Found problems: 3632
1994 AIME Problems, 5
Given a positive integer $n$, let $p(n)$ be the product of the non-zero digits of $n$. (If $n$ has only one digits, then $p(n)$ is equal to that digit.) Let \[ S=p(1)+p(2)+p(3)+\cdots+p(999). \] What is the largest prime factor of $S$?
2022 AMC 10, 15
Quadrilateral $ABCD$ with side lengths $AB=7, BC = 24, CD = 20, DA = 15$ is inscribed in a circle. The area interior to the circle but exterior to the quadrilateral can be written in the form $\frac{a\pi - b}{c}$, where $a, b,$ and $c$ are positive integers such that $a$ and $c$ have no common prime factor. What is $a+b+c$?
$\textbf{(A) } 260 \qquad \textbf{(B) } 855 \qquad \textbf{(C) } 1235 \qquad \textbf{(D) } 1565 \qquad \textbf{(E) } 1997$
1963 AMC 12/AHSME, 33
Given the line $y = \dfrac{3}{4}x + 6$ and a line $L$ parallel to the given line and $4$ units from it. A possible equation for $L$ is:
$\textbf{(A)}\ y = \dfrac{3}{4}x + 1 \qquad
\textbf{(B)}\ y = \dfrac{3}{4}x\qquad
\textbf{(C)}\ y = \dfrac{3}{4}x -\dfrac{2}{3} \qquad$
$
\textbf{(D)}\ y = \dfrac{3}{4}x -1 \qquad
\textbf{(E)}\ y = \dfrac{3}{4}x + 2$
2017 AMC 12/AHSME, 9
Let $S$ be the set of points $(x,y)$ in the coordinate plane such that two of the three quantities $3$, $x+2$, and $y-4$ are equal and the third of the three quantities is no greater than this common value. Which of the following is a correct description of $S$?
$\textbf{(A) } \text{a single point} \qquad \textbf{(B) } \text{two intersecting lines} \\ \\ \textbf{(C) } \text{three lines whose pairwise intersections are three distinct points} \\ \\ \textbf{(D) } \text{a triangle} \qquad \textbf{(E) } \text{three rays with a common endpoint}$
2008 AMC 10, 7
The fraction \[\frac {(3^{2008})^2 - (3^{2006})^2}{(3^{2007})^2 - (3^{2005})^2}\] simplifies to which of the following?
$ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ \frac {9}{4} \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ \frac {9}{2} \qquad \textbf{(E)}\ 9$
2014 AIME Problems, 5
Real numbers $r$ and $s$ are roots of $p(x)=x^3+ax+b$, and $r+4$ and $s-3$ are roots of $q(x)=x^3+ax+b+240$. Find the sum of all possible values of $|b|$.
1990 AMC 12/AHSME, 2
$\left(\dfrac{1}{4}\right)^{-\frac{1}{4}}=$
$\textbf{(A) }-16\qquad
\textbf{(B) }-\sqrt{2}\qquad
\textbf{(C) }-\dfrac{1}{16}\qquad
\textbf{(D) }-\dfrac{1}{256}\qquad
\textbf{(E) }\sqrt{2}$
1993 AMC 8, 7
$3^3+3^3+3^3 = $
$\text{(A)}\ 3^4 \qquad \text{(B)}\ 9^3 \qquad \text{(C)}\ 3^9 \qquad \text{(D)}\ 27^3 \qquad \text{(E)}\ 3^{27}$
2014 AMC 8, 12
A magazine printed photos of three celebrities along with three photos of the celebrities as babies. The baby pictures did not identify the celebrities. Readers were asked to match each celebrity with the correct baby pictures. What is the probability that a reader guessing at random will match all three correctly?
$\textbf{(A) }\frac{1}{9}\qquad\textbf{(B) }\frac{1}{6}\qquad\textbf{(C) }\frac{1}{4}\qquad\textbf{(D) }\frac{1}{3}\qquad \textbf{(E) }\frac{1}{2}$
2021 AMC 10 Fall, 19
A disk of radius $1$ rolls all the way around the inside of a square of side length $s>4$ and sweeps out a region of area $A$. A second disk of radius $1$ rolls all the way around the outside of the same square and sweeps out a region of area $2A$. The value of $s$ can be written as $a+\frac{b\pi}{c}$, where $a,b$, and $c$ are positive integers and $b$ and $c$ are relatively prime. What is $a+b+c$?
