Found problems: 3632
2010 AMC 10, 3
Tyrone had $ 97$ marbles and Eric had $ 11$ marbles. Tyrone then gave some of his marbles to Eric so that Tyrone ended with twice as many marbles as Eric. How many marbles did Tyrone give to Eric?
$ \textbf{(A)}\ 3 \qquad
\textbf{(B)}\ 13 \qquad
\textbf{(C)}\ 18 \qquad
\textbf{(D)}\ 25 \qquad
\textbf{(E)}\ 29$
1988 AIME Problems, 3
Find $(\log_2 x)^2$ if $\log_2 (\log_8 x) = \log_8 (\log_2 x)$.
2010 AIME Problems, 9
Let $ (a,b,c)$ be the real solution of the system of equations $ x^3 \minus{} xyz \equal{} 2$, $ y^3 \minus{} xyz \equal{} 6$, $ z^3 \minus{} xyz \equal{} 20$. The greatest possible value of $ a^3 \plus{} b^3 \plus{} c^3$ can be written in the form $ \frac{m}{n}$, where $ m$ and $ n$ are relatively prime positive integers. Find $ m \plus{} n$.
2022 AIME Problems, 3
In isosceles trapezoid $ABCD$, parallel bases $\overline{AB}$ and $\overline{CD}$ have lengths $500$ and $650$, respectively, and $AD=BC=333$. The angle bisectors of $\angle{A}$ and $\angle{D}$ meet at $P$, and the angle bisectors of $\angle{B}$ and $\angle{C}$ meet at $Q$. Find $PQ$.
1993 AMC 8, 25
A checkerboard consists of one-inch squares. A square card, $1.5$ inches on a side, is placed on the board so that it covers part or all of the area of each of $n$ squares. The maximum possible value of $n$ is
$\text{(A)}\ 4\text{ or }5 \qquad \text{(B)}\ 6\text{ or }7\qquad \text{(C)}\ 8\text{ or }9 \qquad \text{(D)}\ 10\text{ or }11 \qquad \text{(E)}\ 12\text{ or more}$
1993 AMC 12/AHSME, 18
Al and Barb start their new jobs on the same day. Al's schedule is $3$ work-days followed by $1$ rest-day. Barb's schedule is $7$ work-days followed by $3$ rest-days. On how many of their first $1000$ days do both have rest-days on the same day?
$ \textbf{(A)}\ 48 \qquad\textbf{(B)}\ 50 \qquad\textbf{(C)}\ 72 \qquad\textbf{(D)}\ 75 \qquad\textbf{(E)}\ 100 $
2008 AMC 12/AHSME, 11
Three cubes are each formed from the pattern shown. They are then stacked on a table one on top of another so that the $ 13$ visible numbers have the greatest possible sum. What is that sum?
[asy]unitsize(.8cm);
pen p = linewidth(.8pt);
draw(shift(-2,0)*unitsquare,p);
label("1",(-1.5,0.5));
draw(shift(-1,0)*unitsquare,p);
label("2",(-0.5,0.5));
label("32",(0.5,0.5));
draw(shift(1,0)*unitsquare,p);
label("16",(1.5,0.5));
draw(shift(0,1)*unitsquare,p);
label("4",(0.5,1.5));
draw(shift(0,-1)*unitsquare,p);
label("8",(0.5,-0.5));[/asy]$ \textbf{(A)}\ 154 \qquad \textbf{(B)}\ 159 \qquad \textbf{(C)}\ 164 \qquad \textbf{(D)}\ 167 \qquad \textbf{(E)}\ 189$
2014 AMC 10, 21
Positive integers $a$ and $b$ are such that the graphs of $y=ax+5$ and $y=3x+b$ intersect the $x$-axis at the same point. What is the sum of all possible $x$-coordinates of these points of intersection?
$ \textbf{(A)}\ {-20}\qquad\textbf{(B)}\ {-18}\qquad\textbf{(C)}\ {-15}\qquad\textbf{(D)}\ {-12}\qquad\textbf{(E)}\ {-8} $
2017 AMC 10, 18
Amelia has a coin that lands heads with probability $\frac{1}{3}$, and Blaine has a coin that lands on heads with probability $\frac{2}{5}$. Amelia and Blaine alternately toss their coins until someone gets a head; the first one to get a head wins. All coin tosses are independent. Amelia goes first. The probability that Amelia wins is $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. What is $q-p$?
$\textbf{(A) }1\qquad\textbf{(B) }2\qquad\textbf{(C) }3\qquad\textbf{(D) }4\qquad\textbf{(E) }5$
2008 AMC 12/AHSME, 5
Suppose that
\[ \frac {2x}{3} \minus{} \frac {x}{6}
\]is an integer. Which of the following statements must be true about $ x$?
