Found problems: 3632
2008 AMC 12/AHSME, 6
Postman Pete has a pedometer to count his steps. The pedometer records up to $ 99999$ steps, then flips over to $ 00000$ on the next step. Pete plans to determine his mileage for a year. On January $ 1$ Pete sets the pedometer to $ 00000$. During the year, the pedometer flips from $ 99999$ to $ 00000$ forty-four times. On December $ 31$ the pedometer reads $ 50000$. Pete takes $ 1800$ steps per mile. Which of the following is closest to the number of miles Pete walked during the year?
$ \textbf{(A)}\ 2500 \qquad
\textbf{(B)}\ 3000 \qquad
\textbf{(C)}\ 3500 \qquad
\textbf{(D)}\ 4000 \qquad
\textbf{(E)}\ 4500$
2016 AMC 10, 1
What is the value of $\dfrac{11!-10!}{9!}$?
$\textbf{(A)}\ 99\qquad\textbf{(B)}\ 100\qquad\textbf{(C)}\ 110\qquad\textbf{(D)}\ 121\qquad\textbf{(E)}\ 132$
2012 AIME Problems, 13
Equilateral $\triangle ABC$ has side length $\sqrt{111}$. There are four distinct triangles $AD_1E_1$, $AD_1E_2$, $AD_2E_3$, and $AD_2E_4$, each congruent to $\triangle ABC$, with $BD_1 = BD_2=\sqrt{11}$. Find $\sum^4_{k=1}(CE_k)^2$.
1974 AMC 12/AHSME, 26
The number of distinct positive integral divisors of $(30)^4$ excluding $1$ and $(30)^4$ is
$ \textbf{(A)}\ 100 \qquad\textbf{(B)}\ 125 \qquad\textbf{(C)}\ 123 \qquad\textbf{(D)}\ 30 \qquad\textbf{(E)}\ \text{none of these} $
1969 AMC 12/AHSME, 22
Let $K$ be the measure of the area bounded by the $x$-axis, the line $x=8$, and the curve defined by \[f=\{(x,y)\,|\, y=x\text{ when }0\leq x\leq 5,\,y=2x-5\text{ when }5\leq x\leq 8\}.\] Then $K$ is:
$\textbf{(A) }21.5\qquad
\textbf{(B) }36.4\qquad
\textbf{(C) }36.5\qquad
\textbf{(D) }44\qquad$
$\textbf{ (E) }\text{less than 44 but arbitrarily close to it.}$
1987 AMC 12/AHSME, 21
There are two natural ways to inscribe a square in a given isosceles right triangle. If it is done as in Figure 1 below, then one finds that the area of the square is $441 \text{cm}^2$. What is the area (in $\text{cm}^2$) of the square inscribed in the same $\triangle ABC$ as shown in Figure 2 below?
[asy]
draw((0,0)--(10,0)--(0,10)--cycle);
draw((-25,0)--(-15,0)--(-25,10)--cycle);
draw((-20,0)--(-20,5)--(-25,5));
draw((6.5,3.25)--(3.25,0)--(0,3.25)--(3.25,6.5));
label("A", (-25,10), W);
label("B", (-25,0), W);
label("C", (-15,0), E);
label("Figure 1", (-20, -5));
label("Figure 2", (5, -5));
label("A", (0,10), W);
label("B", (0,0), W);
label("C", (10,0), E);
[/asy]
$ \textbf{(A)}\ 378 \qquad\textbf{(B)}\ 392 \qquad\textbf{(C)}\ 400 \qquad\textbf{(D)}\ 441 \qquad\textbf{(E)}\ 484 $
2008 AIME Problems, 4
There exist unique positive integers $ x$ and $ y$ that satisfy the equation $ x^2 \plus{} 84x \plus{} 2008 \equal{} y^2$. Find $ x \plus{} y$.