$\textbf{(A)} ~10\qquad\textbf{(B)} ~11\qquad\textbf{(C)} ~12\qquad\textbf{(D)} ~13\qquad\textbf{(E)} ~14$
2022 AMC 12/AHSME, 19
Suppose that 13 cards numbered $1, 2, 3, \dots, 13$ are arranged in a row. The task is to pick them up in numerically increasing order, working repeatedly from left to right. In the example below, cards 1, 2, 3 are picked up on the first pass, 4 and 5 on the second pass, 6 on the third pass, 7, 8, 9, 10 on the fourth pass, and 11, 12, 13 on the fifth pass. For how many of the $13!$ possible orderings of the cards will the $13$ cards be picked up in exactly two passes?
[asy]
size(11cm);
draw((0,0)--(2,0)--(2,3)--(0,3)--cycle);
label("7", (1,1.5));
draw((3,0)--(5,0)--(5,3)--(3,3)--cycle);
label("11", (4,1.5));
draw((6,0)--(8,0)--(8,3)--(6,3)--cycle);
label("8", (7,1.5));
draw((9,0)--(11,0)--(11,3)--(9,3)--cycle);
label("6", (10,1.5));
draw((12,0)--(14,0)--(14,3)--(12,3)--cycle);
label("4", (13,1.5));
draw((15,0)--(17,0)--(17,3)--(15,3)--cycle);
label("5", (16,1.5));
draw((18,0)--(20,0)--(20,3)--(18,3)--cycle);
label("9", (19,1.5));
draw((21,0)--(23,0)--(23,3)--(21,3)--cycle);
label("12", (22,1.5));
draw((24,0)--(26,0)--(26,3)--(24,3)--cycle);
label("1", (25,1.5));
draw((27,0)--(29,0)--(29,3)--(27,3)--cycle);
label("13", (28,1.5));
draw((30,0)--(32,0)--(32,3)--(30,3)--cycle);
label("10", (31,1.5));
draw((33,0)--(35,0)--(35,3)--(33,3)--cycle);
label("2", (34,1.5));
draw((36,0)--(38,0)--(38,3)--(36,3)--cycle);
label("3", (37,1.5));
[/asy]
$\textbf{(A) }4082\qquad\textbf{(B) }4095\qquad\textbf{(C) }4096\qquad\textbf{(D) }8178\qquad\textbf{(E) }8191$
2015 AMC 12/AHSME, 10
How many noncongruent integer-sided triangles with positive area and perimeter less than $15$ are neither equilateral, isosceles, nor right triangles?
$\textbf{(A) }3\qquad\textbf{(B) }4\qquad\textbf{(C) }5\qquad\textbf{(D) }6\qquad\textbf{(E) }7$
2019 AMC 12/AHSME, 13
How many ways are there to paint each of the integers $2, 3, \dots, 9$ either red, green, or blue so that each number has a different color from each of its proper divisors?
$\textbf{(A)}\ 144\qquad\textbf{(B)}\ 216\qquad\textbf{(C)}\ 256\qquad\textbf{(D)}\ 384\qquad\textbf{(E)}\ 432$
1967 AMC 12/AHSME, 17
If $r_1$ and $r_2$ are the distinct real roots of $x^2+px+8=0$, then it must follow that:
$\textbf{(A)}\ |r_1+r_2|>4\sqrt{2}\qquad
\textbf{(B)}\ |r_1|>3 \; \text{or} \; |r_2| >3 \\
\textbf{(C)}\ |r_1|>2 \; \text{and} \; |r_2|>2\qquad
\textbf{(D)}\ r_1<0 \; \text{and} \; r_2<0\qquad
\textbf{(E)}\ |r_1+r_2|<4\sqrt{2}$
1983 AMC 12/AHSME, 8
Let $f(x) = \frac{x+1}{x-1}$. Then for $x^2 \neq 1$, $f(-x)$ is
$ \textbf{(A)}\ \frac{1}{f(x)}\qquad\textbf{(B)}\ -f(x)\qquad\textbf{(C)}\ \frac{1}{f(-x)}\qquad\textbf{(D)}\ -f(-x)\qquad\textbf{(E)}\ f(x) $
1992 IMTS, 4
In an attempt to copy down from the board a sequence of six positive integers in arithmetic progression, a student wrote down the five numbers, \[ 113,137,149,155,173, \] accidentally omitting one. He later discovered that he also miscopied one of them. Can you help him and recover the original sequence?
