Found problems: 3632
2020 AMC 12/AHSME, 6
For all integers $n \geq 9,$ the value of
$$\frac{(n+2)!-(n+1)!}{n!}$$
is always which of the following?
$\textbf{(A) } \text{a multiple of }4 \qquad \textbf{(B) } \text{a multiple of }10 \qquad \textbf{(C) } \text{a prime number} \\ \textbf{(D) } \text{a perfect square} \qquad \textbf{(E) } \text{a perfect cube}$
1964 AMC 12/AHSME, 26
In a ten-mile race First beats Second by $2$ miles and First beats Third by $4$ miles. If the runners maintain constant speeds throughout the race, by how many miles does Second beat Third?
$ \textbf{(A)}\ 2\qquad\textbf{(B)}\ 2\frac{1}{4}\qquad\textbf{(C)}\ 2\frac{1}{2}\qquad\textbf{(D)}\ 2\frac{3}{4}\qquad\textbf{(E)}\ 3 $
2013 AMC 12/AHSME, 10
Let $S$ be the set of positive integers $n$ for which $\tfrac{1}{n}$ has the repeating decimal representation $0.\overline{ab} = 0.ababab\cdots,$ with $a$ and $b$ different digits. What is the sum of the elements of $S$?
$ \textbf{(A)}\ 11\qquad\textbf{(B)}\ 44\qquad\textbf{(C)}\ 110\qquad\textbf{(D)}\ 143\qquad\textbf{(E)}\ 155\qquad $
2003 AMC 12-AHSME, 5
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$
2023 AMC 12/AHSME, 16
Consider the set of complex numbers $z$ satisfying $|1+z+z^{2}|=4$. The maximum value of the imaginary part of $z$ can be written in the form $\tfrac{\sqrt{m}}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$?
$\textbf{(A)}~20\qquad\textbf{(B)}~21\qquad\textbf{(C)}~22\qquad\textbf{(D)}~23\qquad\textbf{(E)}~24$
1976 AMC 12/AHSME, 24
[asy]
size(150);
pair A=(0,0),B=(1,0),C=(0,1),D=(-1,0),E=(0,.5),F=(sqrt(2)/2,.25);
draw(circle(A,1)^^D--B);
draw(circle(E,.5)^^circle( F ,.25));
label("$A$", D, W);
label("$K$", A, S);
label("$B$", B, dir(0));
label("$L$", E, N);
label("$M$",shift(-.05,.05)*F);
//Credit to Klaus-Anton for the diagram[/asy]
In the adjoining figure, circle $\mathit{K}$ has diameter $\mathit{AB}$; cirlce $\mathit{L}$ is tangent to circle $\mathit{K}$ and to $\mathit{AB}$ at the center of circle $\mathit{K}$; and circle $\mathit{M}$ tangent to circle $\mathit{K}$, to circle $\mathit{L}$ and $\mathit{AB}$. The ratio of the area of circle $\mathit{K}$ to the area of circle $\mathit{M}$ is
$\textbf{(A) }12\qquad\textbf{(B) }14\qquad\textbf{(C) }16\qquad\textbf{(D) }18\qquad \textbf{(E) }\text{not an integer}$
2024 AMC 12/AHSME, 19
Cyclic quadrilateral $ABCD$ has lengths $BC=CD=3$ and $DA=5$ with $\angle CDA=120^\circ$. What is the length of the shorter diagonal of $ABCD$?
$
\textbf{(A) }\frac{31}7 \qquad
\textbf{(B) }\frac{33}7 \qquad
\textbf{(C) }5 \qquad
\textbf{(D) }\frac{39}7 \qquad
\textbf{(E) }\frac{41}7 \qquad
$
2014 AMC 12/AHSME, 5
Doug constructs a square window using $8$ equal-size panes of glass, as shown. The ratio of the height to width for each pane is $5:2$, and the borders around and between the panes are $2$ inches wide. In inches, what is the side length o the square window?
