Found problems: 3632
1966 AMC 12/AHSME, 36
Let $(1+x+x^2)^n=a_0+a_1x+a_2x^2+...+a_{2n}x^{2n}$ be an identity in $x$. If we lt $s=a_0+a_2+a_4+...+a_{2n}$, then $s$ equals:
$\text{(A)}\ 2^n\qquad
\text{(B)}\ 2^n+1\qquad
\text{(C)}\ \dfrac{3^n-1}{2}\qquad
\text{(D)}\ \dfrac{3^n}{2}\qquad
\text{(E)}\ \dfrac{3^n+1}{2}$
1964 AMC 12/AHSME, 37
Given two positive number $a$, $b$ such that $a<b$. Let A.M. be their arithmetic mean and let G.M. be their positive geometric mean. Then A.M. minus G.M. is always less than:
$\textbf{(A) }\dfrac{(b+a)^2}{ab}\qquad\textbf{(B) }\dfrac{(b+a)^2}{8b}\qquad\textbf{(C) }\dfrac{(b-a)^2}{ab}$
$\textbf{(D) }\dfrac{(b-a)^2}{8a}\qquad \textbf{(E) }\dfrac{(b-a)^2}{8b}$
2018 USAJMO, 4
Triangle $ABC$ is inscribed in a circle of radius 2 with $\angle ABC \geq 90^\circ$, and $x$ is a real number satisfying the equation $x^4 + ax^3 + bx^2 + cx + 1 = 0$, where $a=BC$, $b=CA$, $c=AB$. Find all possible values of $x$.
1996 AMC 8, 22
The horizontal and vertical distances between adjacent points equal $1$ unit. The area of triangle $ABC$ is
[asy]
for (int a = 0; a < 5; ++a)
{
for (int b = 0; b < 4; ++b)
{
dot((a,b));
}
}
draw((0,0)--(3,2)--(4,3)--cycle);
label("$A$",(0,0),SW);
label("$B$",(3,2),SE);
label("$C$",(4,3),NE);
[/asy]
$\text{(A)}\ 1/4 \qquad \text{(B)}\ 1/2 \qquad \text{(C)}\ 3/4 \qquad \text{(D)}\ 1 \qquad \text{(E)}\ 5/4$
2018 AIME Problems, 6
Let $N$ be the number of complex numbers $z$ with the properties that $|z|=1$ and $z^{6!}-z^{5!}$ is a real number. Find the remainder when $N$ is divided by $1000$.
2010 AMC 10, 8
A ticket to a school play costs $ x$ dollars, where $ x$ is a whole number. A group of 9th graders buys tickets costing a total of $ \$48$, and a group of 10th graders buys tickets costing a total of $ \$64$. How many values of $ x$ are possible?
$ \textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5$
2015 AIME Problems, 2
The nine delegates to the Economic Cooperation Conference include 2 officials from Mexico, 3 officials from Canada, and 4 officials from the United States. During the opening session, three of the delegates fall asleep. Assuming that the three sleepers were determined randomly, the probability that exactly two of the sleepers are from the same country is $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
2022 AMC 12/AHSME, 1
Define $x\diamond y$ to be $|x-y|$ for all real numbers $x$ and $y$. What is the value of \[(1\diamond(2\diamond3))-((1\diamond2)\diamond3)?\]
$ \textbf{(A)}\ -2 \qquad
\textbf{(B)}\ -1 \qquad
\textbf{(C)}\ 0 \qquad
\textbf{(D)}\ 1 \qquad
\textbf{(E)}\ 2$
2009 AMC 10, 5
What is the sum of the digits of the square of $ 111,111,111$?
$ \textbf{(A)}\ 18 \qquad
\textbf{(B)}\ 27 \qquad
\textbf{(C)}\ 45 \qquad
\textbf{(D)}\ 63 \qquad
\textbf{(E)}\ 81$
2017 AMC 12/AHSME, 9
A circle has center $ (-10,-4) $ and radius $13$. Another circle has center $(3,9) $ and radius $\sqrt{65}$. The line passing through the two points of intersection of the two circles has equation $x+y=c$. What is $c$?
