Found problems: 3632
2009 AMC 10, 9
Positive integers $ a$, $ b$, and $ 2009$, with $ a<b<2009$, form a geometric sequence with an integer ratio. What is $ a$?
$ \textbf{(A)}\ 7 \qquad
\textbf{(B)}\ 41 \qquad
\textbf{(C)}\ 49 \qquad
\textbf{(D)}\ 289 \qquad
\textbf{(E)}\ 2009$
1986 AMC 12/AHSME, 22
Six distinct integers are picked at random from $\{1,2,3,\ldots,10\}$. What is the probability that, among those selected, the second smallest is $3$?
$ \textbf{(A)}\ \frac{1}{60}\qquad\textbf{(B)}\ \frac{1}{6}\qquad\textbf{(C)}\ \frac{1}{3}\qquad\textbf{(D)}\ \frac{1}{2}\qquad\textbf{(E)}\ \text{none of these} $
2018 AMC 10, 7
For how many (not necessarily positive) integer values of $n$ is the value of $4000\cdot \left(\tfrac{2}{5}\right)^n$ an integer?
$
\textbf{(A) }3 \qquad
\textbf{(B) }4 \qquad
\textbf{(C) }6 \qquad
\textbf{(D) }8 \qquad
\textbf{(E) }9 \qquad
$
2014 AMC 10, 14
Danica drove her new car on a trip for a whole number of hours, averaging $55$ miles per hour. At the beginning of the trip, $abc$ miles were displayed on the odometer, where $abc$ is a 3-digit number with $a \ge 1$ and $a+b+c \le 7$. At the end of the trip, where the odometer showed $cba$ miles. What is $a^2+b^2+c^2$?
$ \textbf{(A) } 26 \qquad\textbf{(B) }27\qquad\textbf{(C) }36\qquad\textbf{(D) }37\qquad\textbf{(E) }41\qquad $
2009 AMC 10, 13
As shown below, convex pentagon $ ABCDE$ has sides $ AB \equal{} 3$, $ BC \equal{} 4$, $ CD \equal{} 6$, $ DE \equal{} 3$, and $ EA \equal{} 7$. The pentagon is originally positioned in the plane with vertex $ A$ at the origin and vertex $ B$ on the positive $ x$-axis. The pentagon is then rolled clockwise to the right along the $ x$-axis. Which side will touch the point $ x \equal{} 2009$ on the $ x$-axis?
[asy]size(250);
defaultpen(linewidth(.8pt)+fontsize(8pt));
dotfactor=4;
pair A=(0,0), Ep=7*dir(105), B=3*dir(0);
pair D=Ep+B;
pair C=intersectionpoints(Circle(D,6),Circle(B,4))[1];
pair[] ds={A,B,C,D,Ep};
dot(ds);
draw(B--C--D--Ep--A);
draw((6,6)..(8,4)..(8,3),EndArrow(3));
xaxis("$x$",-8,14,EndArrow(3));
label("$E$",Ep,NW);
label("$D$",D,NE);
label("$C$",C,E);
label("$B$",B+(.2,.1),ENE);
label("$A$",A+(-.1,.1),WNW);
label("$(0,0)$",A,S);
label("$3$",midpoint(A--B),N);
label("$4$",midpoint(B--C),NW);
label("$6$",midpoint(C--D),NE);
label("$3$",midpoint(D--Ep),S);
label("$7$",midpoint(Ep--A),W);[/asy]$ \textbf{(A)}\ \overline{AB} \qquad \textbf{(B)}\ \overline{BC} \qquad \textbf{(C)}\ \overline{CD} \qquad \textbf{(D)}\ \overline{DE} \qquad \textbf{(E)}\ \overline{EA}$
2023 AMC 12/AHSME, 23
How many ordered pairs of positive real numbers $(a,b)$ satisfy the equation
\[(1+2a)(2+2b)(2a+b) = 32ab?\]
$\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }3\qquad\textbf{(E) }\text{an infinite number}$
2021 AMC 12/AHSME Spring, 6
A deck of cards has only red cards and black cards. The probability of a randomly chosen card being red is $\frac13$. When $4$ black cards are added to the deck, the probability of choosing red becomes $\frac14$. How many cards were in the deck originally.
