Found problems: 3632
2000 Manhattan Mathematical Olympiad, 4
An equilateral triangle $ABC$ is given, together with a point $P$ inside it.
[asy]
draw((0,0)--(4,0)--(2,3.464)--(0,0));
draw((1.3, 1.2)--(0,0));
draw((1.3, 1.2)--(2,3.464));
draw((1.3, 1.2)--(4,0));
label("$A$",(0,0),SW);
label("$B$",(4,0),SE);
label("$C$",(2,3.464),N);
label("$P$",(1.3,1.2),S);
[/asy]
Given that $PA = 3$ cm, $PB = 5$ cm, and $PC = 4$ cm, find the side of the equilateral triangle.
2013 AMC 12/AHSME, 2
A softball team played ten games, scoring $1,2,3,4,5,6,7,8,9$, and $10$ runs. They lost by one run in exactly five games. In each of the other games, they scored twice as many runs as their opponent. How many total runs did their opponents score?
$ \textbf {(A) } 35 \qquad \textbf {(B) } 40 \qquad \textbf {(C) } 45 \qquad \textbf {(D) } 50 \qquad \textbf {(E) } 55 $
2011 AMC 10, 1
What is \[\frac{2+4+6}{1+3+5} - \frac{1+3+5}{2+4+6}?\]
$ \textbf{(A)}\ -1 \qquad
\textbf{(B)}\ \frac{5}{36} \qquad
\textbf{(C)}\ \frac{7}{12} \qquad
\textbf{(D)}\ \frac{147}{60} \qquad
\textbf{(E)}\ \frac{43}{3} $
2023 AMC 10, 7
Square $ABCD$ is rotated $20^\circ$ clockwise about its center to obtain square $EFGH$, as shown below. What is the degree measure of $\angle EAB$?
[asy]
size(170);
defaultpen(linewidth(0.6));
real r = 25;
draw(dir(135)--dir(45)--dir(315)--dir(225)--cycle);
draw(dir(135-r)--dir(45-r)--dir(315-r)--dir(225-r)--cycle);
label("$A$",dir(135),NW);
label("$B$",dir(45),NE);
label("$C$",dir(315),SE);
label("$D$",dir(225),SW);
label("$E$",dir(135-r),N);
label("$F$",dir(45-r),E);
label("$G$",dir(315-r),S);
label("$H$",dir(225-r),W);
[/asy]
$\textbf{(A) }20^\circ\qquad\textbf{(B) }30^\circ\qquad\textbf{(C) }32^\circ\qquad\textbf{(D) }35^\circ\qquad\textbf{(E) }45^\circ$
2011 AMC 12/AHSME, 7
Let $x$ and $y$ be two-digit positive integers with mean 60. What is the maximum value of the ratio $\frac{x}{y}$?
$ \textbf{(A)}\ 3 \qquad
\textbf{(B)}\ \frac{33}{7} \qquad
\textbf{(C)}\ \frac{39}{7} \qquad
\textbf{(D)}\ 9 \qquad
\textbf{(E)}\ \frac{99}{10} $
2014 AMC 12/AHSME, 2
At the theater children get in for half price. The price for $5$ adult tickets and $4$ child tickets is $\$24.50$. How much would $8$ adult tickets and $6$ child tickets cost?
$\textbf{(A) }\$35\qquad
\textbf{(B) }\$38.50\qquad
\textbf{(C) }\$40\qquad
\textbf{(D) }\$42\qquad
\textbf{(E) }\$42.50$
2012 AMC 12/AHSME, 25
Let $f(x)=|2\{x\} -1|$ where $\{x\}$ denotes the fractional part of $x$. The number $n$ is the smallest positive integer such that the equation $$nf(xf(x)) = x$$ has at least $2012$ real solutions $x$. What is $n$?
$\textbf{Note:}$ the fractional part of $x$ is a real number $y= \{x\}$, such that $ 0 \le y < 1$ and $x-y$ is an integer.
