Found problems: 3632
2008 AMC 10, 1
A bakery owner turns on his doughnut machine at 8:30 AM. At 11:10 AM the machine has completed one third of the day's job. At what time will the doughnut machine complete the job?
$ \textbf{(A)}$ 1:50 PM $ \qquad
\textbf{(B)}$ 3:00 PM $ \qquad
\textbf{(C)}$ 3:30 PM $ \qquad
\textbf{(D)}$ 4:30 PM $ \qquad
\textbf{(E)}$ 5:50 PM
2004 AMC 12/AHSME, 3
For how many ordered pairs of positive integers $ (x,y)$ is $ x \plus{} 2y \equal{} 100$?
$ \textbf{(A)}\ 33 \qquad \textbf{(B)}\ 49 \qquad \textbf{(C)}\ 50 \qquad \textbf{(D)}\ 99 \qquad \textbf{(E)}\ 100$
2024 AMC 10, 24
Let
\[P(m)=\frac{m}{2} + \frac{m^2}{4}+ \frac{m^4}{8} + \frac{m^8}{8}.\]
How many of the values of $P(2022)$, $P(2023)$, $P(2024)$, and $P(2025)$ are integers?
$
\textbf{(A) }0 \qquad
\textbf{(B) }1 \qquad
\textbf{(C) }2 \qquad
\textbf{(D) }3 \qquad
\textbf{(E) }4 \qquad
$
2011 AMC 12/AHSME, 8
In the eight-term sequence $A, B, C, D, E, F, G, H$, the value of $C$ is $5$ and the sum of any three consecutive terms is $30$. What is $A + H$?
$ \textbf{(A)}\ 17 \qquad
\textbf{(B)}\ 18 \qquad
\textbf{(C)}\ 25 \qquad
\textbf{(D)}\ 26 \qquad
\textbf{(E)}\ 43
$
2024 AMC 10, 9
In how many ways can $6$ juniors and $6$ seniors form $3$ disjoint teams of $4$ people so that each team has $2$ juniors and $2$ seniors?
$
\textbf{(A) }720 \qquad
\textbf{(B) }1350 \qquad
\textbf{(C) }2700 \qquad
\textbf{(D) }3280 \qquad
\textbf{(E) }8100 \qquad
$
2002 AMC 10, 24
Tina randomly selects two distinct numbers from the set $ \{1,2,3,4,5\}$ and Sergio randomly selects a number from the set $ \{1,2,...,10\}$. The probability that Sergio's number is larger than the sum of the two numbers chosen by Tina is
$ \textbf{(A)}\ 2/5 \qquad \textbf{(B)}\ 9/20 \qquad \textbf{(C)}\ 1/2\qquad \textbf{(D)}\ 11/20 \qquad \textbf{(E)}\ 24/25$
2017 AMC 12/AHSME, 11
Call a positive integer [i]monotonous[/i] if it is a one-digit number or its digits, when read from left to right, form either a strictly increasing or a strictly decreasing sequence. For example, 3, 23578, and 987620 are monotonous, but 88, 7434, and 23557 are not. How many monotonous positive integers are there?
$\textbf{(A)} \text{ 1024} \qquad \textbf{(B)} \text{ 1524} \qquad \textbf{(C)} \text{ 1533} \qquad \textbf{(D)} \text{ 1536} \qquad \textbf{(E)} \text{ 2048}$
2008 AMC 10, 22
Three red beads, two white beads, and one blue bead are placed in a line in random order. What is the probability that no two neighboring beads are the same color?
$ \textbf{(A)}\ \frac{1}{12} \qquad
\textbf{(B)}\ \frac{1}{10} \qquad
\textbf{(C)}\ \frac{1}{6} \qquad
\textbf{(D)}\ \frac{1}{3} \qquad
\textbf{(E)}\ \frac{1}{2}$
1993 AMC 8, 5
Which one of the following bar graphs could represent the data from the circle graph?