$\textbf{(A) }6 \qquad \textbf{(B) }9 \qquad \textbf{(C) }12 \qquad \textbf{(D) }15 \qquad \textbf{(E) }18$
2001 AIME Problems, 6
A fair die is rolled four times. The probability that each of the final three rolls is at least as large as the roll preceding it may be expressed in the form $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$
2017 AMC 12/AHSME, 20
Real numbers $x$ and $y$ are chosen independently and uniformly at random from the interval $(0,1)$. What is the probability that $\lfloor \log_2{x} \rfloor=\lfloor \log_2{y} \rfloor$, where $\lfloor r \rfloor$ denotes the greatest integer less than or equal to the real number $r$?
$\textbf{(A)}\ \frac{1}{8}\qquad\textbf{(B)}\ \frac{1}{6}\qquad\textbf{(C)}\ \frac{1}{4}\qquad\textbf{(D)}\ \frac{1}{3}\qquad\textbf{(E)}\ \frac{1}{2}$
2023 AMC 10, 10
You are playing a game. A $2 \times 1$ rectangle covers two adjacent squares (oriented either horizontally or vertically) of a $3 \times 3$ grid of squares, but you are not told which two squares are covered. Your goal is to find at least one square that is covered by the rectangle. A "turn" consists of you guessing a square, after which you are told whether that square is covered by the hidden rectangle. What is the minimum number of turns you need to ensue that at least one of your guessed squares is covered by the rectangle?
$\textbf{(A)}~3\qquad\textbf{(B)}~5\qquad\textbf{(C)}~4\qquad\textbf{(D)}~8\qquad\textbf{(E)}~6$
2024 AIME, 1
Every morning, Aya does a $9$ kilometer walk, and then finishes at the coffee shop. One day, she walks at $s$ kilometers per hour, and the walk takes $4$ hours, including $t$ minutes at the coffee shop. Another morning, she walks at $s+2$ kilometers per hour, and the walk takes $2$ hours and $24$ minutes, including $t$ minutes at the coffee shop. This morning, if she walks at $s+\frac12$ kilometers per hour, how many minutes will the walk take, including the $t$ minutes at the coffee shop?
2006 AMC 12/AHSME, 11
Joe and JoAnn each bought 12 ounces of coffee in a 16-ounce cup. Joe drank 2 ounces of his coffee and then added 2 ounces of cream. JoAnn added 2 ounces of cream, stirred the coffee well, and then drank 2 ounces. What is the resulting ratio of the amount of cream in Joe's coffee to that in JoAnn's coffee?
$ \textbf{(A) } \frac 67 \qquad \textbf{(B) } \frac {13}{14} \qquad \textbf{(C) } 1 \qquad \textbf{(D) } \frac {14}{13} \qquad \textbf{(E) } \frac 76$
2023 AMC 12/AHSME, 7
A digital display shows the current date as an $8$-digit integer consisting of a $4$-digit year, followed by a $2$-digit month, followed by a $2$-digit date within the month. For example, Arbor Day this year is displayed as 20230428. For how many dates in $2023$ will each digit appear an even number of times in the 8-digital display for that date?
$\textbf{(A)}~5\qquad\textbf{(B)}~6\qquad\textbf{(C)}~7\qquad\textbf{(D)}~8\qquad\textbf{(E)}~9$
2008 AIME Problems, 15
Find the largest integer $ n$ satisfying the following conditions:
(i) $ n^2$ can be expressed as the difference of two consecutive cubes;
(ii) $ 2n\plus{}79$ is a perfect square.
2022 AMC 10, 9
A rectangle is partitioned into 5 regions as shown. Each region is to be painted a solid color - red, orange, yellow, blue, or green - so that regions that touch are painted different colors, and colors can be used more than once. How many different colorings are possible?
