Found problems: 3632
1985 USAMO, 4
There are $n$ people at a party. Prove that there are two people such that, of the remaining $n-2$ people, there are at least $\left\lfloor\frac{n}{2}\right\rfloor-1$ of them, each of whom either knows both or else knows neither of the two. Assume that knowing is a symmetric relation, and that $\lfloor x\rfloor$ denotes the greatest integer less than or equal to $x$.
2022 AMC 10, 8
Consider the following $100$ sets of $10$ elements each:
\begin{align*}
&\{1,2,3,\cdots,10\}, \\
&\{11,12,13,\cdots,20\},\\
&\{21,22,23,\cdots,30\},\\
&\vdots\\
&\{991,992,993,\cdots,1000\}.
\end{align*}
How many of these sets contain exactly two multiples of $7$?
$\textbf{(A)} 40\qquad\textbf{(B)} 42\qquad\textbf{(C)} 43\qquad\textbf{(D)} 49\qquad\textbf{(E)} 50$
1975 AMC 12/AHSME, 30
Let $x=\cos 36^{\circ} - \cos 72^{\circ}$. Then $x$ equals
$ \textbf{(A)}\ \frac{1}{3} \qquad\textbf{(B)}\ \frac{1}{2} \qquad\textbf{(C)}\ 3-\sqrt{6} \qquad\textbf{(D)}\ 2\sqrt{3}-3 \qquad\textbf{(E)}\ \text{none of these} $
2009 AMC 12/AHSME, 24
The [i]tower function of twos[/i] is defined recursively as follows: $ T(1) \equal{} 2$ and $ T(n \plus{} 1) \equal{} 2^{T(n)}$ for $ n\ge1$. Let $ A \equal{} (T(2009))^{T(2009)}$ and $ B \equal{} (T(2009))^A$. What is the largest integer $ k$ such that
\[ \underbrace{\log_2\log_2\log_2\ldots\log_2B}_{k\text{ times}}
\]is defined?
$ \textbf{(A)}\ 2009\qquad \textbf{(B)}\ 2010\qquad \textbf{(C)}\ 2011\qquad \textbf{(D)}\ 2012\qquad \textbf{(E)}\ 2013$
1988 AMC 12/AHSME, 4
The slope of the line $\frac{x}{3} + \frac{y}{2} = 1$ is
$ \textbf{(A)}\ -\frac{3}{2}\qquad\textbf{(B)}\ -\frac{2}{3}\qquad\textbf{(C)}\ \frac{1}{3}\qquad\textbf{(D)}\ \frac{2}{3}\qquad\textbf{(E)}\ \frac{3}{2} $
2013 AMC 10, 4
When counting from $3$ to $201$, $53$ is the $51^{\text{st}}$ number counted. When counting backwards from $201$ to $3$, $53$ is the $n^{\text{th}}$ number counted. What is $n$?
$\textbf{(A) }146\qquad \textbf{(B) } 147\qquad\textbf{(C) } 148\qquad\textbf{(D) }149\qquad\textbf{(E) }150$
2021 AMC 12/AHSME Spring, 10
Two right circular cones with vertices facing down as shown in the figure below contain the same amount of liquid. The radii of the tops of the liquid surfaces are $3 \text{ cm}$ and $6 \text{ cm}$. Into each cone is dropped a spherical marble of radius $1 \text{ cm}$, which sinks to the bottom and is completely submerged without spilling any liquid. What is the ratio of the rise of the liquid level in the narrow cone to the rise of the liquid level in the wide cone?
