Found problems: 3632
2007 AMC 12/AHSME, 24
How many pairs of positive integers $ (a,b)$ are there such that $ \gcd(a,b) \equal{} 1$ and
\[ \frac {a}{b} \plus{} \frac {14b}{9a}
\]is an integer?
$ \textbf{(A)}\ 4 \qquad \textbf{(B)}\ 6 \qquad \textbf{(C)}\ 9 \qquad \textbf{(D)}\ 12 \qquad \textbf{(E)}\ \text{infinitely many}$
1979 AMC 12/AHSME, 2
For all non-zero real numbers $x$ and $y$ such that $x-y=xy$, $\frac{1}{x}-\frac{1}{y}$ equals
$\textbf{(A) }\frac{1}{xy}\qquad\textbf{(B) }\frac{1}{x-y}\qquad\textbf{(C) }0\qquad\textbf{(D) }-1\qquad\textbf{(E) }y-x$
1963 AMC 12/AHSME, 35
The lengths of the sides of a triangle are integers, and its area is also an integer. One side is $21$ and the perimeter is $48$. The shortest side is:
$\textbf{(A)}\ 8 \qquad
\textbf{(B)}\ 10\qquad
\textbf{(C)}\ 12 \qquad
\textbf{(D)}\ 14 \qquad
\textbf{(E)}\ 16$
2013 AMC 12/AHSME, 11
Two bees start at the same spot and fly at the same rate in the following directions. Bee $A$ travels $1$ foot north, then $1$ foot east, then $1$ foot upwards, and then continues to repeat this pattern. Bee $B$ travels $1$ foot south, then $1$ foot west, and then continues to repeat this pattern. In what directions are the bees traveling when they are exactly $10$ feet away from each other?
$\textbf{(A) }A \text{ east}, B \text{ west} \qquad \textbf{(B) } A\text{ north}, B\text{ south} \qquad \textbf{(C) } A\text{ north}, B\text{ west} \qquad \textbf{(D) } A\text{ up}, B\text{ south} \qquad \textbf{(E) } A\text{ up}, B\text{ west}$
2009 AMC 10, 4
A rectangular yard contains two flower beds in the shape of congruent isosceles right triangles. THe remainder of the yard has a trapezoidal shape, as shown. The parallel sides of the trapezoid have lengths $ 15$ and $ 25$ meters. What fraction of the yard is occupied by the flower beds?
[asy]unitsize(2mm);
defaultpen(linewidth(.8pt));
fill((0,0)--(0,5)--(5,5)--cycle,gray);
fill((25,0)--(25,5)--(20,5)--cycle,gray);
draw((0,0)--(0,5)--(25,5)--(25,0)--cycle);
draw((0,0)--(5,5));
draw((20,5)--(25,0));[/asy]$ \textbf{(A)}\ \frac18\qquad
\textbf{(B)}\ \frac16\qquad
\textbf{(C)}\ \frac15\qquad
\textbf{(D)}\ \frac14\qquad
\textbf{(E)}\ \frac13$
1967 AMC 12/AHSME, 36
Given a geometric progression of five terms, each a positive integer less than $100$. The sum of the five terms is $211$. If $S$ is the sum of those terms in the progression which are squares of integers, then $S$ is:
$\textbf{(A)}\ 0\qquad
\textbf{(B)}\ 91\qquad
\textbf{(C)}\ 133\qquad
\textbf{(D)}\ 195\qquad
\textbf{(E)}\ 211$
2020 AMC 12/AHSME, 17
The vertices of a quadrilateral lie on the graph of $y = \ln x$, and the $x$-coordinates of these vertices are consecutive positive integers. The area of the quadrilateral is $\ln \frac{91}{90}$. What is the $x$-coordinate of the leftmost vertex?
$\textbf{(A)}\ 6\qquad\textbf{(B)}\ 7\qquad\textbf{(C)}\ 10\qquad\textbf{(D)}\ 12\qquad\textbf{(E)}\ 13$
2022 AIME Problems, 1
Adults made up $\frac5{12}$ of the crowd of people at a concert. After a bus carrying $50$ more people arrived, adults made up $\frac{11}{25}$ of the people at the concert. Find the minimum number of adults who could have been at the concert after the bus arrived.
2013 USAJMO, 6
Find all real numbers $x,y,z\geq 1$ satisfying \[\min(\sqrt{x+xyz},\sqrt{y+xyz},\sqrt{z+xyz})=\sqrt{x-1}+\sqrt{y-1}+\sqrt{z-1}.\]
2001 USAMO, 6
Each point in the plane is assigned a real number such that, for any triangle, the number at the center of its inscribed circle is equal to the arithmetic mean of the three numbers at its vertices. Prove that all points in the plane are assigned the same number.
