Found problems: 3632
2021 AMC 10 Fall, 6
Elmer the emu takes $44$ equal strides to walk between consecutive telephone poles on a rural road. Oscar the ostrich can cover the same distance in $12$ equal leaps. The telephone poles are evenly spaced, and the $41$st pole along this road is exactly one mile ($5280$ feet) from the first pole. How much longer, in feet, is Oscar's leap than Elmer's stride?
$\textbf{(A) }6\qquad\textbf{(B) }8\qquad\textbf{(C) }10\qquad\textbf{(D) }11\qquad\textbf{(E) }15$
1964 AMC 12/AHSME, 38
The sides $PQ$ and $PR$ of triangle $PQR$ are respectively of lengths $4$ inches, and $7$ inches. The median $PM$ is $3\frac{1}{2}$ inches. Then $QR$, in inches, is:
$\textbf{(A) }6\qquad\textbf{(B) }7\qquad\textbf{(C) }8\qquad\textbf{(D) }9\qquad \textbf{(E) }10$
2021 AMC 10 Fall, 10
Fourty slips of paper numbered $1$ to $40$ are placed in a hat. Alice and Bob each draw one number from the hat without replacement, keeping their numbers hidden from each other. Alice says, "I can't tell who has the larger number." Then Bob says, "I know who has the larger number." Alice says, "You do? Is your number prime?" Bob replies, "Yes." Alice says, "In that case, if I multiply your number by $100$ and add my number, the result is a perfect square. " What is the sum of the two numbers drawn from the hat?
$\textbf{(A) }27\qquad\textbf{(B) }37\qquad\textbf{(C) }47\qquad\textbf{(D) }57\qquad\textbf{(E) }67$
2019 AMC 12/AHSME, 7
What is the sum of all real numbers $x$ for which the median of the numbers $4,6,8,17,$ and $x$ is equal to the mean of those five numbers?
$\textbf{(A) } -5 \qquad\textbf{(B) } 0 \qquad\textbf{(C) } 5 \qquad\textbf{(D) } \frac{15}{4} \qquad\textbf{(E) } \frac{35}{4}$
2008 AMC 10, 14
Older television screens have an aspect ratio of $ 4: 3$. That is, the ratio of the width to the height is $ 4: 3$. The aspect ratio of many movies is not $ 4: 3$, so they are sometimes shown on a television screen by 'letterboxing' - darkening strips of equal height at the top and bottom of the screen, as shown. Suppose a movie has an aspect ratio of $ 2: 1$ and is shown on an older television screen with a $ 27$-inch diagonal. What is the height, in inches, of each darkened strip?
[asy]unitsize(1mm);
defaultpen(linewidth(.8pt));
filldraw((0,0)--(21.6,0)--(21.6,2.7)--(0,2.7)--cycle,grey,black);
filldraw((0,13.5)--(21.6,13.5)--(21.6,16.2)--(0,16.2)--cycle,grey,black);
draw((0,2.7)--(0,13.5));
draw((21.6,2.7)--(21.6,13.5));[/asy]$ \textbf{(A)}\ 2 \qquad \textbf{(B)}\ 2.25 \qquad \textbf{(C)}\ 2.5 \qquad \textbf{(D)}\ 2.7 \qquad \textbf{(E)}\ 3$
2024 AMC 12/AHSME, 24
What is the number of ordered triples $(a,b,c)$ of positive integers, with $a\le b\le c\le 9$, such that there exists a (non-degenerate) triangle $\triangle ABC$ with an integer inradius for which $a$, $b$, and $c$ are the lengths of the altitudes from $A$ to $\overline{BC}$, $B$ to $\overline{AC}$, and $C$ to $\overline{AB}$, respectively? (Recall that the inradius of a triangle is the radius of the largest possible circle that can be inscribed in the triangle.)