$\textbf{(A)} \text{ 3} \qquad \textbf{(B)} \text{ } 3 \sqrt{3} \qquad \textbf{(C)} \text{ } 4\sqrt{2} \qquad \textbf{(D)} \text{ 6} \qquad \textbf{(E)} \text{ }\frac{13}{2}$
1964 AMC 12/AHSME, 7
Let $n$ be the number of real values of $p$ for which the roots of
\[ x^2-px+p=0 \]
are equal. Then $n$ equals:
${{ \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ \text{a finite number greater than 2} }\qquad\textbf{(E)}\ \text{an infinitely large number} } $
2011 AMC 12/AHSME, 25
For every $m$ and $k$ integers with $k$ odd, denote by $[\frac{m}{k}]$ the integer closest to $\frac{m}{k}$. For every odd integer $k$, let $P\left(k\right)$ be the probability that \[ [\frac{n}{k}] + [\frac{100-n}{k}] = [\frac{100}{k}] \] for an integer $n$ randomly chosen from the interval $1 \le n \le 99!$. What is the minimum possible value of $P\left(k\right)$ over the odd integers $k$ in the interval $1 \le k \le 99$?
$ \textbf{(A)}\ \frac{1}{2} \qquad
\textbf{(B)}\ \frac{50}{99} \qquad
\textbf{(C)}\ \frac{44}{87} \qquad
\textbf{(D)}\ \frac{34}{67} \qquad
\textbf{(E)}\ \frac{7}{13} $
2021 AMC 12/AHSME Fall, 11
Consider two concentric circles of radius $17$ and $19.$ The larger circle has a chord, half of which lies inside the smaller circle. What is the length of the chord in the larger circle?
$\textbf{(A)}\ 12\sqrt{2} \qquad\textbf{(B)}\ 10\sqrt{3} \qquad\textbf{(C)}\ \sqrt{17 \cdot 19} \qquad\textbf{(D)}\
18 \qquad\textbf{(E)}\ 8\sqrt{6}$
2006 AMC 12/AHSME, 7
Mary is $ 20\%$ older than Sally, and Sally is $ 40\%$ younger than Danielle. The sum of their ages is 23.2 years. How old will Mary be on her next birthday?
$ \textbf{(A) } 7\qquad \textbf{(B) } 8\qquad \textbf{(C) } 9\qquad \textbf{(D) } 10\qquad \textbf{(E) } 11$
1998 AMC 8, 11
Harry has $3$ sisters and $5$ brothers. His sister Harriet has $X$ sisters and $Y$ brothers. What is the product of $X$ and $Y$?
$ \text{(A)}\ 8\qquad\text{(B)}\ 10\qquad\text{(C)}\ 12\qquad\text{(D)}\ 15\qquad\text{(E)}\ 18 $
1959 AMC 12/AHSME, 39
Let $S$ be the sum of the first nine terms of the sequence \[x+a, x^2+2a, x^3+3a, \cdots.\]
Then $S$ equals:
$ \textbf{(A)}\ \frac{50a+x+x^8}{x+1} \qquad\textbf{(B)}\ 50a-\frac{x+x^{10}}{x-1}\qquad\textbf{(C)}\ \frac{x^9-1}{x+1}+45a\qquad$$\textbf{(D)}\ \frac{x^{10}-x}{x-1}+45a\qquad\textbf{(E)}\ \frac{x^{11}-x}{x-1}+45a$
1972 AMC 12/AHSME, 30
[asy]
real h = 7;
real t = asin(6/h)/2;
real x = 6-h*tan(t);
real y = x*tan(2*t);
draw((0,0)--(0,h)--(6,h)--(x,0)--cycle);
draw((x,0)--(0,y)--(6,h));
draw((6,h)--(6,0)--(x,0),dotted);
label("L",(3.75,h/2),W);
label("$\theta$",(6,h-1.5),W);draw(arc((6,h),2,270,270-degrees(t)),Arrow(2mm));
label("6''",(3,0),S);
draw((2.5,-.5)--(0,-.5),Arrow(2mm));
draw((3.5,-.5)--(6,-.5),Arrow(2mm));
draw((0,-.25)--(0,-.75));draw((6,-.25)--(6,-.75));
//Credit to Zimbalono for the diagram[/asy]
A rectangular piece of paper $6$ inches wide is folded as in the diagram so that one corner touches the opposite side. The length in inches of the crease $L$ in terms of angle $\theta$ is
$\textbf{(A) }3\sec ^2\theta\csc\theta\qquad\textbf{(B) }6\sin\theta\sec\theta\qquad\textbf{(C) }3\sec\theta\csc\theta\qquad\textbf{(D) }6\sec\theta\csc ^2\theta\qquad \textbf{(E) }\text{None of these}$
2024 AMC 10, 15
A list of 9 real numbers consists of $1$, $2.2 $, $3.2 $, $5.2 $, $6.2 $, $7$, as well as $x, y,z$ with $x\leq y\leq z$. The range of the list is $7$, and the mean and median are both positive integers. How many ordered triples $(x,y,z)$ are possible?