$\textbf{(A) }1:1 \qquad \textbf{(B) }47:43 \qquad \textbf{(C) }2:1 \qquad \textbf{(D) }40:13 \qquad \textbf{(E) }4:1$
[asy]
size(350);
defaultpen(linewidth(0.8));
real h1 = 10, r = 3.1, s=0.75;
pair P = (r,h1), Q = (-r,h1), Pp = s * P, Qp = s * Q;
path e = ellipse((0,h1),r,0.9), ep = ellipse((0,h1*s),r*s,0.9);
draw(ellipse(origin,r*(s-0.1),0.8));
fill(ep,gray(0.8));
fill(origin--Pp--Qp--cycle,gray(0.8));
draw((-r,h1)--(0,0)--(r,h1)^^e);
draw(subpath(ep,0,reltime(ep,0.5)),linetype("4 4"));
draw(subpath(ep,reltime(ep,0.5),reltime(ep,1)));
draw(Qp--(0,Qp.y),Arrows(size=8));
draw(origin--(0,12),linetype("4 4"));
draw(origin--(r*(s-0.1),0));
label("$3$",(-0.9,h1*s),N,fontsize(10));
real h2 = 7.5, r = 6, s=0.6, d = 14;
pair P = (d+r-0.05,h2-0.15), Q = (d-r+0.05,h2-0.15), Pp = s * P + (1-s)*(d,0), Qp = s * Q + (1-s)*(d,0);
path e = ellipse((d,h2),r,1), ep = ellipse((d,h2*s+0.09),r*s,1);
draw(ellipse((d,0),r*(s-0.1),0.8));
fill(ep,gray(0.8));
fill((d,0)--Pp--Qp--cycle,gray(0.8));
draw(P--(d,0)--Q^^e);
draw(subpath(ep,0,reltime(ep,0.5)),linetype("4 4"));
draw(subpath(ep,reltime(ep,0.5),reltime(ep,1)));
draw(Qp--(d,Qp.y),Arrows(size=8));
draw((d,0)--(d,10),linetype("4 4"));
draw((d,0)--(d+r*(s-0.1),0));
label("$6$",(d-r/4,h2*s-0.06),N,fontsize(10));
[/asy]
2023 AMC 12/AHSME, 12
What is the value of
\[ 2^3 - 1^2 + 4^3 - 3^3 + 6^3 - 5^3 + \dots + 18^3 - 17^3?\]
$\textbf{(A) } 2023 \qquad\textbf{(B) } 2679 \qquad\textbf{(C) } 2941 \qquad\textbf{(D) } 3159 \qquad\textbf{(E) } 3235$
1979 AMC 12/AHSME, 3
[asy]
real s=sqrt(3)/2;
draw(box((0,0),(1,1)));
draw((1+s,0.5)--(1,1));
draw((1+s,0.5)--(1,0));
draw((0,1)--(1+s,0.5));
label("$A$",(1,1),N);
label("$B$",(1,0),S);
label("$C$",(0,0),W);
label("$D$",(0,1),W);
label("$E$",(1+s,0.5),E);
//Credit to TheMaskedMagician for the diagram
[/asy]
In the adjoining figure, $ABCD$ is a square, $ABE$ is an equilateral triangle and point $E$ is outside square $ABCD$. What is the measure of $\measuredangle AED$ in degrees?
$\textbf{(A) }10\qquad\textbf{(B) }12.5\qquad\textbf{(C) }15\qquad\textbf{(D) }20\qquad\textbf{(E) }25$
1990 AIME Problems, 6
A biologist wants to calculate the number of fish in a lake. On May 1 she catches a random sample of 60 fish, tags them, and releases them. On September 1 she catches a random sample of 70 fish and finds that 3 of them are tagged. To calculate the number of fish in the lake on May 1, she assumes that $25\%$ of these fish are no longer in the lake on September 1 (because of death and emigrations), that $40\%$ of the fish were not in the lake May 1 (because of births and immigrations), and that the number of untagged fish and tagged fish in the September 1 sample are representative of the total population. What does the biologist calculate for the number of fish in the lake on May 1?
1970 AMC 12/AHSME, 17
If $r\ge 0$, then for all $p$ and $q$ such that $pq\neq 0$ and $pr>qr$, we have
$\textbf{(A) }-p>-q\qquad\textbf{(B) }-p>q\qquad\textbf{(C) }1>-q/p\qquad$
$\textbf{(D) }1<q/p\qquad \textbf{(E) }\text{None of These}$
2012 AMC 8, 11
The mean, median, and unique mode of the positive integers 3, 4, 5, 6, 6, 7, $x$ are all equal. What is the value of $x$?
$\textbf{(A)}\hspace{.05in}5 \qquad \textbf{(B)}\hspace{.05in}6 \qquad \textbf{(C)}\hspace{.05in}7 \qquad \textbf{(D)}\hspace{.05in}11 \qquad \textbf{(E)}\hspace{.05in}12 $
2021 USAJMO, 5
A finite set $S$ of positive integers has the property that, for each $s \in S,$ and each positive integer divisor $d$ of $s$, there exists a unique element $t \in S$ satisfying $\text{gcd}(s, t) = d$. (The elements $s$ and $t$ could be equal.)
Given this information, find all possible values for the number of elements of $S$.
1981 AMC 12/AHSME, 20
A ray of light originates from point $A$ and and travels in a plane, being reflected $n$ times between lines $AD$ and $CD$, before striking a point $B$ (which may be on $AD$ or $CD$) perpendicularly and retracing its path to $A$. (At each point of reflection the light makes two equal angles as indicated in the adjoining figure. The figure shows the light path for $n = 3.$) If $\measuredangle CDA = 8^\circ$, what is the largest value $n$ can have?
