Found problems: 3632
2021 AMC 12/AHSME Fall, 9
Triangle $ABC$ is equilateral with side length $6$. Suppose that $O$ is the center of the inscribed circle of this triangle. What is the area of the circle passing through $A$, $O$, and $C$?
$\textbf{(A)}\ 9\pi \qquad\textbf{(B)}\ 12\pi \qquad\textbf{(C)}\ 18\pi \qquad\textbf{(D)}\
24\pi \qquad\textbf{(E)}\ 27\pi$
2019 AMC 12/AHSME, 2
Suppose $a$ is $150\%$ of $b$. What percent of $a$ is $3b$?
$\textbf{(A) } 50 \qquad \textbf{(B) } 66\frac{2}{3} \qquad \textbf{(C) } 150 \qquad \textbf{(D) } 200 \qquad \textbf{(E) } 450$
1997 AMC 8, 7
The area of the smallest square that will contain a circle of radius 4 is
$\textbf{(A)}\ 8 \qquad \textbf{(B)}\ 16 \qquad \textbf{(C)}\ 32 \qquad \textbf{(D)}\ 64 \qquad \textbf{(E)}\ 128$
2024 AMC 10, 25
The figure below shows a dotted grid $8$ cells wide and $3$ cells tall consisting of $1''\times1''$ squares. Carl places $1$-inch toothpicks along some of the sides of the squares to create a closed loop that does not intersect itself. The numbers in the cells indicate the number of sides of that square that are to be covered by toothpicks, and any number of toothpicks are allowed if no number is written. In how many ways can Carl place the toothpicks? [asy]
size(6cm);
for (int i=0; i<9; ++i) {
draw((i,0)--(i,3),dotted);
}
for (int i=0; i<4; ++i){
draw((0,i)--(8,i),dotted);
}
for (int i=0; i<8; ++i) {
for (int j=0; j<3; ++j) {
if (j==1) {
label("1",(i+0.5,1.5));
}}}
[/asy] $\textbf{(A) }130\qquad\textbf{(B) }144\qquad\textbf{(C) }146\qquad\textbf{(D) }162\qquad\textbf{(E) }196$
1990 AMC 12/AHSME, 26
Ten people form a circle. Each picks a number and tells it to the two neighbors adjacent to him in the circle. Then each person computes and announces the average of the numbers of his two neighbors. The figure shows the average announced by each person ([u]not[/u] the original number the person picked). The number picked by the person who announced the average $6$ was
[asy]
label("(1)", (0,.9));
label("(2)", (.4,.65));
label("(3)", (.8,.25));
label("(4)", (.8,-.2));
label("(5)", (.4,-.65));
label("(6)", (0,-.9));
label("(7)", (-.4,-.65));
label("(8)", (-.8,-.2));
label("(9)", (-.8,.25));
label("(10)", (-.4,.65));
[/asy]
$ \textbf{(A)}\ 1 \qquad\textbf{(B)}\ 5 \qquad\textbf{(C)}\ 6 \qquad\textbf{(D)}\ 10 \qquad\textbf{(E)}\ \text{not uniquely determined from the given information} $
2024 AMC 10, 11
How many ordered pairs of integers $(m, n)$ satisfy $\sqrt{n^2 - 49} = m$?
$
\textbf{(A) }1 \qquad
\textbf{(B) }2 \qquad
\textbf{(C) }3 \qquad
\textbf{(D) }4 \qquad
\textbf{(E) } \text{Infinitely many} \qquad
$
2008 AMC 10, 20
The faces of a cubical die are marked with the numbers $ 1$, $ 2$, $ 2$, $ 3$, $ 3$, and $ 4$. The faces of a second cubical die are marked with the numbers $ 1$, $ 3$, $ 4$, $ 5$, $ 6$, and $ 8$. Both dice are thrown. What is the probability that the sum of the two top numbers will be $ 5$, $ 7$, or $ 9$ ?
$ \textbf{(A)}\ \frac {5}{18} \qquad \textbf{(B)}\ \frac {7}{18} \qquad \textbf{(C)}\ \frac {11}{18} \qquad \textbf{(D)}\ \frac {3}{4} \qquad \textbf{(E)}\ \frac {8}{9}$
1997 AMC 8, 4
Julie is preparing a speech for her class. Her speech must last between one-half hour and three-quarters of an hour. The ideal rate of speech is 150 words per minute. If Julie speaks at the ideal rate, which of the following number of words would be an appropriate length for her speech?