$
\textbf{(A) }1 \qquad
\textbf{(B) }2 \qquad
\textbf{(C) }3 \qquad
\textbf{(D) }4 \qquad
\textbf{(E) infinitely many}\qquad
$
2020 AMC 12/AHSME, 22
What is the maximum value of $\frac{(2^t-3t)t}{4^t}$ for real values of $t?$
$\textbf{(A)}\ \frac{1}{16} \qquad\textbf{(B)}\ \frac{1}{15} \qquad\textbf{(C)}\ \frac{1}{12} \qquad\textbf{(D)}\ \frac{1}{10} \qquad\textbf{(E)}\ \frac{1}{9}$
1972 AMC 12/AHSME, 23
[asy]
draw((0,0)--(0,1)--(2,1)--(2,0)--cycle^^(.5,1)--(.5,2)--(1.5,2)--(1.5,1)--(.5,2)^^(.5,1)--(1.5,2)^^(1,2)--(1,0));
//Credit to Zimbalono for the diagram[/asy]
The radius of the smallest circle containing the symmetric figure composed of the $3$ unit squares shown above is
$\textbf{(A) }\sqrt{2}\qquad\textbf{(B) }\sqrt{1.25}\qquad\textbf{(C) }1.25\qquad\textbf{(D) }\frac{5\sqrt{17}}{16}\qquad \textbf{(E) }\text{None of these}$
2022 AIME Problems, 2
Find the three-digit positive integer $\underline{a} \ \underline{b} \ \underline{c}$ whose representation in base nine is $\underline{b} \ \underline{c} \ \underline{a}_{\hspace{.02in}\text{nine}}$, where $a$, $b$, and $c$ are (not necessarily distinct) digits.
1996 AMC 8, 5
The letters $P$, $Q$, $R$, $S$, and $T$ represent numbers located on the number line as shown.
[asy]
unitsize(36);
draw((-4,0)--(4,0));
draw((-3.9,0.1)--(-4,0)--(-3.9,-0.1));
draw((3.9,0.1)--(4,0)--(3.9,-0.1));
for (int i = -3; i <= 3; ++i)
{
draw((i,-0.1)--(i,0));
}
label("$-3$",(-3,-0.1),S);
label("$-2$",(-2,-0.1),S);
label("$-1$",(-1,-0.1),S);
label("$0$",(0,-0.1),S);
label("$1$",(1,-0.1),S);
label("$2$",(2,-0.1),S);
label("$3$",(3,-0.1),S);
draw((-3.7,0.1)--(-3.6,0)--(-3.5,0.1));
draw((-3.6,0)--(-3.6,0.25));
label("$P$",(-3.6,0.25),N);
draw((-1.3,0.1)--(-1.2,0)--(-1.1,0.1));
draw((-1.2,0)--(-1.2,0.25));
label("$Q$",(-1.2,0.25),N);
draw((0.1,0.1)--(0.2,0)--(0.3,0.1));
draw((0.2,0)--(0.2,0.25));
label("$R$",(0.2,0.25),N);
draw((0.8,0.1)--(0.9,0)--(1,0.1));
draw((0.9,0)--(0.9,0.25));
label("$S$",(0.9,0.25),N);
draw((1.4,0.1)--(1.5,0)--(1.6,0.1));
draw((1.5,0)--(1.5,0.25));
label("$T$",(1.5,0.25),N);
[/asy]
Which of the following expressions represents a negative number?
$\text{(A)}\ P-Q \qquad \text{(B)}\ P\cdot Q \qquad \text{(C)}\ \dfrac{S}{Q}\cdot P \qquad \text{(D)}\ \dfrac{R}{P\cdot Q} \qquad \text{(E)}\ \dfrac{S+T}{R}$
1971 AMC 12/AHSME, 25
A teen age boy wrote his own age after his father's. From this new four place number, he subtracted the absolute value of the difference of their ages to get $4,289$. The sum of their ages was
$\textbf{(A) }48\qquad\textbf{(B) }52\qquad\textbf{(C) }56\qquad\textbf{(D) }59\qquad \textbf{(E) }64$
2016 AMC 12/AHSME, 2
For what value of $x$ does $10^{x}\cdot 100^{2x}=1000^{5}$?
$\textbf{(A)}\ 1 \qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5$
2006 AMC 10, 6
What non-zero real value for $ x$ satisfies $ (7x)^{14} \equal{} (14x)^7$?
$ \textbf{(A) } \frac 17 \qquad \textbf{(B) } \frac 27 \qquad \textbf{(C) } 1 \qquad \textbf{(D) } 7 \qquad \textbf{(E) } 14$