$
\textbf{(A) }2\qquad
\textbf{(B) }3\qquad
\textbf{(C) }4\qquad
\textbf{(D) }5\qquad
\textbf{(E) }6\qquad
$
1966 AMC 12/AHSME, 31
Triangle $ABC$ is inscribed in a circle with center $O'$. A circle with center $O$ is inscribed in triangle $ABC$. $AO$ is drawn, and extended to intersect the larger circle in $D$. Then, we must have:
$\text{(A)}\ CD=BD=O'D \qquad
\text{(B)}\ AO=CO=OD \qquad
\text{(C)}\ CD=CO=BD \qquad\\
\text{(D)}\ CD=OD=BD \qquad
\text{(E)}\ O'B=O'C=OD $
[asy]
size(200);
defaultpen(linewidth(0.8)+fontsize(12pt));
pair A=origin,B=(15,0),C=(5,9),O=incenter(A,B,C),Op=circumcenter(A,B,C);
path incirc = incircle(A,B,C),circumcirc = circumcircle(A,B,C),line=A--3*O;
pair D[]=intersectionpoints(circumcirc,line);
draw(A--B--C--A--D[0]^^incirc^^circumcirc);
dot(O^^Op,linewidth(4));
label("$A$",A,dir(185));
label("$B$",B,dir(355));
label("$C$",C,dir(95));
label("$D$",D[0],dir(O--D[0]));
label("$O$",O,NW);
label("$O'$",Op,E);[/asy]
2023 AIME, 1
The number of apples growing on each of six apple trees form an arithmetic sequence where the greatest number of apples growing on any of the six trees is double the least number of apples growing on any of the six trees. The total number of apples growing on all six trees is $990$. Find the greatest number of apples growing on any of the six trees.
2024 AIME, 10
Let $\triangle ABC$ have circumcenter $O$ and incenter $I$ with $\overline{IA}\perp\overline{OI}$, circumradius $13$, and inradius $6$. Find $AB\cdot AC$.
2008 AMC 12/AHSME, 14
What is the area of the region defined by the inequality $ |3x\minus{}18|\plus{}|2y\plus{}7|\le 3$?
$ \textbf{(A)}\ 3 \qquad
\textbf{(B)}\ \frac{7}{2} \qquad
\textbf{(C)}\ 4 \qquad
\textbf{(D)}\ \frac{9}{2} \qquad
\textbf{(E)}\ 5$
1961 AMC 12/AHSME, 9
Let $r$ be the result of doubling both the base and exponent of $a^b$, $b\neq 0$. If $r$ equals the product of $a^b$ by $x^b$, then $x$ equals:
${{ \textbf{(A)} a\qquad\textbf{(B)}\ 2a \qquad\textbf{(C)}\ 4a \qquad\textbf{(D)}\ 2}\qquad\textbf{(E)}\ 4} $
2003 AMC 10, 11
The sum of the two $ 5$-digit numbers $ AMC10$ and $ AMC12$ is $ 123422$. What is $ A\plus{}M\plus{}C$?
$ \textbf{(A)}\ 10 \qquad
\textbf{(B)}\ 11 \qquad
\textbf{(C)}\ 12 \qquad
\textbf{(D)}\ 13 \qquad
\textbf{(E)}\ 14$
1972 AMC 12/AHSME, 33
The minimum value of the quotient of a (base ten) number of three different nonzero digits divided by the sum of its digits is
$\textbf{(A) }9.7\qquad\textbf{(B) }10.1\qquad\textbf{(C) }10.5\qquad\textbf{(D) }10.9\qquad \textbf{(E) }20.5$
2015 AIME Problems, 3
There is a prime number $p$ such that $16p+1$ is the cube of a positive integer. Find $p$.
2009 AIME Problems, 2
There is a complex number $ z$ with imaginary part $ 164$ and a positive integer $ n$ such that
\[ \frac {z}{z \plus{} n} \equal{} 4i.
\]Find $ n$.
1989 AMC 12/AHSME, 20
Let $x$ be a real number selected uniformly at random between 100 and 200. If $\lfloor {\sqrt{x}} \rfloor = 12$, find the probability that $\lfloor {\sqrt{100x}} \rfloor = 120$. ($\lfloor {v} \rfloor$ means the greatest integer less than or equal to $v$.)
$\text{(A)} \ \frac{2}{25} \qquad \text{(B)} \ \frac{241}{2500} \qquad \text{(C)} \ \frac{1}{10} \qquad \text{(D)} \ \frac{96}{625} \qquad \text{(E)} \ 1$
2013 AMC 12/AHSME, 12
Cities $A$, $B$, $C$, $D$, and $E$ are connected by roads $\widetilde{AB}$, $\widetilde{AD}$, $\widetilde{AE}$, $\widetilde{BC}$, $\widetilde{BD}$, $\widetilde{CD}$, $\widetilde{DE}$. How many different routes are there from $A$ to $B$ that use each road exactly once? (Such a route will necessarily visit cities more than once.)