$\textbf{(A)}\ 2250 \qquad \textbf{(B)}\ 3000 \qquad \textbf{(C)}\ 4200 \qquad \textbf{(D)}\ 4350 \qquad \textbf{(E)}\ 5650$
2012 Hanoi Open Mathematics Competitions, 11
[Help me] Suppose that the equation $x^3+px^2+qx+r = 0$ has 3 real roots $x_1; x_2; x_3$; where p; q; r are integer numbers. Put $S_n = x_1^n+x_2^n+x_3^n$ ; n = 1; 2; : : : Prove that $S_{2012}$ is an integer.
2005 AIME Problems, 8
The equation \[2^{333x-2}+2^{111x+2}=2^{222x+1}+1\] has three real roots. Given that their sum is $m/n$ where $m$ and $n$ are relatively prime positive integers, find $m+n$.
1971 AMC 12/AHSME, 31
[asy]
size(2.5inch);
pair A = (-2,0), B = 2dir(150), D = (2,0), C;
draw(A..(0,2)..D--cycle);
C = intersectionpoint(A..(0,2)..D,Circle(B,arclength(A--B)));
draw(A--B--C--D--cycle);
label("$A$",A,W);
label("$B$",B,NW);
label("$C$",C,N);
label("$D$",D,E);
label("$4$",A--D,S);
label("$1$",A--B,E);
label("$1$",B--C,SE);
//Credit to chezbgone2 for the diagram[/asy]
Quadrilateral $ABCD$ is inscribed in a circle with side $AD$, a diameter of length $4$. If sides $AB$ and $BC$ each have length $1$, then side $CD$ has length
$\textbf{(A) }\frac{7}{2}\qquad\textbf{(B) }\frac{5\sqrt{2}}{2}\qquad\textbf{(C) }\sqrt{11}\qquad\textbf{(D) }\sqrt{13}\qquad \textbf{(E) }2\sqrt{3}$
2006 AMC 10, 5
Doug and Dave shared a pizza with $ 8$ equally-sized slices. Doug wanted a plain pizza, but Dave wanted anchovies on half the pizza. The cost of a plain pizza was $ \$8$, and there was an additional cost of $ \$2$ for putting anchovies on one half. Dave ate all the slices of anchovy pizza and one plain slice. Doug ate the remainder. Each paid for what he had eaten. How many more dollars did Dave pay than Doug?
$ \textbf{(A) } 1\qquad \textbf{(B) } 2\qquad \textbf{(C) } 3\qquad \textbf{(D) } 4\qquad \textbf{(E) } 5$
1961 AMC 12/AHSME, 26
For a given arithmetic series the sum of the first $50$ terms is $200$, and the sum of the next $50$ terms is $2700$. The first term in the series is:
${{ \textbf{(A)}\ -1221 \qquad\textbf{(B)}\ -21.5 \qquad\textbf{(C)}\ -20.5 \qquad\textbf{(D)}\ 3 }\qquad\textbf{(E)}\ 3.5 } $
2008 AIME Problems, 6
A triangular array of numbers has a first row consisting of the odd integers $ 1,3,5,\ldots,99$ in increasing order. Each row below the first has one fewer entry than the row above it, and the bottom row has a single entry. Each entry in any row after the top row equals the sum of the two entries diagonally above it in the row immediately above it. How many entries in the array are multiples of $ 67$?
[asy]size(200);
defaultpen(fontsize(10));
label("1", origin);
label("3", (2,0));
label("5", (4,0));
label("$\cdots$", (6,0));
label("97", (8,0));
label("99", (10,0));
label("4", (1,-1));
label("8", (3,-1));
label("12", (5,-1));
label("196", (9,-1));
label(rotate(90)*"$\cdots$", (6,-2));[/asy]
1970 AMC 12/AHSME, 6
The smallest value of $x^2+8x$ for real values of $x$ is:
$\textbf{(A) }-16.25\qquad\textbf{(B) }-16\qquad\textbf{(C) }-15\qquad\textbf{(D) }-8\qquad \textbf{(E) }\text{None of these}$
2024 AMC 10, 25
Each of $27$ bricks (right rectangular prisms) has dimensions $a \times b \times c$, where $a$, $b$, and $c$ are pairwise relatively prime positive integers. These bricks are arranged to form a $3 \times 3 \times 3$ block, as shown on the left below. A $28$[sup]th[/sup] brick with the same dimensions is introduced, and these bricks are reconfigured into a $2 \times 2 \times 7$ block, shown on the right. The new block is $1$ unit taller, $1$ unit wider, and $1$ unit deeper than the old one. What is $a + b + c$?