2020 AMC 12/AHSME, 24
Suppose that $\triangle ABC$ is an equilateral triangle of side length $s$, with the property that there is a unique point $P$ inside the triangle such that $AP = 1$, $BP = \sqrt{3}$, and $CP = 2$. What is $s?$
$\textbf{(A) } 1 + \sqrt{2} \qquad \textbf{(B) } \sqrt{7} \qquad \textbf{(C) } \frac{8}{3} \qquad \textbf{(D) } \sqrt{5 + \sqrt{5}} \qquad \textbf{(E) } 2\sqrt{2}$
2012 AMC 12/AHSME, 25
Let $S=\{(x,y) : x \in \{0,1,2,3,4\}, y \in \{0,1,2,3,4,5\}$, and $(x,y) \neq (0,0) \}$. Let $T$ be the set of all right triangles whose vertices are in $S$. For every right triangle $t=\triangle ABC$ with vertices $A$, $B$, and $C$ in counter-clockwise order and right angle at $A$, let $f(t)= \tan (\angle CBA)$. What is
\[ \displaystyle \prod_{t \in T} f(t) \text{?} \]
[asy]
size((120));
dot((1,0));
dot((2,0));
dot((3,0));
dot((4,0));
dot((0,1));
dot((0,2));
dot((0,3));
dot((0,4));
dot((0,5));
dot((1,1));
dot((1,2));
dot((1,3));
dot((1,4));
dot((1,5));
dot((2,1));
dot((2,2));
dot((2,3));
dot((2,4));
dot((2,5));
dot((3,1));
dot((3,2));
dot((3,3));
dot((3,4));
dot((3,5));
dot((4,1));
dot((4,2));
dot((4,3));
dot((4,4));
dot((4,5));
label("$\circ$", (0,0));
label("$S$", (-.7,2.5));
[/asy]
$\textbf{(A)}\ 1 \qquad \textbf{(B)}\ \frac{625}{144} \qquad \textbf{(C)}\ \frac{125}{24} \qquad \textbf{(D)}\ 6 \qquad \textbf{(E)}\ \frac{625}{24}$
2011 AMC 10, 3
Suppose $[a \,\,\, b]$ denotes the average of $a$ and $b$, and $\{a\,\,\,b\,\,\,c\}$ denotes the average of $a$, $b$, and $c$. What is $\{\{1\,\,\, 1\,\,\, 0\}\,\,\, [0\,\,\, 1]\,\,\, 0\}$?
$ \textbf{(A)}\ \frac{2}{9} \qquad\textbf{(B)}\ \frac{5}{18} \qquad\textbf{(C)}\ \frac{1}{3} \qquad\textbf{(D)}\ \frac{7}{18} \qquad\textbf{(E)}\ \frac{2}{3} $
2018 AMC 12/AHSME, 25
For a positive integer $n$ and nonzero digits $a$, $b$, and $c$, let $A_n$ be the $n$-digit integer each of whose digits is equal to $a$; let $B_n$ be the $n$-digit integer each of whose digits is equal to $b$, and let $C_n$ be the $2n$-digit (not $n$-digit) integer each of whose digits is equal to $c$. What is the greatest possible value of $a + b + c$ for which there are at least two values of $n$ such that $C_n - B_n = A_n^2$?
$\textbf{(A)} \text{ 12} \qquad \textbf{(B)} \text{ 14} \qquad \textbf{(C)} \text{ 16} \qquad \textbf{(D)} \text{ 18} \qquad \textbf{(E)} \text{ 20}$
1992 AMC 12/AHSME, 1
$6^{6} + 6^{6} + 6^{6} + 6^{6} + 6^{6} + 6^{6} = $
$ \textbf{(A)}\ 6^{6}\qquad\textbf{(B)}\ 6^{7}\qquad\textbf{(C)}\ 36^{6}\qquad\textbf{(D)}\ 6^{36}\qquad\textbf{(E)}\ 36^{36} $
2019 AMC 10, 11
Two jars each contain the same number of marbles, and every marble is either blue or green. In Jar 1 the ratio of blue to green marbles is 9:1, and the ratio of blue to green marbles in Jar 2 is 8:1. There are 95 green marbles in all. How many more blue marbles are in Jar 1 than in Jar 2?