[asy]
unitsize(36);
draw(circle((0,0),1),gray);
fill((0,0)--arc((0,0),(0,-1),(1,0))--cycle,gray);
fill((0,0)--arc((0,0),(1,0),(0,1))--cycle,black);
[/asy]
[asy]
unitsize(4);
fill((1,0)--(1,15)--(5,15)--(5,0)--cycle,gray);
fill((6,0)--(6,15)--(10,15)--(10,0)--cycle,black);
draw((11,0)--(11,20)--(15,20)--(15,0));
fill((26,0)--(26,15)--(30,15)--(30,0)--cycle,gray);
fill((31,0)--(31,15)--(35,15)--(35,0)--cycle,black);
draw((36,0)--(36,15)--(40,15)--(40,0));
fill((51,0)--(51,10)--(55,10)--(55,0)--cycle,gray);
fill((56,0)--(56,10)--(60,10)--(60,0)--cycle,black);
draw((61,0)--(61,20)--(65,20)--(65,0));
fill((76,0)--(76,10)--(80,10)--(80,0)--cycle,gray);
fill((81,0)--(81,15)--(85,15)--(85,0)--cycle,black);
draw((86,0)--(86,20)--(90,20)--(90,0));
fill((101,0)--(101,15)--(105,15)--(105,0)--cycle,gray);
fill((106,0)--(106,10)--(110,10)--(110,0)--cycle,black);
draw((111,0)--(111,20)--(115,20)--(115,0));
for(int a = 0; a < 5; ++a)
{
draw((25*a,21)--(25*a,0)--(25*a+16,0));
}
label("(A)",(8,21),N);
label("(B)",(33,21),N);
label("(C)",(58,21),N);
label("(D)",(83,21),N);
label("(E)",(108,21),N);
[/asy]
2011 USAMO, 5
Let $P$ be a given point inside quadrilateral $ABCD$. Points $Q_1$ and $Q_2$ are located within $ABCD$ such that
\[\angle Q_1BC=\angle ABP,\quad\angle Q_1CB=\angle DCP,\quad\angle Q_2AD=\angle BAP,\quad\angle Q_2DA=\angle CDP.\] Prove that $\overline{Q_1Q_2}\parallel\overline{AB}$ if and only if $\overline{Q_1Q_2}\parallel\overline{CD}$.
2014 AMC 8, 24
One day the Beverage Barn sold $252$ cans of soda to $100$ customers, and every customer bought at least one can of soda. What is the maximum possible median number of cans of soda bought per customer on that day?
$\textbf{(A) }2.5\qquad\textbf{(B) }3.0\qquad\textbf{(C) }3.5\qquad\textbf{(D) }4.0\qquad \textbf{(E) }4.5$
2024 AMC 10, 1
In a long line of people, the 1013th person from the left is also the 1010th person from the right. How many people are in the line?
$
\textbf{(A) }2021 \qquad
\textbf{(B) }2022 \qquad
\textbf{(C) }2023 \qquad
\textbf{(D) }2024 \qquad
\textbf{(E) }2025 \qquad
$
2011 AMC 10, 8
At a certain beach if it is at least $80 ^\circ F$ and sunny, then the beach will be crowded. On June 10 the beach was not crowded. What can be said about the weather conditions on June 10?
$(A)$ The temperature was cooler than $80 ^\circ F$ and it was not sunny.
$(B)$ The temperature was cooler than $80 ^\circ F$ or it was not sunny.
$(C)$ If the temperature was at least $80 ^\circ F$, then it was sunny.
$(D)$ If the temperature was cooler than $80 ^\circ F$, then it was sunny.
$(E)$ If the temperature was cooler than $80 ^\circ F$, then it was not sunny.
1969 AMC 12/AHSME, 6
The area of the ring between two concentric circles is $12\tfrac12\pi$ square inches. The length of a chord of the larger circle tangent to the smaller circle, in inches, is:
$\textbf{(A) }\dfrac5{\sqrt2}\qquad
\textbf{(B) }5\qquad
\textbf{(C) }5\sqrt2\qquad
\textbf{(D) }10\qquad
\textbf{(E) }10\sqrt2$
2010 AMC 10, 9
Lucky Larry's teacher asked him to substitute numbers for $ a$, $ b$, $ c$, $ d$, and $ e$ in the expression $ a\minus{}(b\minus{}(c\minus{}(d\plus{}e)))$ and evaluate the result. Larry ignored the parentheses but added and subtracted correctly and obtained the correct result by coincedence. The numbers Larry substituted for $ a$, $ b$, $ c$, and $ d$ were $ 1$, $ 2$, $ 3$, and $ 4$, respectively. What number did Larry substitute for $ e$?
$ \textbf{(A)}\ \minus{}5\qquad\textbf{(B)}\ \minus{}3\qquad\textbf{(C)}\ 0\qquad\textbf{(D)}\ 3\qquad\textbf{(E)}\ 5$
2013 AMC 8, 1
Danica wants to arrange her model cars in rows with exactly 6 cars in each row. She now has 23 model cars. What is the smallest number of additional cars she must buy in order to be able to arrange all her cars this way?
$\textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 5$
2017 AMC 10, 3
Tamara has three rows of two 6-feet by 2-feet flower beds in her garden. The beds are separated and also surrounded by 1-foot-wide walkways, as shown on the diagram. What is the total area of the walkways, in square feet?