$ \textbf{(A)}\ \text{It is negative.} \qquad \textbf{(B)}\ \text{It is even, but not necessarily a multiple of }3\text{.}$
$ \textbf{(C)}\ \text{It is a multiple of }3\text{, but not necessarily even.}$
$ \textbf{(D)}\ \text{It is a multiple of }6\text{, but not necessarily a multiple of }12\text{.}$
$ \textbf{(E)}\ \text{It is a multiple of }12\text{.}$
2014 AIME Problems, 4
The repeating decimals $0.abab\overline{ab}$ and $0.abcabc\overline{abc}$ satisfy
\[0.abab\overline{ab}+0.abcabc\overline{abc}=\frac{33}{37},\]
where $a,b$, and $c$ are (not necessarily distinct) digits. Find the three-digit number $abc$.
1980 USAMO, 4
The inscribed sphere of a given tetrahedron touches all four faces of the tetrahedron at their respective centroids. Prove that the tetrahedron is regular.
1997 AMC 8, 20
A pair of 8-sided dice have sides numbered 1 through 8. Each side has the same probability (chance) of landing face up. The probability that the product of the two numbers that land face-up exceeds 36 is
$\textbf{(A)}\ \dfrac{5}{32} \qquad \textbf{(B)}\ \dfrac{11}{64} \qquad \textbf{(C)}\ \dfrac{3}{16} \qquad \textbf{(D)}\ \dfrac{1}{4} \qquad \textbf{(E)}\ \dfrac{1}{2}$
2010 AMC 10, 2
Four identical squares and one rectangle are placed together to form one large square as shown. The length of the rectangle is how many times as large as its width?
[asy]unitsize(8mm);
defaultpen(linewidth(.8pt));
draw(scale(4)*unitsquare);
draw((0,3)--(4,3));
draw((1,3)--(1,4));
draw((2,3)--(2,4));
draw((3,3)--(3,4));[/asy]$ \textbf{(A)}\ \frac {5}{4} \qquad \textbf{(B)}\ \frac {4}{3} \qquad \textbf{(C)}\ \frac {3}{2} \qquad \textbf{(D)}\ 2 \qquad \textbf{(E)}\ 3$
2014 AIME Problems, 4
Jon and Steve ride their bicycles on a path that parallels two side-by-side train tracks running in the east/west direction. Jon rides east at 20 miles per hour, and Steve rides west at 20 miles per hour. Two trains of equal length traveling in opposite directions at constant but different speeds, each pass the two riders. Each train takes exactly 1 minute to go past Jon. The westbound train takes 10 times as long as the eastbound train to go past Steve. The length of each train is $\tfrac mn$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
2018 AIME Problems, 3
Kathy has $5$ red cards and $5$ green cards. She shuffles the $10$ cards and lays out $5$ of the cards in a row in a random order. She will be happy if and only if all the red cards laid out are adjacent and all the green cards laid out are adjacent. For example, card orders RRGGG, GGGGR, or RRRRR will make Kathy happy, but RRRGR will not. The probability that Kathy will be happy is $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
1968 AMC 12/AHSME, 19
Let $n$ be the number of ways that $10$ dollars can be changed into dimes and quarters, with at least one of each coin being used. Then $n$ equals:
$\textbf{(A)}\ 40 \qquad
\textbf{(B)}\ 38 \qquad
\textbf{(C)}\ 21 \qquad
\textbf{(D)}\ 20 \qquad
\textbf{(E)}\ 19 $
2017 AMC 10, 16
How many of the base-ten numerals for the positive integers less than or equal to 2017 contain the digit 0?
$\textbf{(A)} \text{ 469} \qquad \textbf{(B)} \text{ 471} \qquad \textbf{(C)} \text{ 475} \qquad \textbf{(D)} \text{ 478} \qquad \textbf{(E)} \text{ 481}$
2017 AMC 10, 23
Let $N = 123456789101112\dots4344$ be the $79$-digit number obtained that is formed by writing the integers from $1$ to $44$ in order, one after the other. What is the remainder when $N$ is divided by $45$?
$\textbf{(A)}\ 1 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 9 \qquad\textbf{(D)}\ 18 \qquad\textbf{(E)}\ 44$
2006 AMC 12/AHSME, 13
The vertices of a $ 3 \minus{} 4 \minus{} 5$ right triangle are the centers of three mutually externally tangent circles, as shown. What is the sum of the areas of the three circles?