$\text{(A)} \ 6 \qquad \text{(B)} \ 10 \qquad \text{(C)} \ 38 \qquad \text{(D)} \ 98 \qquad \text{(E)} \ \text{There is no largest value.}$
2021 AMC 12/AHSME Fall, 8
Let $M$ be the least common multiple of all the integers $10$ through $30,$ inclusive. Let $N$ be the least common multiple of $M,$ $32,$ $33,$ $34,$ $35,$ $36,$ $37,$ $38,$ $39,$ and $40.$ What is the value of $\frac{N}{M}?$
$(\textbf{A})\: 1\qquad(\textbf{B}) \: 2\qquad(\textbf{C}) \: 37\qquad(\textbf{D}) \: 74\qquad(\textbf{E}) \: 2886$
1972 AMC 12/AHSME, 34
Three times Dick's age plus Tom's age equals twice Harry's age. Double the cube of Harry's age is equal to three times the cube of Dick's age added to the cube of Tom's age. Their respective ages are relatively prime to each other. The sum of the squares of their ages is
$\textbf{(A) }42\qquad\textbf{(B) }46\qquad\textbf{(C) }122\qquad\textbf{(D) }290\qquad \textbf{(E) }326$
2015 AMC 10, 22
Eight people are sitting around a circular table, each holding a fair coin. All eight people flip their coins and those who flip heads stand while those who flip tails remain seated. What is the probability that no two adjacent people will stand?
$\textbf{(A) }\dfrac{47}{256}\qquad\textbf{(B) }\dfrac{3}{16}\qquad\textbf{(C) }\dfrac{49}{256}\qquad\textbf{(D) }\dfrac{25}{128}\qquad\textbf{(E) }\dfrac{51}{256}$
2010 Contests, 3
The 2010 positive numbers $a_1, a_2, \ldots , a_{2010}$ satisfy the inequality $a_ia_j \le i+j$ for all distinct indices $i, j$. Determine, with proof, the largest possible value of the product $a_1a_2\ldots a_{2010}$.
2012 AIME Problems, 11
Let $f_1(x) = \frac{2}{3}-\frac{3}{3x+1}$, and for $n \ge 2$, define $f_n(x) = f_1(f_{n-1} (x))$. The value of x that satisfies $f_{1001}(x) = x - 3$ can be expressed in the form $\frac{m}{n}$,
where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
2006 AMC 12/AHSME, 17
For a particular peculiar pair of dice, the probabilities of rolling 1, 2, 3, 4, 5 and 6 on each die are in the ratio $ 1: 2: 3: 4: 5: 6$. What is the probability of rolling a total of 7 on the two dice?
$ \textbf{(A) } \frac 4{63} \qquad \textbf{(B) } \frac 18 \qquad \textbf{(C) } \frac 8{63} \qquad \textbf{(D) } \frac 16 \qquad \textbf{(E) } \frac 27$
2000 AIME Problems, 13
In the middle of a vast prairie, a firetruck is stationed at the intersection of two perpendicular straight highways. The truck travels at $50$ miles per hour along the highways and at $14$ miles per hour across the prairie. Consider the set of points that can be reached by the firetruck within six minutes. The area of this region is $m/n$ square miles, where $m$ and $n$ are relatively prime positive integers. Find $m+n.$
2021 AMC 12/AHSME Fall, 13
The angle bisector of the acute angle formed at the origin by the graphs of the lines $y=x$ and $y=3x$ has equation $y=kx$. What is $k$?
$\textbf{(A)} \: \frac{1+\sqrt{5}}{2} \qquad \textbf{(B)} \: \frac{1+\sqrt{7}}{2} \qquad \textbf{(C)} \: \frac{2+\sqrt{3}}{2} \qquad \textbf{(D)} \: 2\qquad \textbf{(E)} \: \frac{2+\sqrt{5}}{2}$
2010 AMC 10, 3
A drawer contains red, green, blue, and white socks with at least 2 of each color. What is the minimum number of socks that must be pulled from the drawer to guarantee a matching pair?
$ \textbf{(A)}\ 3 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 8 \qquad \textbf{(E)}\ 9$
2017 AMC 12/AHSME, 3
Suppose that $x$ and $y$ are nonzero real numbers such that \[\frac{3x+y}{x-3y}= -2.\] What is the value of \[\frac{x+3y}{3x-y}?\]
$\textbf{(A) } {-3} \qquad \textbf{(B) } {-1} \qquad \textbf{(C) } 1 \qquad \textbf{(D) }2 \qquad \textbf{(E) } 3$
2008 AMC 12/AHSME, 20
Michael walks at the rate of $ 5$ feet per second on a long straight path. Trash pails are located every $ 200$ feet along the path. A garbage truck travels at $ 10$ feet per second in the same direction as Michael and stops for $ 30$ seconds at each pail. As Michael passes a pail, he notices the truck ahead of him just leaving the next pail. How many times will Michael and the truck meet?
$ \textbf{(A)}\ 4\qquad \textbf{(B)}\ 5\qquad \textbf{(C)}\ 6\qquad \textbf{(D)}\ 7\qquad \textbf{(E)}\ 8$