$\textbf{(A) } 5 \qquad\textbf{(B) } 10 \qquad\textbf{(C) } 25 \qquad\textbf{(D) } 45 \qquad\textbf{(E) } 50$
2011 AMC 12/AHSME, 12
A dart board is a regular octagon divided into regions as shown. Suppose that a dart thrown at the board is equally likely to land anywhere on the board. What is probability that the dart lands within the center square?
[asy]
unitsize(10mm);
defaultpen(linewidth(.8pt)+fontsize(10pt));
dotfactor=4;
pair A=(0,1), B=(1,0), C=(1+sqrt(2),0), D=(2+sqrt(2),1), E=(2+sqrt(2),1+sqrt(2)), F=(1+sqrt(2),2+sqrt(2)), G=(1,2+sqrt(2)), H=(0,1+sqrt(2));
draw(A--B--C--D--E--F--G--H--cycle);
draw(A--D);
draw(B--G);
draw(C--F);
draw(E--H);
[/asy]
$ \textbf{(A)}\ \frac{\sqrt{2} - 1}{2} \qquad\textbf{(B)}\ \frac{1}{4} \qquad\textbf{(C)}\ \frac{2 - \sqrt{2}}{2} \qquad\textbf{(D)}\ \frac{\sqrt{2}}{4} \qquad\textbf{(E)}\ 2 - \sqrt{2}$
1993 AMC 8, 24
What number is directly above $142$ in this array of numbers?
\[\begin{array}{cccccc}
& & & 1 & & \\
& & 2 & 3 & 4 & \\
& 5 & 6 & 7 & 8 & 9 \\
10 & 11 & 12 & \cdots & & \\
\end{array}\]
$\textbf{(A)}\ 99 \qquad \textbf{(B)}\ 119 \qquad \textbf{(C)}\ 120 \qquad \textbf{(D)}\ 121 \qquad \textbf{(E)}\ 122$
2024 AIME, 2
Real numbers $x$ and $y$ with $x,y>1$ satisfy $\log_x(y^x)=\log_y(x^{4y})=10.$ What is the value of $xy$?
2020 AMC 12/AHSME, 23
Jason rolls three fair standard six-sided dice. Then he looks at the rolls and chooses a subset of the dice (possibly empty, possibly all three dice) to reroll. After rerolling, he wins if and only if the sum of the numbers face up on the three dice is exactly $7$. Jason always plays to optimize his chances of winning. What is the probability that he chooses to reroll exactly two of the dice?
$\textbf{(A) } \frac{7}{36} \qquad\textbf{(B) } \frac{5}{24} \qquad\textbf{(C) } \frac{2}{9} \qquad\textbf{(D) } \frac{17}{72} \qquad\textbf{(E) } \frac{1}{4}$
2021 AMC 10 Fall, 22
For each integer $ n\geq 2 $, let $ S_n $ be the sum of all products $ jk $, where $ j $ and $ k $ are integers and $ 1\leq j<k\leq n $. What is the sum of the 10 least values of $ n $ such that $ S_n $ is divisible by $ 3 $?
$\textbf{(A) }196\qquad\textbf{(B) }197\qquad\textbf{(C) }198\qquad\textbf{(D) }199\qquad\textbf{(E) }200$
2014 AMC 10, 15
David drives from his home to the airport to catch a flight. He drives $35$ miles in the first hour, but realizes that he will be $1$ hour late if he continues at this speed. He increases his speed by $15$ miles per hour for the rest of the way to the airport and arrives $30$ minutes early. How many miles is the airport from his home?
