Found problems: 3632
1987 AMC 12/AHSME, 5
A student recorded the exact percentage frequency distribution for a set of measurements, as shown below. However, the student neglected to indicate $N$, the total number of measurements. What is the smallest possible value of $N$?
\[ \begin{tabular}{c c}
\text{measured value} & \text{percent frequency} \\
\hline
0 & 12.5 \\
1 & 0\\
2 & 50\\
3 & 25 \\
4 & 12.5 \\ \hline
\ & 100 \\
\end{tabular}
\]
$ \textbf{(A)}\ 5 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C)}\ 16 \qquad\textbf{(D)}\ 25 \qquad\textbf{(E)}\ 50 $
1960 AMC 12/AHSME, 29
Five times $A$'s money added to $B$'s money is more than $\$51.00$. Three times $A$'s money minus $B$'s money is $\$21.00$. If $a$ represents $A$'s money in dollars and $b$represents $B$'s money in dollars, then:
$ \textbf{(A)}\ a>9, b>6 \qquad\textbf{(B)}\ a>9, b<6 \qquad\textbf{(C)}\ a>9, b=6\qquad$
$\textbf{(D)}\ a>9, \text{but we can put no bounds on} \text{ } b\qquad\textbf{(E)}\ 2a=3b $
2007 AMC 10, 19
A paint brush is swept along both diagonals of a square to produce the symmetric painted area, as shown. Half the area of the square is painted. What is the ratio of the side length of the square to the brush width?
[asy]unitsize(15mm);
defaultpen(linewidth(.8pt));
path P=(-sqrt(2)/2,1)--(0,1-sqrt(2)/2)--(sqrt(2)/2,1);
path Pc=(-sqrt(2)/2,1)--(0,1-sqrt(2)/2)--(sqrt(2)/2,1)--cycle;
path S=(-1,-1)--(-1,1)--(1,1)--(1,-1)--cycle;
fill(S,gray);
for(int i=0;i<4;++i)
{
fill(rotate(90*i)*Pc,white);
draw(rotate(90*i)*P);
}
draw(S);[/asy]$ \textbf{(A)}\ 2\sqrt {2} \plus{} 1 \qquad \textbf{(B)}\ 3\sqrt {2}\qquad \textbf{(C)}\ 2\sqrt {2} \plus{} 2 \qquad \textbf{(D)}\ 3\sqrt {2} \plus{} 1 \qquad \textbf{(E)}\ 3\sqrt {2} \plus{} 2$
2016 AMC 10, 5
The mean age of Amanda's $4$ cousins is $8$, and their median age is $5$. What is the sum of the ages of Amanda's youngest and oldest cousins?
$\textbf{(A)}\ 13\qquad\textbf{(B)}\ 16\qquad\textbf{(C)}\ 19\qquad\textbf{(D)}\ 22\qquad\textbf{(E)}\ 25$
1984 USAMO, 3
$P, A, B, C,$ and $D$ are five distinct points in space such that $\angle APB = \angle BPC = \angle CPD = \angle DPA = \theta$, where $\theta$ is a given acute angle. Determine the greatest and least values of $\angle APC + \angle BPD$.
2019 AIME Problems, 5
A moving particle starts at the point $\left(4,4\right)$ and moves until it hits one of the coordinate axes for the first time. When the particle is at the point $\left(a,b\right)$, it moves at random to one of the points $\left(a-1,b\right)$, $\left(a,b-1\right)$, or $\left(a-1,b-1\right)$, each with probability $\tfrac{1}{3}$, independently of its previous moves. The probability that it will hit the coordinate axes at $\left(0,0\right)$ is $\tfrac{m}{3^n}$, where $m$ and $n$ are positive integers, and $m$ is not divisible by $3$. Find $m+n$.
2024 AMC 12/AHSME, 6
The product of three integers is $60$. What is the least possible positive sum of the three integers?
