonnegative integers $n.$
Suppose that $\max{\{|a_n|, |b_n|, |c_n|\}} \leq 2022$ for all $n.$ Prove that $$|a_0|^2 + |b_0|^2 + |c_0|^2 \leq 1.$$
2010 AMC 10, 22
Seven distinct pieces of candy are to be distributed among three bags. The red bag and the blue bag must each receive at least one piece of candy; the white bag may remain empty. How many arrangements are possible?
$ \textbf{(A)}\ 1930\qquad\textbf{(B)}\ 1931\qquad\textbf{(C)}\ 1932\qquad\textbf{(D)}\ 1933\qquad\textbf{(E)}\ 1934$
2018 AMC 10, 5
How many subsets of $\{2,3,4,5,6,7,8,9\}$ contain at least one prime number?
$\textbf{(A)} \text{ 128} \qquad \textbf{(B)} \text{ 192} \qquad \textbf{(C)} \text{ 224} \qquad \textbf{(D)} \text{ 240} \qquad \textbf{(E)} \text{ 256}$
2013 AMC 12/AHSME, 18
Barbara and Jenna play the following game, in which they take turns. A number of coins lie on a table. When it is Barbara's turn, she must remove $2$ or $4$ coins, unless only one coin remains, in which case she loses her turn. When it is Jenna's turn, she must remove $1$ or $3$ coins. A coin flip determines who goes first. Whoever removes the last coin wins the game. Assume both players use their best strategy. Who will win when the game starts with $2013$ coins and when the game starts with $2014$ coins?
$\textbf{(A)}$ Barbara will win with $2013$ coins, and Jenna will win with $2014$ coins.
$\textbf{(B)}$ Jenna will win with $2013$ coins, and whoever goes first will win with $2014$ coins.
$\textbf{(C)}$ Barbara will win with $2013$ coins, and whoever goes second will win with $2014$ coins.
$\textbf{(D)}$ Jenna will win with $2013$ coins, and Barbara will win with $2014$ coins.
$\textbf{(E)}$ Whoever goes first will win with $2013$ coins, and whoever goes second will win with $2014$ coins.
2009 AIME Problems, 4
In parallelogram $ ABCD$, point $ M$ is on $ \overline{AB}$ so that $ \frac{AM}{AB} \equal{} \frac{17}{1000}$ and point $ N$ is on $ \overline{AD}$ so that $ \frac{AN}{AD} \equal{} \frac{17}{2009}$. Let $ P$ be the point of intersection of $ \overline{AC}$ and $ \overline{MN}$. Find $ \frac{AC}{AP}$.
1971 AMC 12/AHSME, 26
[asy]
size(2.5inch);
pair A, B, C, E, F, G;
A = (0,3);
B = (-1,0);
C = (3,0);
E = (0,0);
F = (1,2);
G = intersectionpoint(B--F,A--E);
draw(A--B--C--cycle);
draw(A--E);
draw(B--F);
label("$A$",A,N);
label("$B$",B,W);
label("$C$",C,dir(0));
label("$E$",E,S);
label("$F$",F,NE);
label("$G$",G,SE);
//Credit to chezbgone2 for the diagram[/asy]
In triangle $ABC$, point $F$ divides side $AC$ in the ratio $1:2$. Let $E$ be the point of intersection of side $BC$ and $AG$ where $G$ is the midpoints of $BF$. The point $E$ divides side $BC$ in the ratio
$\textbf{(A) }1:4\qquad\textbf{(B) }1:3\qquad\textbf{(C) }2:5\qquad\textbf{(D) }4:11\qquad \textbf{(E) }3:8$
2016 AMC 10, 2
If $n\heartsuit m=n^3m^2$, what is $\frac{2\heartsuit 4}{4\heartsuit 2}$?
$\textbf{(A)}\ \frac{1}{4}\qquad\textbf{(B)}\ \frac{1}{2}\qquad\textbf{(C)}\ 1\qquad\textbf{(D)}\ 2\qquad\textbf{(E)}\ 4$
2011 AMC 12/AHSME, 8
Keiko walks once around a track at exactly the same constant speed every day. The sides of the track are straight, and the ends are semicircles. The track has width 6 meters, and it takes her 36 seconds longer to walk around the outside edge of the track than around the inside edge. What is Keiko's speed in meters per second?
$ \textbf{(A)}\ \frac{\pi}{3} \qquad
\textbf{(B)}\ \frac{2\pi}{3} \qquad
\textbf{(C)}\ \pi \qquad
\textbf{(D)}\ \frac{4\pi}{3} \qquad
\textbf{(E)}\ \frac{5\pi}{3} $
2000 Korea - Final Round, 2
Determine all function $f$ from the set of real numbers to itself such that for every $x$ and $y$,
\[f(x^2-y^2)=(x-y)(f(x)+f(y))\]
2008 AMC 10, 9
A quadratic equation $ ax^2\minus{}2ax\plus{}b\equal{}0$ has two real solutions. What is the average of the solutions?
$ \textbf{(A)}\ 1 \qquad
\textbf{(B)}\ 2 \qquad
\textbf{(C)}\ \frac{b}{a} \qquad
\textbf{(D)}\ \frac{2b}{a} \qquad
\textbf{(E)}\ \sqrt{2b\minus{}a}$
2023 AMC 12/AHSME, 4
How many digits are in the base-ten representation of $8^5 \cdot 5^{10} \cdot 15^5$?
$\textbf{(A)}~14\qquad\textbf{(B)}~15\qquad\textbf{(C)}~16\qquad\textbf{(D)}~17\qquad\textbf{(E)}~18\qquad$