[asy]unitsize(10mm);
defaultpen(linewidth(1.2pt)+fontsize(10pt));
dotfactor=4;
pair A=(1,0), B=(4.24,0), C=(5.24,3.08), D=(2.62,4.98), E=(0,3.08);
dot (A);
dot (B);
dot (C);
dot (D);
dot (E);
label("$A$",A,S);
label("$B$",B,SE);
label("$C$",C,E);
label("$D$",D,N);
label("$E$",E,W);
guide squiggly(path g, real stepsize, real slope=45)
{
real len = arclength(g);
real step = len / round(len / stepsize);
guide squig;
for (real u = 0; u < len; u += step){
real a = arctime(g, u);
real b = arctime(g, u + step / 2);
pair p = point(g, a);
pair q = point(g, b);
pair np = unit( rotate(slope) * dir(g,a));
pair nq = unit( rotate(0 - slope) * dir(g,b));
squig = squig .. p{np} .. q{nq};
}
squig = squig .. point(g, length(g)){unit(rotate(slope)*dir(g,length(g)))};
return squig;
}
pen pp = defaultpen + 2.718;
draw(squiggly(A--B, 4.04, 30), pp);
draw(squiggly(A--D, 7.777, 20), pp);
draw(squiggly(A--E, 5.050, 15), pp);
draw(squiggly(B--C, 5.050, 15), pp);
draw(squiggly(B--D, 4.04, 20), pp);
draw(squiggly(C--D, 2.718, 20), pp);
draw(squiggly(D--E, 2.718, -60), pp);
[/asy]
$ \textbf{(A)}\ 7 \qquad \textbf{(B)}\ 9\qquad\textbf{(C)}\ 12\qquad\textbf{(D)}\ 16\qquad\textbf{(E)}\ 18 $
2006 AMC 10, 22
Two farmers agree that pigs are worth $ \$300$ and that goats are worth $ \$210$. When one farmer owes the other money, he pays the debt in pigs or goats, with ``change'' received in the form of goats or pigs as necessary. (For example, a $ \$390$ debt could be paid with two pigs, with one goat received in change.) What is the amount of the smallest positive debt that can be resolved in this way?
$ \textbf{(A) } \$5\qquad \textbf{(B) } \$10\qquad \textbf{(C) } \$30\qquad \textbf{(D) } \$90\qquad \textbf{(E) } \$210$
2008 AMC 12/AHSME, 2
A $ 4\times 4$ block of calendar dates is shown. The order of the numbers in the second row is to be reversed. Then the order of the numbers in the fourth row is to be reversed. Finally, the numbers on each diagonal are to be added. What will be the positive difference between the two diagonal sums?
\[ \setlength{\unitlength}{5mm} \begin{picture}(4,4)(0,0) \multiput(0,0)(0,1){5}{\line(1,0){4}} \multiput(0,0)(1,0){5}{\line(0,1){4}} \put(0,3){\makebox(1,1){\footnotesize{1}}} \put(1,3){\makebox(1,1){\footnotesize{2}}} \put(2,3){\makebox(1,1){\footnotesize{3}}} \put(3,3){\makebox(1,1){\footnotesize{4}}} \put(0,2){\makebox(1,1){\footnotesize{8}}} \put(1,2){\makebox(1,1){\footnotesize{9}}} \put(2,2){\makebox(1,1){\footnotesize{10}}} \put(3,2){\makebox(1,1){\footnotesize{11}}} \put(0,1){\makebox(1,1){\footnotesize{15}}} \put(1,1){\makebox(1,1){\footnotesize{16}}} \put(2,1){\makebox(1,1){\footnotesize{17}}} \put(3,1){\makebox(1,1){\footnotesize{18}}} \put(0,0){\makebox(1,1){\footnotesize{22}}} \put(1,0){\makebox(1,1){\footnotesize{23}}} \put(2,0){\makebox(1,1){\footnotesize{24}}} \put(3,0){\makebox(1,1){\footnotesize{25}}} \end{picture}
\]$ \textbf{(A)}\ 2 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 6 \qquad \textbf{(D)}\ 8 \qquad \textbf{(E)}\ 10$
2016 AMC 10, 12
Three distinct integers are selected at random between $1$ and $2016$, inclusive. Which of the following is a correct statement about the probability $p$ that the product of the three integers is odd?