$ \textbf{(A)}\ 30\qquad\textbf{(B)}\ 31\qquad\textbf{(C)}\ 32\qquad\textbf{(D)}\ 62\qquad\textbf{(E)}\ 64 $
2014 USAJMO, 1
Let $a$, $b$, $c$ be real numbers greater than or equal to $1$. Prove that
\[ \min \left(\frac{10a^2-5a+1}{b^2-5b+10},\frac{10b^2-5b+1}{c^2-5c+10},\frac{10c^2-5c+1}{a^2-5a+10}\right )\leq abc. \]
2011 AMC 12/AHSME, 11
A frog located at $(x,y)$, with both $x$ and $y$ integers, makes successive jumps of length $5$ and always lands on points with integer coordinates. Suppose that the frog starts at $(0,0)$ and ends at $(1,0)$. What is the smallest possible number of jumps the frog makes?
$ \textbf{(A)}\ 2 \qquad
\textbf{(B)}\ 3 \qquad
\textbf{(C)}\ 4 \qquad
\textbf{(D)}\ 5 \qquad
\textbf{(E)}\ 6 $
PEN O Problems, 54
Let $S$ be a subset of $\{1, 2, 3, \cdots, 1989 \}$ in which no two members differ by exactly $4$ or by exactly $7$. What is the largest number of elements $S$ can have?
2008 AMC 12/AHSME, 22
A round table has radius $ 4$. Six rectangular place mats are placed on the table. Each place mat has width $ 1$ and length $ x$ as shown. They are positioned so that each mat has two corners on the edge of the table, these two corners being end points of the same side of length $ x$. Further, the mats are positioned so that the inner corners each touch an inner corner of an adjacent mat. What is $ x$?
[asy]unitsize(4mm);
defaultpen(linewidth(.8)+fontsize(8));
draw(Circle((0,0),4));
path mat=(-2.687,-1.5513)--(-2.687,1.5513)--(-3.687,1.5513)--(-3.687,-1.5513)--cycle;
draw(mat);
draw(rotate(60)*mat);
draw(rotate(120)*mat);
draw(rotate(180)*mat);
draw(rotate(240)*mat);
draw(rotate(300)*mat);
label("$x$",(-2.687,0),E);
label("$1$",(-3.187,1.5513),S);[/asy]$ \textbf{(A)}\ 2\sqrt {5} \minus{} \sqrt {3} \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ \frac {3\sqrt {7} \minus{} \sqrt {3}}{2} \qquad \textbf{(D)}\ 2\sqrt {3} \qquad \textbf{(E)}\ \frac {5 \plus{} 2\sqrt {3}}{2}$
2002 AIME Problems, 7
The Binomial Expansion is valid for exponents that are not integers. That is, for all real numbers $ x, y,$ and $ r$ with $ |x| > |y|,$
\[ (x \plus{} y)^r \equal{} x^r \plus{} rx^{r \minus{} 1}y \plus{} \frac {r(r \minus{} 1)}2x^{r \minus{} 2}y^2 \plus{} \frac {r(r \minus{} 1)(r \minus{} 2)}{3!}x^{r \minus{} 3}y^3 \plus{} \cdots
\]
What are the first three digits to the right of the decimal point in the decimal representation of $ \left(10^{2002} \plus{} 1\right)^{10/7}?$
1983 USAMO, 1
On a given circle, six points $A$, $B$, $C$, $D$, $E$, and $F$ are chosen at random, independently and uniformly with respect to arc length. Determine the probability that the two triangles $ABC$ and $DEF$ are disjoint, i.e., have no common points.
2015 AMC 12/AHSME, 2
Two of the three sides of a triangle are $20$ and $15$. Which of the following numbers is not a possible perimeter of the triangle?
$\textbf{(A) }52\qquad\textbf{(B) }57\qquad\textbf{(C) }62\qquad\textbf{(D) }67\qquad\textbf{(E) }72$
2006 AMC 12/AHSME, 3
The ratio of Mary's age to Alice's age is $ 3: 5$. Alice is $ 30$ years old. How old is Mary?
$ \textbf{(A) } 15\qquad \textbf{(B) } 18\qquad \textbf{(C) } 20\qquad \textbf{(D) } 24\qquad \textbf{(E) } 50$
1987 AIME Problems, 4
Find the area of the region enclosed by the graph of $|x-60|+|y|=|x/4|.$
2018 AMC 10, 25
Let $\lfloor x\rfloor$ denote the greatest integer less than or equal to $x$. How many real numbers $x$ satisfy the equation $x^2 + 10{,}000\lfloor x \rfloor = 10{,}000x$?