[img]https://cdn.artofproblemsolving.com/attachments/2/d/b18d3d0a9e5005c889b34e79c6dab3aaefeffd.png[/img]
$
\textbf{(A) }88 \qquad
\textbf{(B) }89 \qquad
\textbf{(C) }90 \qquad
\textbf{(D) }91 \qquad
\textbf{(E) }92 \qquad
$
2016 AMC 12/AHSME, 6
A triangular array of $2016$ coins has $1$ coin in the first row, $2$ coins in the second row, $3$ coins in the third row, and so on up to $N$ coins in the $N$th row. What is the sum of the digits of $N$?
$\textbf{(A)}\ 6\qquad\textbf{(B)}\ 7\qquad\textbf{(C)}\ 8\qquad\textbf{(D)}\ 9\qquad\textbf{(E)}\ 10$
2020 AMC 10, 12
The decimal representation of $$\dfrac{1}{20^{20}}$$ consists of a string of zeros after the decimal point, followed by a 9 and then several more digits. How many zeros are in that initial string of zeros after the decimal point?
$\textbf{(A) }23\qquad\textbf{(B) }24\qquad\textbf{(C) }25\qquad\textbf{(D) }26\qquad\textbf{(E) }27$
2013 AMC 10, 10
A flower bouquet contains pink roses, red roses, pink carnations, and red carnations. One third of the pink flowers are roses, three fourths of the red flowers are carnations, and six tenths of the flowers are pink. What percent of the flowers are carnations?
$ \textbf{(A)}\ 15\qquad\textbf{(B)}\ 30\qquad\textbf{(C)}\ 40\qquad\textbf{(D)}\ 60\qquad\textbf{(E)}\ 70 $
2015 AMC 12/AHSME, 22
Six chairs are evenly spaced around a circular table. One person is seated in each chair. Each person gets up and sits down in a chair that is not the same chair and is not adjacent to the chair he or she originally occupied, so that again one person is seated in each chair. In how many ways can this be done?
$ \textbf{(A) }14\qquad\textbf{(B) }16\qquad\textbf{(C) }18\qquad\textbf{(D) }20\qquad\textbf{(E) }24 $
2012 AMC 10, 22
The sum of the first $m$ positive odd integers is $212$ more than the sum of the first $n$ positive even integers. What is the sum of all possible values of $n$?
$ \textbf{(A)}\ 255
\qquad\textbf{(B)}\ 256
\qquad\textbf{(C)}\ 257
\qquad\textbf{(D)}\ 258
\qquad\textbf{(E)}\ 259
$
2014 Contests, 2
Let $\mathbb{Z}$ be the set of integers. Find all functions $f : \mathbb{Z} \rightarrow \mathbb{Z}$ such that \[xf(2f(y)-x)+y^2f(2x-f(y))=\frac{f(x)^2}{x}+f(yf(y))\] for all $x, y \in \mathbb{Z}$ with $x \neq 0$.
2009 AMC 10, 14
On Monday, Millie puts a quart of seeds, $ 25\%$ of which are millet, into a bird feeder. On each successive day she adds another quart of the same mix of seeds without removing any seeds that are left. Each day the birds eat only $ 25\%$ of the millet in the feeder, but they eat all of the other seeds. On which day, just after Millie has placed the seeds, will the birds find that more than half the seeds in the feeder are millet?
$ \textbf{(A)}\ \text{Tuesday}\qquad \textbf{(B)}\ \text{Wednesday}\qquad \textbf{(C)}\ \text{Thursday} \qquad \textbf{(D)}\ \text{Friday}\qquad \textbf{(E)}\ \text{Saturday}$
2013 Purple Comet Problems, 15
Let $a$, $b$, and $c$ be positive real numbers such that $a^2+b^2+c^2=989$ and $(a+b)^2+(b+c)^2+(c+a)^2=2013$. Find $a+b+c$.
2010 AMC 10, 1
What is $ 100(100\minus{}3) \minus{} (100 \cdot 100 \minus{} 3)$?
$ \textbf{(A)}\ \minus{}20,000 \qquad \textbf{(B)}\ \minus{}10,000 \qquad \textbf{(C)}\ \minus{}297 \qquad \textbf{(D)}\ \minus{}6 \qquad \textbf{(E)}\ 0$