[asy]
size(5.5cm);
draw((0,0)--(0,2)--(2,2)--(2,0)--cycle);
draw((2,0)--(8,0)--(8,2)--(2,2)--cycle);
draw((8,0)--(12,0)--(12,2)--(8,2)--cycle);
draw((0,2)--(6,2)--(6,4)--(0,4)--cycle);
draw((6,2)--(12,2)--(12,4)--(6,4)--cycle);
[/asy]
$\textbf{(A) }120\qquad\textbf{(B) }270\qquad\textbf{(C) }360\qquad\textbf{(D) }540\qquad\textbf{(E) }720$
2006 AMC 10, 10
In a triangle with integer side lengths, one side is three times as long as a second side, and the length of the third side is 15. What is the greatest possible perimeter of the triangle?
$ \textbf{(A) } 43 \qquad \textbf{(B) } 44 \qquad \textbf{(C) } 45 \qquad \textbf{(D) } 46 \qquad \textbf{(E) } 47$
1986 AMC 12/AHSME, 25
If $\lfloor x\rfloor$ is the greatest integer less than or equal to $x$, then \[\displaystyle\sum_{N=1}^{1024} \lfloor \log_{2}N\rfloor = \]
$ \textbf{(A)}\ 8192\qquad\textbf{(B)}\ 8204\qquad\textbf{(C)}\ 9218\qquad\textbf{(D)}\ \lfloor \log_{2}(1024!)\rfloor\qquad\textbf{(E)}\ \text{none of these} $
2022 AMC 12/AHSME, 15
One of the following numbers is not divisible by any prime number less than 10. Which is it?
(A) $2^{606} - 1 \ \ $ (B) $2^{606} + 1 \ \ $ (C) $2^{607} - 1 \ \ $ (D) $2^{607} + 1 \ \ $ (E) $2^{607} + 3^{607} \ \ $
2012 AMC 10, 17
Jesse cuts a circular paper disk of radius $12$ along two radii to form two sectors, the smaller having a central angle of $120$ degrees. He makes two circular cones, using each sector to form the lateral surface of a cone. What is the ratio of the volume of the smaller cone to that of the larger?
$ \textbf{(A)}\ \frac{1}{8} \qquad\textbf{(B)}\ \frac{1}{4} \qquad\textbf{(C)}\ \frac{\sqrt{10}}{10} \qquad\textbf{(D)}\ \frac{\sqrt{5}}{6} \qquad\textbf{(E)}\ \frac{\sqrt{10}}{5} $
2022 AMC 12/AHSME, 25
A circle with integer radius $r$ is centered at $(r, r)$. Distinct line segments of length $c_i$ connect points $(0, a_i)$ to $(b_i, 0)$ for $1 \le i \le 14$ and are tangent to the circle, where $a_i$, $b_i$, and $c_i$ are all positive integers and $c_1 \le c_2 \le \cdots \le c_{14}$. What is the ratio $\frac{c_{14}}{c_1}$ for the least possible value of $r$?
$\textbf{(A)} ~\frac{21}{5} \qquad\textbf{(B)} ~\frac{85}{13} \qquad\textbf{(C)} ~7 \qquad\textbf{(D)} ~\frac{39}{5} \qquad\textbf{(E)} ~17 $
1963 AMC 12/AHSME, 20
Two men at points $R$ and $S$, $76$ miles apart, set out at the same time to walk towards each other. The man at $R$ walks uniformly at the rate of $4\dfrac{1}{2}$ miles per hour; the man at $S$ walks at the constant rate of $3\dfrac{1}{4}$ miles per hour for the first hour, at $3\dfrac{3}{4}$ miles per hour for the second hour,
and so on, in arithmetic progression. If the men meet $x$ miles nearer $R$ than $S$ in an integral number of hours, then $x$ is:
$\textbf{(A)}\ 10 \qquad
\textbf{(B)}\ 8 \qquad
\textbf{(C)}\ 6 \qquad
\textbf{(D)}\ 4 \qquad
\textbf{(E)}\ 2$
2013 AMC 10, 2
Alice is making a batch of cookies and needs $2 \frac{1}{2}$ cups of sugar. Unforunately, her measuring cup holds only $\frac{1}{4}$ cup of sugar. How many times must she fill that cup to get the correct amount of sugar?