[asy]unitsize(5mm);
defaultpen(fontsize(10pt)+linewidth(.8pt));
pair B=(0,0), C=(5,0);
pair A=intersectionpoints(Circle(B,3),Circle(C,4))[0];
draw(A--B--C--cycle);
draw(Circle(C,3));
draw(Circle(A,1));
draw(Circle(B,2));
label("$A$",A,N);
label("$B$",B,W);
label("$C$",C,E);
label("3",midpoint(B--A),NW);
label("4",midpoint(A--C),NE);
label("5",midpoint(B--C),S);[/asy]$ \textbf{(A) } 12\pi\qquad \textbf{(B) } \frac {25\pi}{2}\qquad \textbf{(C) } 13\pi\qquad \textbf{(D) } \frac {27\pi}{2}\qquad \textbf{(E) } 14\pi$
2013 AMC 12/AHSME, 11
Triangle $ABC$ is equilateral with $AB=1$. Points $E$ and $G$ are on $\overline{AC}$ and points $D$ and $F$ are on $\overline{AB}$ such that both $\overline{DE}$ and $\overline{FG}$ are parallel to $\overline{BC}$. Furthermore, triangle $ADE$ and trapezoids $DFGE$ and $FBCG$ all have the same perimeter. What is $DE+FG$?
[asy]
size(180);
pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps);
real s=1/2,m=5/6,l=1;
pair A=origin,B=(l,0),C=rotate(60)*l,D=(s,0),E=rotate(60)*s,F=m,G=rotate(60)*m;
draw(A--B--C--cycle^^D--E^^F--G);
dot(A^^B^^C^^D^^E^^F^^G);
label("$A$",A,SW);
label("$B$",B,SE);
label("$C$",C,N);
label("$D$",D,S);
label("$E$",E,NW);
label("$F$",F,S);
label("$G$",G,NW);
[/asy]
$\textbf{(A) }1\qquad
\textbf{(B) }\dfrac{3}{2}\qquad
\textbf{(C) }\dfrac{21}{13}\qquad
\textbf{(D) }\dfrac{13}{8}\qquad
\textbf{(E) }\dfrac{5}{3}\qquad$
1972 AMC 12/AHSME, 29
If $f(x)=\log \left(\frac{1+x}{1-x}\right)$ for $-1<x<1$, then $f\left(\frac{3x+x^3}{1+3x^2}\right)$ in terms of $f(x)$ is
$\textbf{(A) }-f(x)\qquad\textbf{(B) }2f(x)\qquad\textbf{(C) }3f(x)\qquad$
$\textbf{(D) }\left[f(x)\right]^2\qquad \textbf{(E) }[f(x)]^3-f(x)$
1983 AIME Problems, 14
In the adjoining figure, two circles of radii 6 and 8 are drawn with their centers 12 units apart. At $P$, one of the points of intersection, a line is drawn in such a way that the chords $QP$ and $PR$ have equal length. Find the square of the length of $QP$.
[asy]unitsize(2.5mm);
defaultpen(linewidth(.8pt)+fontsize(12pt));
dotfactor=3;
pair O1=(0,0), O2=(12,0);
path C1=Circle(O1,8), C2=Circle(O2,6);
pair P=intersectionpoints(C1,C2)[0];
path C3=Circle(P,sqrt(130));
pair Q=intersectionpoints(C3,C1)[0];
pair R=intersectionpoints(C3,C2)[1];
draw(C1);
draw(C2);
//draw(O2--O1);
//dot(O1);
//dot(O2);
draw(Q--R);
label("$Q$",Q,N);
label("$P$",P,dir(80));
label("$R$",R,E);
//label("12",waypoint(O1--O2,0.4),S);[/asy]
2012 AMC 10, 1
Cagney can frost a cupcake every $20$ seconds and Lacey can frost a cupcake every $30$ seconds. Working together, how many cupcakes can they frost in $5$ minutes?
$ \textbf{(A)}\ 10
\qquad\textbf{(B)}\ 15
\qquad\textbf{(C)}\ 20
\qquad\textbf{(D)}\ 25
\qquad\textbf{(E)}\ 30
$
2020 AMC 10, 3
The ratio of $w$ to $x$ is $4 : 3$, the ratio of $y$ to $z$ is $3 : 2$, and the ratio of $z$ to $x$ is $1 : 6$. What is the ratio of $w$ to $y$?
$\textbf{(A) }4:3 \qquad \textbf{(B) }3:2 \qquad \textbf{(C) } 8:3 \qquad \textbf{(D) } 4:1 \qquad \textbf{(E) } 16:3 $