[asy]
fill((0,0)--(25,0)--(25,25)--(0,25)--cycle,grey);
for(int i = 0; i < 4; ++i){
for(int j = 0; j < 2; ++j){
fill((6*i+2,11*j+3)--(6*i+5,11*j+3)--(6*i+5,11*j+11)--(6*i+2,11*j+11)--cycle,white);
}
}[/asy]
$\textbf{(A) }26\qquad\textbf{(B) }28\qquad\textbf{(C) }30\qquad\textbf{(D) }32\qquad\textbf{(E) }34$
2012 USAJMO, 3
Let $a,b,c$ be positive real numbers. Prove that $\frac{a^3+3b^3}{5a+b}+\frac{b^3+3c^3}{5b+c}+\frac{c^3+3a^3}{5c+a} \geq \frac{2}{3}(a^2+b^2+c^2)$.
2019 AMC 12/AHSME, 3
A box contains $28$ red balls, $20$ green balls, $19$ yellow balls, $13$ blue balls, $11$ white balls, and $9$ black balls. What is the minimum number of balls that must be drawn from the box without replacement to guarantee that at least $15$ balls of a single color will be drawn$?$
$\textbf{(A) } 75 \qquad\textbf{(B) } 76 \qquad\textbf{(C) } 79 \qquad\textbf{(D) } 84 \qquad\textbf{(E) } 91$
2016 AMC 12/AHSME, 21
Let $ABCD$ be a unit square. Let $Q_1$ be the midpoint of $\overline{CD}$. For $i=1,2,\dots,$ let $P_i$ be the intersection of $\overline{AQ_i}$ and $\overline{BD}$, and let $Q_{i+1}$ be the foot of the perpendicular from $P_i$ to $\overline{CD}$. What is
$$\sum_{i=1}^{\infty} \text{Area of } \triangle DQ_i P_i \, ?$$
$\textbf{(A)}\ \frac{1}{6} \qquad
\textbf{(B)}\ \frac{1}{4} \qquad
\textbf{(C)}\ \frac{1}{3} \qquad
\textbf{(D)}\ \frac{1}{2} \qquad
\textbf{(E)}\ 1$
1960 AMC 12/AHSME, 14
If $a$ and $b$ are real numbers, the equation $3x-5+a=bx+1$ has a unique solution $x$ [The symbol $a \neq 0$ means that $a$ is different from zero]:
$ \textbf{(A) }\text{for all a and b} \qquad\textbf{(B) }\text{if a }\neq\text{2b}\qquad\textbf{(C) }\text{if a }\neq 6\qquad$
$\textbf{(D) }\text{if b }\neq 0\qquad\textbf{(E) }\text{if b }\neq 3 $
2009 AMC 12/AHSME, 14
Five unit squares are arranged in the coordinate plane as shown, with the lower left corner at the origin. The slanted line, extending from $ (a,0)$ to $ (3,3)$, divides the entire region into two regions of equal area. What is $ a$?
[asy]size(200);
defaultpen(linewidth(.8pt)+fontsize(8pt));
fill((2/3,0)--(3,3)--(3,1)--(2,1)--(2,0)--cycle,gray);
xaxis("$x$",-0.5,4,EndArrow(HookHead,4));
yaxis("$y$",-0.5,4,EndArrow(4));
draw((0,1)--(3,1)--(3,3)--(2,3)--(2,0));
draw((1,0)--(1,2)--(3,2));
draw((2/3,0)--(3,3));
label("$(a,0)$",(2/3,0),S);
label("$(3,3)$",(3,3),NE);[/asy]$ \textbf{(A)}\ \frac12\qquad
\textbf{(B)}\ \frac35\qquad
\textbf{(C)}\ \frac23\qquad
\textbf{(D)}\ \frac34\qquad
\textbf{(E)}\ \frac45$
1963 AMC 12/AHSME, 9
In the expansion of $\left(a-\dfrac{1}{\sqrt{a}}\right)^7$ the coefficient of $a^{-\dfrac{1}{2}}$ is:
$\textbf{(A)}\ -7 \qquad
\textbf{(B)}\ 7 \qquad
\textbf{(C)}\ -21 \qquad
\textbf{(D)}\ 21 \qquad
\textbf{(E)}\ 35$
2019 AIME Problems, 10
There is a unique angle $\theta$ between $0^{\circ}$ and $90^{\circ}$ such that for nonnegative integers $n$, the value of $\tan{\left(2^{n}\theta\right)}$ is positive when $n$ is a multiple of $3$, and negative otherwise. The degree measure of $\theta$ is $\tfrac{p}{q}$, where $p$ and $q$ are relatively prime integers. Find $p+q$.