$ \textbf{(A)}\ 8 \qquad\textbf{(B)}\ 10 \qquad\textbf{(C)}\ 12 \qquad\textbf{(D)}\ 16 \qquad\textbf{(E)}\ 20$
1978 USAMO, 2
$ABCD$ and $A'B'C'D'$ are square maps of the same region, drawn to different scales and superimposed as shown in the figure. Prove that there is only one point $O$ on the small map that lies directly over point $O'$ of the large map such that $O$ and $O'$ each represent the same place of the country. Also, give a Euclidean construction (straight edge and compass) for $O$.
[asy]
size(200);
defaultpen(linewidth(0.7)+fontsize(10));
real theta = -100, r = 0.3; pair D2 = (0.3,0.76);
string[] lbl = {'A', 'B', 'C', 'D'}; draw(unitsquare); draw(shift(D2)*rotate(theta)*scale(r)*unitsquare);
for(int i = 0; i < lbl.length; ++i) {
pair Q = dir(135-90*i), P = (.5,.5)+Q/2^.5;
label("$"+lbl[i]+"'$", P, Q);
label("$"+lbl[i]+"$",D2+rotate(theta)*(r*P), rotate(theta)*Q);
}[/asy]
1976 AMC 12/AHSME, 11
Which of the following statements is (are) equivalent to the statement "If the pink elephant on planet alpha has purple eyes, then the wild pig on planet beta does not have a long nose"?
$\textbf{I. }$ "If the wild pig on planet beta has a long nose, then the pink elephant on planet alpha has purple eyes."
$\textbf{II. }$ "If the pink elephant on planet alpha does not have purple eyes, then the wild pig on planet beta does not have a long nose.
$\textbf{III. }$ "If the wild pig on planet beta has a long nose, then the pink elephant on planet alpha does not have purple eyes."
$ \textbf{IV. }$ "The pink elephant on planet alpha does not have purple eyes, or the wild pig on planet beta does not have a long nose."
$\textbf{(A) }\textbf{I. }\text{and }\textbf{II. }\text{only}\qquad\textbf{(B) }\textbf{III. }\text{and }\textbf{IV. }\text{only}\qquad\textbf{(C) }\textbf{II. }\text{and }\textbf{IV. }\text{only}\qquad\textbf{(D) }\textbf{II. }\text{and }\textbf{III. }\text{only}\qquad \textbf{(E) }\text{and }\textbf{III. }\text{only}$
2009 AMC 12/AHSME, 13
A ship sails $ 10$ miles in a straight line from $ A$ to $ B$, turns through an angle between $ 45^{\circ}$ and $ 60^{\circ}$, and then sails another $ 20$ miles to $ C$. Let $ AC$ be measured in miles. Which of the following intervals contains $ AC^2$?
[asy]unitsize(2mm);
defaultpen(linewidth(.8pt)+fontsize(10pt));
dotfactor=4;
pair B=(0,0), A=(-10,0), C=20*dir(50);
draw(A--B--C);
draw(A--C,linetype("4 4"));
dot(A);
dot(B);
dot(C);
label("$10$",midpoint(A--B),S);
label("$20$",midpoint(B--C),SE);
label("$A$",A,SW);
label("$B$",B,SE);
label("$C$",C,NE);[/asy]$ \textbf{(A)}\ [400,500] \qquad \textbf{(B)}\ [500,600] \qquad \textbf{(C)}\ [600,700] \qquad \textbf{(D)}\ [700,800]$
$ \textbf{(E)}\ [800,900]$