1964 AMC 12/AHSME, 2
The graph of $x^2-4y^2=0$ is:
${{ \textbf{(A)}\ \text{a parabola} \qquad\textbf{(B)}\ \text{an ellipse} \qquad\textbf{(C)}\ \text{a pair of straight lines} \qquad\textbf{(D)}\ \text{a point} }\qquad\textbf{(E)}\ \text{none of these} } $
2007 AIME Problems, 9
Rectangle $ABCD$ is given with $AB=63$ and $BC=448.$ Points $E$ and $F$ lie on $AD$ and $BC$ respectively, such that $AE=CF=84.$ The inscribed circle of triangle $BEF$ is tangent to $EF$ at point $P,$ and the inscribed circle of triangle $DEF$ is tangent to $EF$ at point $Q.$ Find $PQ.$
1967 AMC 12/AHSME, 32
In quadrilateral $ABCD$ with diagonals $\overline{AC}$ and $\overline{BD}$ intersecting at $O$, $\overline{BO}=4$, $\overline{AO}=8$, $\overline{OC}=3$, and $\overline{AB}=6$. The length of $\overline{AD}$ is:
$\textbf{(A)}\ 9\qquad
\textbf{(B)}\ 10\qquad
\textbf{(C)}\ 6\sqrt{3}\qquad
\textbf{(D)}\ 8\sqrt{2}\qquad
\textbf{(E)}\ \sqrt{166}$
2013 AMC 10, 16
In $\triangle ABC$, medians $\overline{AD}$ and $\overline{CE}$ intersect at $P$, $PE=1.5$, $PD=2$, and $DE=2.5$. What is the area of $AEDC?$
[asy]
unitsize(75);
pathpen = black; pointpen=black;
pair A = MP("A", D((0,0)), dir(200));
pair B = MP("B", D((2,0)), dir(-20));
pair C = MP("C", D((1/2,1)), dir(100));
pair D = MP("D", D(midpoint(B--C)), dir(30));
pair E = MP("E", D(midpoint(A--B)), dir(-90));
pair P = MP("P", D(IP(A--D, C--E)), dir(150)*2.013);
draw(A--B--C--cycle);
draw(A--D--E--C);
[/asy]
$\textbf{(A)}\ 13 \qquad
\textbf{(B)}\ 13.5 \qquad
\textbf{(C)}\ 14 \qquad
\textbf{(D)}\ 14.5 \qquad
\textbf{(E)}\ 15 $
2012 AIME Problems, 5
Let $B$ be the set of all binary integers that can be written using exactly 5 zeros and 8 ones where leading zeros are allowed. If all possible subtractions are performed in which one element of $B$ is subtracted from another, find the number of times the answer 1 is obtained.
2008 AMC 12/AHSME, 7
While Steve and LeRoy are fishing $ 1$ mile from shore, their boat springs a leak, and water comes in at a constant rate of $ 10$ gallons per minute. The boat will sink if it takes in more than $ 30$ gallons of water. Steve starts rowing toward the shore at a constant rate of $ 4$ miles per hour while LeRoy bails water out of the boat. What is the slowest rate, in gallons per minute, at which LeRoy can bail if they are to reach the shore without sinking?
$ \textbf{(A)}\ 2 \qquad
\textbf{(B)}\ 4 \qquad
\textbf{(C)}\ 6 \qquad
\textbf{(D)}\ 8 \qquad
\textbf{(E)}\ 10$
2006 AMC 10, 7
Which of the following is equivalent to $ \displaystyle \sqrt {\frac {x}{1 \minus{} \frac {x \minus{} 1}{x}}}$ when $ x < 0$?
$ \textbf{(A) } \minus{} x \qquad \textbf{(B) } x \qquad \textbf{(C) } 1 \qquad \textbf{(D) } \sqrt {\frac x2} \qquad \textbf{(E) } x\sqrt { \minus{} 1}$
2012 AMC 10, 7
In a bag of marbles, $\frac{3}{5}$ of the marbles are blue and the rest are red. If the number of red marbles is doubled and the number of blue marbles stays the same, what fraction of the marbles will be red?
$ \textbf{(A)}\ \dfrac{2}{5}
\qquad\textbf{(B)}\ \dfrac{3}{7}
\qquad\textbf{(C)}\ \dfrac{4}{7}
\qquad\textbf{(D)}\ \dfrac{3}{5}
\qquad\textbf{(E)}\ \dfrac{4}{5}
$
1961 AMC 12/AHSME, 10
Each side of triangle $ABC$ is $12$ units. $D$ is the foot of the perpendicular dropped from $A$ on $BC$, and $E$ is the midpoint of $AD$. The length of $BE$, in the same unit, is:
${{ \textbf{(A)}\ \sqrt{18} \qquad\textbf{(B)}\ \sqrt{28} \qquad\textbf{(C)}\ 6 \qquad\textbf{(D)}\ \sqrt{63} }\qquad\textbf{(E)}\ \sqrt{98} } $