$\textbf{(A) } 197 \qquad \textbf{(B) } 198 \qquad \textbf{(C) } 199 \qquad \textbf{(D) } 200 \qquad \textbf{(E) } 201$
1976 AMC 12/AHSME, 3
The sum of the distances from one vertex of a square with sides of length two to the midpoints of each of the sides of the square is
$\textbf{(A) }2\sqrt{5}\qquad\textbf{(B) }2+\sqrt{3}\qquad\textbf{(C) }2+2\sqrt{3}\qquad\textbf{(D) }2+\sqrt{5}\qquad \textbf{(E) }2+2\sqrt{5}$
1990 USAMO, 1
A certain state issues license plates consisting of six digits (from 0 to 9). The state requires that any two license plates differ in at least two places. (For instance, the numbers 027592 and 020592 cannot both be used.) Determine, with proof, the maximum number of distinct license plates that the state can use.
2011 USAJMO, 3
For a point $P = (a,a^2)$ in the coordinate plane, let $l(P)$ denote the line passing through $P$ with slope $2a$. Consider the set of triangles with vertices of the form $P_1 = (a_1, a_1^2), P_2 = (a_2, a_2^2), P_3 = (a_3, a_3^2)$, such that the intersection of the lines $l(P_1), l(P_2), l(P_3)$ form an equilateral triangle $\triangle$. Find the locus of the center of $\triangle$ as $P_1P_2P_3$ ranges over all such triangles.
2010 AMC 10, 4
A book that is to be recorded onto compact discs takes $ 412$ minutes to read aloud. Each disc can hold up to $ 56$ minutes of reading. Assume that the smallest possible number of discs is used and that each disc contains the same length of reading. How many minutes of reading will each disc contain?
$ \textbf{(A)}\ 50.2 \qquad
\textbf{(B)}\ 51.5 \qquad
\textbf{(C)}\ 52.4 \qquad
\textbf{(D)}\ 53.8 \qquad
\textbf{(E)}\ 55.2$
1979 USAMO, 5
A certain organization has $n$ members, and it has $n\plus{}1$ three-member committees, no two of which have identical member-ship. Prove that there are two committees which share exactly one member.
1994 AMC 12/AHSME, 26
A regular polygon of $m$ sides is exactly enclosed (no overlaps, no gaps) by $m$ regular polygons of $n$ sides each. (Shown here for $m=4, n=8$.) If $m=10$, what is the value of $n$?
[asy]
size(200);
defaultpen(linewidth(0.8));
draw(unitsquare);
path p=(0,1)--(1,1)--(1+sqrt(2)/2,1+sqrt(2)/2)--(1+sqrt(2)/2,2+sqrt(2)/2)--(1,2+sqrt(2))--(0,2+sqrt(2))--(-sqrt(2)/2,2+sqrt(2)/2)--(-sqrt(2)/2,1+sqrt(2)/2)--cycle;
draw(p);
draw(shift((1+sqrt(2)/2,-sqrt(2)/2-1))*p);
draw(shift((0,-2-sqrt(2)))*p);
draw(shift((-1-sqrt(2)/2,-sqrt(2)/2-1))*p);[/asy]
$ \textbf{(A)}\ 5 \qquad\textbf{(B)}\ 6 \qquad\textbf{(C)}\ 14 \qquad\textbf{(D)}\ 20 \qquad\textbf{(E)}\ 26 $
2020 AMC 10, 1
What is the value of $$1-(-2)-3-(-4)-5-(-6)?$$
$\textbf{(A) } -20 \qquad\textbf{(B) } -3 \qquad\textbf{(C) } 3 \qquad\textbf{(D) } 5 \qquad\textbf{(E) } 21$
2012 Hanoi Open Mathematics Competitions, 6
For every n = 2; 3; : : : , we put
$$A_n = \left(1 - \frac{1}{1+2}\right) X \left(1 - \frac{1}{1+2+3}\right)X \left(1 - \frac{1}{1+2+3+...+n}\right) $$
Determine all positive integer $ n (n \geq 2)$ such that $\frac{1}{A_n}$ is an integer.