$\textbf{(A)}\ p<\dfrac{1}{8}\qquad\textbf{(B)}\ p=\dfrac{1}{8}\qquad\textbf{(C)}\ \dfrac{1}{8}<p<\dfrac{1}{3}\qquad\textbf{(D)}\ p=\dfrac{1}{3}\qquad\textbf{(E)}\ p>\dfrac{1}{3}$
1970 AMC 12/AHSME, 2
A square and a circle have equal perimeters. The ratio of the area of the circle to the area of the square is:
$\textbf{(A) }\frac{4}{\pi}\qquad\textbf{(B) }\frac{\pi}{\sqrt{2}}\qquad\textbf{(C) }\frac{4}{1}\qquad\textbf{(D) }\frac{\sqrt{2}}{\pi}\qquad \textbf{(E) }\frac{\pi}{4}$
2000 AMC 10, 8
At Olympic High School, $\frac25$ of the freshmen and $\frac45$ of the sophomores took the AMC-10. Given that the number of freshmen and sophomore contestants was the same, which of the following must be true?
$\text{(A)}$ There are five times as many sophomores as freshmen.
$\text{(B)}$ There are twice as many sophomores as freshmen.
$\text{(C)}$ There are as many freshmen as sophomores.
$\text{(D)}$ There are twice as many freshmen as sophomores.
$\text{(E)}$ There are five times as many freshmen as sophomores.
2019 AMC 10, 24
Let $p$, $q$, and $r$ be the distinct roots of the polynomial $x^3 - 22x^2 + 80x - 67$. It is given that there exist real numbers $A$, $B$, and $C$ such that \[\dfrac{1}{s^3 - 22s^2 + 80s - 67} = \dfrac{A}{s-p} + \dfrac{B}{s-q} + \frac{C}{s-r}\] for all $s\not\in\{p,q,r\}$. What is $\tfrac1A+\tfrac1B+\tfrac1C$?
$\textbf{(A) }243\qquad\textbf{(B) }244\qquad\textbf{(C) }245\qquad\textbf{(D) }246\qquad\textbf{(E) } 247$
1976 USAMO, 1
(a) Suppose that each square of a 4 x 7 chessboard is colored either black or white. Prove that with [i]any[/i] such coloring, the board must contain a rectangle (formed by the horizontal and vertical lines of the board) whose four distinct unit corner squares are all of the same color.
(b) Exhibit a black-white coloring of a 4 x6 board in which the four corner squares of every rectangle, as described above, are not all of the same color.
2013 AMC 8, 18
Isabella uses one-foot cubical blocks to build a rectangular fort that is 12 feet long, 10 feet wide, and 5 feet high. The floor and the four walls are all one foot thick. How many blocks does the fort contain?
[asy]
import three;
size(3inch);
currentprojection=orthographic(-8,15,15);
triple A,B,C,D,E,F,G,H,I,J,K,L,M,N,O,P;
A = (0,0,0);
B = (0,10,0);
C = (12,10,0);
D = (12,0,0);
E = (0,0,5);
F = (0,10,5);
G = (12,10,5);
H = (12,0,5);
I = (1,1,1);
J = (1,9,1);
K = (11,9,1);
L = (11,1,1);
M = (1,1,5);
N = (1,9,5);
O = (11,9,5);
P = (11,1,5);
//outside box far
draw(surface(A--B--C--D--cycle),white,nolight);
draw(A--B--C--D--cycle);
draw(surface(E--A--D--H--cycle),white,nolight);
draw(E--A--D--H--cycle);
draw(surface(D--C--G--H--cycle),white,nolight);
draw(D--C--G--H--cycle);
//inside box far
draw(surface(I--J--K--L--cycle),white,nolight);
draw(I--J--K--L--cycle);
draw(surface(I--L--P--M--cycle),white,nolight);
draw(I--L--P--M--cycle);
draw(surface(L--K--O--P--cycle),white,nolight);
draw(L--K--O--P--cycle);
//inside box near
draw(surface(I--J--N--M--cycle),white,nolight);
draw(I--J--N--M--cycle);
draw(surface(J--K--O--N--cycle),white,nolight);
draw(J--K--O--N--cycle);
//outside box near
draw(surface(A--B--F--E--cycle),white,nolight);
draw(A--B--F--E--cycle);
draw(surface(B--C--G--F--cycle),white,nolight);
draw(B--C--G--F--cycle);
//top
draw(surface(E--H--P--M--cycle),white,nolight);
draw(surface(E--M--N--F--cycle),white,nolight);
draw(surface(F--N--O--G--cycle),white,nolight);
draw(surface(O--G--H--P--cycle),white,nolight);
draw(M--N--O--P--cycle);
draw(E--F--G--H--cycle);
label("10",(A--B),SE);
label("12",(C--B),SW);
label("5",(F--B),W);[/asy]
$\textbf{(A)}\ 204 \qquad \textbf{(B)}\ 280 \qquad \textbf{(C)}\ 320 \qquad \textbf{(D)}\ 340 \qquad \textbf{(E)}\ 600$