$\textbf{(A) } 2 \qquad \textbf{(B) } 3 \qquad \textbf{(C) } 5 \qquad \textbf{(D) } 6 \qquad \textbf{(E) } 13$
2014 Contests, 1
What is $10 \cdot \left(\tfrac{1}{2} + \tfrac{1}{5} + \tfrac{1}{10}\right)^{-1}?$
${ \textbf{(A)}\ 3\qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ \frac{25}{2}\qquad\textbf{(D)}}\ \frac{170}{3}\qquad\textbf{(E)}\ 170$
1969 AMC 12/AHSME, 1
When $x$ is added to both the numerator and the denominator of the fraction $a/b, a\neq b, b\neq 0$, the value of the fraction is changed to $c/d$. Then $x$ equals:
$\textbf{(A) }\dfrac1{c-d}\qquad
\textbf{(B) }\dfrac{ad-bc}{c-d}\qquad
\textbf{(C) }\dfrac{ad-bc}{c+d}\qquad
\textbf{(D) }\dfrac{bc-ad}{c-d}\qquad
\textbf{(E) }\dfrac{bc-ad}{c+d}$
1964 AMC 12/AHSME, 23
Two numbers are such that their difference, their sum, and their product are to one another as $1:7:24$. The product of the two numbers is:
$ \textbf{(A)}\ 6\qquad\textbf{(B)}\ 12\qquad\textbf{(C)}\ 24\qquad\textbf{(D)}\ 48\qquad\textbf{(E)}\ 96 $
1985 USAMO, 2
Determine each real root of \[x^4-(2\cdot10^{10}+1)x^2-x+10^{20}+10^{10}-1=0\] correct to four decimal places.
2024 AMC 8 -, 15
Let the letters $F$, $L$, $Y$, $B$, $U$, $G$ represent different digits. Suppose $\underline{F}\underline{L}\underline{Y}\underline{F}\underline{L}\underline{Y}$ is the largest number that satisfies the equation $$8 \cdot \underline{F}\underline{L}\underline{Y}\underline{F}\underline{L}\underline{Y} = \underline{B}\underline{U}\underline{G}\underline{B}\underline{U}\underline{G}.$$ What is the value of $\underline{F}\underline{L}\underline{Y} + \underline{B}\underline{U}\underline{G}$?
$\textbf{(A) } 1089\qquad\textbf{(B) } 1098\qquad\textbf{(C) } 1107\qquad\textbf{(D) } 1116\qquad\textbf{(E) } 1125$
1967 AMC 12/AHSME, 3
The side of an equilateral triangle is $s$. A circle is inscribed in the triangle and a square is inscribed in the circle. The area of the square is:
$ \text{(A)}\ \frac{s^2}{24}\qquad\text{(B)}\ \frac{s^2}{6}\qquad\text{(C)}\ \frac{s^2\sqrt{2}}{6}\qquad\text{(D)}\ \frac{s^2\sqrt{3}}{6}\qquad\text{(E)}\ \frac{s^2}{3} $
1996 AMC 8, 19
The pie charts below indicate the percent of students who prefer golf, bowling, or tennis at East Junior High School and West Middle School. The total number of students at East is $2000$ and at West, $2500$. In the two schools combined, the percent of students who prefer tennis is
[asy]
unitsize(18);
draw(circle((0,0),4));
draw(circle((9,0),4));
draw((-4,0)--(0,0)--4*dir(352.8));
draw((0,0)--4*dir(100.8));
draw((5,0)--(9,0)--(4*dir(324)+(9,0)));
draw((9,0)--(4*dir(50.4)+(9,0)));
label("$48\%$",(0,-1),S);
label("bowling",(0,-2),S);
label("$30\%$",(1.5,1.5),N);
label("golf",(1.5,0.5),N);
label("$22\%$",(-2,1.5),N);
label("tennis",(-2,0.5),N);
label("$40\%$",(8.5,-1),S);
label("tennis",(8.5,-2),S);
label("$24\%$",(10.5,0.5),E);
label("golf",(10.5,-0.5),E);
label("$36\%$",(7.8,1.7),N);
label("bowling",(7.8,0.7),N);
label("$\textbf{East JHS}$",(0,-4),S);
label("$\textbf{2000 students}$",(0,-5),S);
label("$\textbf{West MS}$",(9,-4),S);
label("$\textbf{2500 students}$",(9,-5),S);
[/asy]
$\text{(A)}\ 30\% \qquad \text{(B)}\ 31\% \qquad \text{(C)}\ 32\% \qquad \text{(D)}\ 33\% \qquad \text{(E)}\ 34\%$
2019 AMC 10, 17
A child builds towers using identically shaped cubes of different color. How many different towers with a height $8$ cubes can the child build with $2$ red cubes, $3$ blue cubes, and $4$ green cubes? (One cube will be left out.)