2011 India IMO Training Camp, 1
Find all positive integer $n$ satisfying the conditions
$a)n^2=(a+1)^3-a^3$
$b)2n+119$ is a perfect square.
2005 AIME Problems, 6
Let $P$ be the product of the nonreal roots of $x^4-4x^3+6x^2-4x=2005$. Find $\lfloor P\rfloor$.
2009 AMC 10, 17
Rectangle $ ABCD$ has $ AB \equal{} 4$ and $ BC \equal{} 3$. Segment $ EF$ is constructed through $ B$ so that $ EF$ is perpendicular to $ DB$, and $ A$ and $ C$ lie on $ DE$ and $ DF$, respectively. What is $ EF$?
$ \textbf{(A)}\ 9\qquad \textbf{(B)}\ 10\qquad \textbf{(C)}\ \frac {125}{12}\qquad \textbf{(D)}\ \frac {103}{9}\qquad \textbf{(E)}\ 12$
2025 AIME, 4
Find the number of ordered pairs $(x,y)$, where both $x$ and $y$ are integers between $-100$ and $100$ inclusive, such that $12x^2-xy-6y^2=0$.
1964 AMC 12/AHSME, 9
A jobber buys an article at $\$24$ less $12\frac{1}{2}\%$. He then wishes to sell the article at a gain of $33\frac{1}{3}\%$ of his cost after allowing a $20\%$ discount on his marked price. At what price, in dollars, should the article be marked?
${{ \textbf{(A)}\ 25.20 \qquad\textbf{(B)}\ 30.00 \qquad\textbf{(C)}\ 33.60 \qquad\textbf{(D)}\ 40.00 }\qquad\textbf{(E)}\ \text{none of these} } $
2011 NIMO Problems, 8
Triangle $ABC$ with $\measuredangle A = 90^\circ$ has incenter $I$. A circle passing through $A$ with center $I$ is drawn, intersecting $\overline{BC}$ at $E$ and $F$ such that $BE < BF$. If $\tfrac{BE}{EF} = \tfrac{2}{3}$, then $\tfrac{CF}{FE} = \tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
[i]Proposed by Lewis Chen
[/i]
1991 AIME Problems, 7
Find $A^2$, where $A$ is the sum of the absolute values of all roots of the following equation: \begin{eqnarray*}x &=& \sqrt{19} + \frac{91}{{\displaystyle \sqrt{19}+\frac{91}{{\displaystyle \sqrt{19}+\frac{91}{{\displaystyle \sqrt{19}+\frac{91}{{\displaystyle \sqrt{19}+\frac{91}{x}}}}}}}}}\end{eqnarray*}
2010 AMC 12/AHSME, 10
The average of the numbers $ 1,2,3,...,98,99$, and $ x$ is $ 100x$. What is $ x$?
$ \textbf{(A)}\ \frac{49}{101} \qquad\textbf{(B)}\ \frac{50}{101} \qquad\textbf{(C)}\ \frac12 \qquad\textbf{(D)}\ \frac{51}{101} \qquad\textbf{(E)}\ \frac{50}{99}$
2006 AIME Problems, 13
For each even positive integer $x$, let $g(x)$ denote the greatest power of $2$ that divides $x$. For example, $g(20)=4$ and $g(16)=16$. For each positive integer $n$, let $S_n=\sum_{k=1}^{2^{n-1}}g(2k).$ Find the greatest integer $n$ less than $1000$ such that $S_n$ is a perfect square.
1971 AMC 12/AHSME, 6
Let $\ast$ be the symbol denoting the binary operation on the set $S$ of all non-zero real numbers as follows: For any two numbers $a$ and $b$ of $S$, $a\ast b=2ab$. Then the one of the following statements which is not true, is
$\textbf{(A) }\ast\text{ is commutative over }S \qquad\textbf{(B) }\ast\text{ is associative over }S\qquad$
$\textbf{(C) }\frac{1}{2}\text{ is an identity element for }\ast\text{ in }S\qquad\textbf{(D) }\text{Every element of }S\text{ has an inverse for }\ast\qquad$
$\textbf{(E) }\dfrac{1}{2a}\text{ is an inverse for }\ast\text{ of the element }a\text{ of }S$