[asy]
unitsize(0.7cm);
path p1 = (0,0)--(15,0)--(15,10)--(0,10)--cycle;
fill(p1,lightgray);
draw(p1);
for (int i = 1; i <= 8; i += 7) {
for (int j = 1; j <= 7; j += 3 ) {
path p2 = (i,j)--(i+6,j)--(i+6,j+2)--(i,j+2)--cycle;
draw(p2);
fill(p2,white);
}
}
draw((0,8)--(1,8),Arrows);
label("1",(0.5,8),S);
draw((7,8)--(8,8),Arrows);
label("1",(7.5,8),S);
draw((14,8)--(15,8),Arrows);
label("1",(14.5,8),S);
draw((11,0)--(11,1),Arrows);
label("1",(11,0.5),W);
draw((11,3)--(11,4),Arrows);
label("1",(11,3.5),W);
draw((11,6)--(11,7),Arrows);
label("1",(11,6.5),W);
draw((11,9)--(11,10),Arrows);
label("1",(11,9.5),W);
label("6",(4,1),N);
label("2",(1,2),E);
[/asy]
$\textbf{(A) }72 \qquad
\textbf{(B) }78 \qquad
\textbf{(C) }90 \qquad
\textbf{(D) }120 \qquad
\textbf{(E) }150 $
2010 AMC 12/AHSME, 12
In a magical swamp there are two species of talking amphibians: toads, whose statements are always true, and frogs, whose statements are always false. Four amphibians, Brian, Chris, LeRoy, and Mike live together in the swamp, and they make the following statements:
Brian: "Mike and I are different species."
Chris: "LeRoy is a frog."
LeRoy: "Chris is a frog."
Mike: "Of the four of us, at least two are toads."
How many of these amphibians are frogs?
$ \textbf{(A)}\ 0\qquad\textbf{(B)}\ 1\qquad\textbf{(C)}\ 2\qquad\textbf{(D)}\ 3\qquad\textbf{(E)}\ 4$
1988 AMC 12/AHSME, 24
An isosceles trapezoid is circumscribed around a circle. The longer base of the trapezoid is $16$, and one of the base angles is $\arcsin(.8)$. Find the area of the trapezoid.
$ \textbf{(A)}\ 72\qquad\textbf{(B)}\ 75\qquad\textbf{(C)}\ 80\qquad\textbf{(D)}\ 90\qquad\textbf{(E)}\ \text{not uniquely determined} $
1963 AMC 12/AHSME, 30
Let \[F=\log\dfrac{1+x}{1-x}.\] Find a new function $G$ by replacing each $x$ in $F$ by \[\dfrac{3x+x^3}{1+3x^2},\] and simplify. The simplified expression $G$ is equal to:
$\textbf{(A)}\ -F \qquad
\textbf{(B)}\ F\qquad
\textbf{(C)}\ 3F \qquad
\textbf{(D)}\ F^3 \qquad
\textbf{(E)}\ F^3-F$
1960 AMC 12/AHSME, 6
The circumference of a circle is $100$ inches. The side of a square inscribed in this circle, expressed in inches, is:
$ \textbf{(A) }\frac{25\sqrt{2}}{\pi} \qquad\textbf{(B) }\frac{50\sqrt{2}}{\pi}\qquad\textbf{(C) }\frac{100}{\pi}\qquad\textbf{(D) }\frac{100\sqrt{2}}{\pi}\qquad\textbf{(E) }50\sqrt{2} $
2008 AIME Problems, 4
There exist $ r$ unique nonnegative integers $ n_1 > n_2 > \cdots > n_r$ and $ r$ unique integers $ a_k$ ($ 1\le k\le r$) with each $ a_k$ either $ 1$ or $ \minus{} 1$ such that
\[ a_13^{n_1} \plus{} a_23^{n_2} \plus{} \cdots \plus{} a_r3^{n_r} \equal{} 2008.
\]Find $ n_1 \plus{} n_2 \plus{} \cdots \plus{} n_r$.
2006 AIME Problems, 10
Seven teams play a soccer tournament in which each team plays every other team exactly once. No ties occur, each team has a $50\%$ chance of winning each game it plays, and the outcomes of the games are independent. In each game, the winner is awarded a point and the loser gets 0 points. The total points are accumilated to decide the ranks of the teams. In the first game of the tournament, team $A$ beats team $B$. The probability that team $A$ finishes with more points than team $B$ is $m/n$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
1997 AMC 8, 2
Ahn chooses a two-digit integer, subtracts it from 200, and doubles the result. What is the largest number Ahn can get?
$\textbf{(A)}\ 200 \qquad \textbf{(B)}\ 202 \qquad \textbf{(C)}\ 220 \qquad \textbf{(D)}\ 380 \qquad \textbf{(E)}\ 398$
2013 AMC 8, 7
Trey and his mom stopped at a railroad crossing to let a train pass. As the train began to pass, Trey counted 6 cars in the first 10 seconds. It took the train 2 minutes and 45 seconds to clear the crossing at a constant speed. Which of the following was the most likely number of cars in the train?
$\textbf{(A)}\ 60 \qquad \textbf{(B)}\ 80 \qquad \textbf{(C)}\ 100 \qquad \textbf{(D)}\ 120 \qquad \textbf{(E)}\ 140$