$\textbf{(A) }140\qquad
\textbf{(B) }175\qquad
\textbf{(C) }210\qquad
\textbf{(D) }245\qquad
\textbf{(E) }280\qquad$
2021 AMC 12/AHSME Spring, 19
How many solutions does the equation $\sin \left( \frac{\pi}2 \cos x\right)=\cos \left( \frac{\pi}2 \sin x\right)$ have in the closed interval $[0,\pi]$?
$\textbf{(A) }0 \qquad \textbf{(B) }1 \qquad \textbf{(C) }2 \qquad \textbf{(D) }3\qquad \textbf{(E) }4$
2023 AMC 12/AHSME, 11
What is the maximum area of an isosceles trapezoid that has legs of length $1$ and one base twice as long as the other?
$
\textbf{(A) }\frac 54 \qquad \textbf{(B) } \frac 87 \qquad \textbf{(C)} \frac{5\sqrt2}4 \qquad \textbf{(D) } \frac 32 \qquad \textbf{(E) } \frac{3\sqrt3}4 $
2021 AMC 12/AHSME Fall, 14
In the figure, equilateral hexagon $ABCDEF$ has three nonadjacent acute interior angles that each measure $30^\circ$. The enclosed area of the hexagon is $6\sqrt{3}$. What is the perimeter of the hexagon?
[asy]
size(6cm);
pen p=black+linewidth(1),q=black+linewidth(5);
pair C=(0,0),D=(cos(pi/12),sin(pi/12)),E=rotate(150,D)*C,F=rotate(-30,E)*D,A=rotate(150,F)*E,B=rotate(-30,A)*F;
draw(C--D--E--F--A--B--cycle,p);
dot(A,q);
dot(B,q);
dot(C,q);
dot(D,q);
dot(E,q);
dot(F,q);
label("$C$",C,2*S);
label("$D$",D,2*S);
label("$E$",E,2*S);
label("$F$",F,2*dir(0));
label("$A$",A,2*N);
label("$B$",B,2*W);
[/asy]
$(\textbf{A})\: 4\qquad(\textbf{B}) \: 4\sqrt3\qquad(\textbf{C}) \: 12\qquad(\textbf{D}) \: 18\qquad(\textbf{E}) \: 12\sqrt3$
2002 USAMO, 3
Prove that any monic polynomial (a polynomial with leading coefficient 1) of degree $n$ with real coefficients is the average of two monic polynomials of degree $n$ with $n$ real roots.
2018 AMC 10, 8
Sara makes a staircase out of toothpicks as shown:[asy]
size(150);
defaultpen(linewidth(0.8));
path h = ellipse((0.5,0),0.45,0.015), v = ellipse((0,0.5),0.015,0.45);
for(int i=0;i<=2;i=i+1)
{
for(int j=0;j<=3-i;j=j+1)
{
filldraw(shift((i,j))*h,black);
filldraw(shift((j,i))*v,black);
}
}[/asy]
This is a 3-step staircase and uses 18 toothpicks. How many steps would be in a staircase that used 180 toothpicks?
$\textbf{(A)}\ 10\qquad\textbf{(B)}\ 11\qquad\textbf{(C)}\ 12\qquad\textbf{(D)}\ 24\qquad\textbf{(E)}\ 30$
1993 AMC 8, 12
If each of the three operation signs, $+$, $-$, $\times $, is used exactly ONCE in one of the blanks in the expression
\[5\hspace{1 mm}\underline{\hspace{4 mm}}\hspace{1 mm}4\hspace{1 mm}\underline{\hspace{4 mm}}\hspace{1 mm}6\hspace{1 mm}\underline{\hspace{4 mm}}\hspace{1 mm}3\]
then the value of the result could equal
$\text{(A)}\ 9 \qquad \text{(B)}\ 10 \qquad \text{(C)}\ 15 \qquad \text{(D)}\ 16 \qquad \text{(E)}\ 19$