$\textbf{(A) } 24 \qquad\textbf{(B) } 288 \qquad\textbf{(C) } 312 \qquad\textbf{(D) } 1,260 \qquad\textbf{(E) } 40,320$
1959 AMC 12/AHSME, 44
The roots of $x^2+bx+c=0$ are both real and greater than $1$. Let $s=b+c+1$. Then $s:$
$ \textbf{(A)}\ \text{may be less than zero}\qquad\textbf{(B)}\ \text{may be equal to zero}\qquad$ $\textbf{(C)}\ \text{must be greater than zero}\qquad\textbf{(D)}\ \text{must be less than zero}\qquad $
$\textbf{(E)}\text{ must be between -1 and +1} $
2021 AIME Problems, 8
Find the number of integers $c$ such that the equation $$\left||20|x|-x^2|-c\right|=21$$ has $12$ distinct real solutions.
2006 AMC 12/AHSME, 2
Define $ x\otimes y \equal{} x^3 \minus{} y$. What is $ h\otimes (h\otimes h)$?
$ \textbf{(A) } \minus{} h\qquad \textbf{(B) } 0\qquad \textbf{(C) } h\qquad \textbf{(D) } 2h\qquad \textbf{(E) } h^3$
2013 AMC 10, 6
Joey and his five brothers are ages $3,5,7,9,11,$ and $13$. One afternoon two of his brothers whose ages sum to $16$ went to the movies, two brothers younger than $10$ went to play baseball, and Joey and the 5-year-old stayed home. How old is Joey?
$ \textbf{(A)}\ 3\qquad\textbf{(B)}\ 7\qquad\textbf{(C)}\ 9\qquad\textbf{(D)}\ 11\qquad\textbf{(E)}\ 13 $
2006 AMC 12/AHSME, 10
In a triangle with integer side lengths, one side is three times as long as a second side, and the length of the third side is 15. What is the greatest possible perimeter of the triangle?
$ \textbf{(A) } 43 \qquad \textbf{(B) } 44 \qquad \textbf{(C) } 45 \qquad \textbf{(D) } 46 \qquad \textbf{(E) } 47$
2024 AMC 10, 14
One side of an equilateral triangle of height $24$ lies on line $\ell.$ A circle of radius $12$ is tangent to $\ell$ and is externally tangent to the triangle. The area of the region exterior to the triangle and the circle and bounded by the triangle, the circle, and line $\ell$ can be written as $a\sqrt{b} - c\pi,$ where $a,$ $b,$ and $c$ are positive integers and $b$ is not divisible by the square of any prime. What is $a+b+c\,?$
$\phantom{boo}$
$\displaystyle
\textbf{(A)}\; 72 \quad
\textbf{(B)}\; 73 \quad
\textbf{(C)}\; 74 \quad
\textbf{(D)}\; 75 \quad
\textbf{(E)}\; 76
$
2008 AMC 10, 16
Two fair coins are to be tossed once. For each head that results, one fair die is to be rolled. What is the probability that the sum of the die rolls is odd? (Note that if no die is rolled, their sum is $ 0$.)
$ \textbf{(A)}\ \frac{3}{8} \qquad
\textbf{(B)}\ \frac{1}{2} \qquad
\textbf{(C)}\ \frac{43}{72} \qquad
\textbf{(D)}\ \frac{5}{8} \qquad
\textbf{(E)}\ \frac{2}{3}$
2021 AIME Problems, 5
Call a three-term strictly increasing arithmetic sequence of integers [i]special[/i] if the sum of the squares of the three terms equals the product of the middle term and the square of the common difference. Find the sum of the third terms of all special sequences.
1979 AMC 12/AHSME, 11
Find a positive integral solution to the equation
\[\frac{1+3+5+\dots+(2n-1)}{2+4+6+\dots+2n}=\frac{115}{116}\]
$\textbf{(A) }110\qquad\textbf{(B) }115\qquad\textbf{(C) }116\qquad\textbf{(D) }231\qquad\textbf{(E) }\text{The equation has no positive integral solutions.}$
2020 AMC 10, 16
Bela and Jenn play the following game on the closed interval $[0, n]$ of the real number line, where $n$ is a fixed integer greater than $4$. They take turns playing, with Bela going first. At his first turn, Bela chooses any real number in the interval $[0, n]$. Thereafter, the player whose turn it is chooses a real number that is more than one unit away from all numbers previously chosen by either player. A player unable to choose such a number loses. Using optimal strategy, which player will win the game?
$\textbf{(A) } \text{Bela will always win.}$
$\textbf{(B) } \text{Jenn will always win.} $
$\textbf{(C) } \text{Bela will win if and only if }n \text{ is odd.}$
$\textbf{(D) } \text{Jenn will win if and only if }n \text{ is odd.} $
$\textbf{(E) } \text{Jenn will win if and only if }n > 8.$