2008 AMC 10, 6
Points $ B$ and $ C$ lie on $ \overline{AD}$. The length of $ \overline{AB}$ is $ 4$ times the length of $ \overline{BD}$, and the length of $ \overline{AC}$ is $ 9$ times the length of $ \overline{CD}$. The length of $ \overline{BC}$ is what fraction of the length of $ \overline{AD}$?
$ \textbf{(A)}\ \frac{1}{36} \qquad
\textbf{(B)}\ \frac{1}{13} \qquad
\textbf{(C)}\ \frac{1}{10} \qquad
\textbf{(D)}\ \frac{5}{36} \qquad
\textbf{(E)}\ \frac{1}{5}$
1992 AMC 12/AHSME, 10
The number of positive integers $k$ for which the equation $kx - 12 = 3k$ has an integer solution for $x$ is
$ \textbf{(A)}\ 3\qquad\textbf{(B)}\ 4\qquad\textbf{(C)}\ 5\qquad\textbf{(D)}\ 6\qquad\textbf{(E)}\ 7 $
1961 AMC 12/AHSME, 3
If the graphs of $2y+x+3=0$ and $3y+ax+2=0$ are to meet at right angles, the value of $a$ is:
${{ \textbf{(A)}\ \pm \frac{2}{3} \qquad\textbf{(B)}\ -\frac{2}{3}\qquad\textbf{(C)}\ -\frac{3}{2} \qquad\textbf{(D)}\ 6}\qquad\textbf{(E)}\ -6} $
1994 AMC 12/AHSME, 22
Nine chairs in a row are to be occupied by six students and Professors Alpha, Beta and Gamma. These three professors arrive before the six students and decide to choose their chairs so that each professor will be between two students. In how many ways can Professors Alpha, Beta and Gamma choose their chairs?
$ \textbf{(A)}\ 12 \qquad\textbf{(B)}\ 36 \qquad\textbf{(C)}\ 60 \qquad\textbf{(D)}\ 84 \qquad\textbf{(E)}\ 630 $
2001 AIME Problems, 15
The numbers 1, 2, 3, 4, 5, 6, 7, and 8 are randomly written on the faces of a regular octahedron so that each face contains a different number. The probability that no two consecutive numbers, where 8 and 1 are considered to be consecutive, are written on faces that share an edge is $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$
2019 AMC 12/AHSME, 6
In a given plane, points $A$ and $B$ are $10$ units apart. How many points $C$ are there in the plane such that the perimeter of $\triangle ABC$ is $50$ units and the area of $\triangle ABC$ is $100$ square units?
$\textbf{(A) }0\qquad\textbf{(B) }2\qquad\textbf{(C) }4\qquad\textbf{(D) }8\qquad\textbf{(E) }\text{infinitely many}$
2019 AMC 12/AHSME, 12
Positive real numbers $x \neq 1$ and $y \neq 1$ satisfy $\log_2{x} = \log_y{16}$ and $xy = 64$. What is $(\log_2{\tfrac{x}{y}})^2$?
$\textbf{(A) } \frac{25}{2} \qquad\textbf{(B) } 20 \qquad\textbf{(C) } \frac{45}{2} \qquad\textbf{(D) } 25 \qquad\textbf{(E) } 32$
1997 AMC 8, 11
Let $\boxed{N}$ mean the number of whole number divisors of $N$. For example, $\boxed{3}=2$ because 3 has two divisors, 1 and 3. Find the value of
\[\boxed{\boxed{11}\times\boxed{20}}.\]
$\textbf{(A)}\ 6 \qquad \textbf{(B)}\ 8 \qquad \textbf{(C)}\ 12 \qquad \textbf{(D)}\ 16 \qquad \textbf{(E)}\ 24$
2006 AMC 12/AHSME, 11
Which of the following describes the graph of the equation $ (x \plus{} y)^2 \equal{} x^2 \plus{} y^2$?
$ \textbf{(A)}\text{ the empty set}\qquad \textbf{(B)}\text{ one point}\qquad \textbf{(C)}\text{ two lines}$
$\textbf{(D)}\text{ a circle}\qquad \textbf{(E)}\text{ the entire plane}$
2010 Contests, 1
Maya lists all the positive divisors of $ 2010^2$. She then randomly selects two distinct divisors from this list. Let $ p$ be the probability that exactly one of the selected divisors is a perfect square. The probability $ p$ can be expressed in the form $ \frac{m}{n}$, where $ m$ and $ n$ are relatively prime positive integers. Find $ m \plus{} n$.
2013 AMC 10, 23
In $ \bigtriangleup ABC $, $ AB = 86 $, and $ AC = 97 $. A circle with center $ A $ and radius $ AB $ intersects $ \overline{BC} $ at points $ B $ and $ X $. Moreover $ \overline{BX} $ and $ \overline{CX} $ have integer lengths. What is $ BC $?
$ \textbf{(A)} \ 11 \qquad \textbf{(B)} \ 28 \qquad \textbf{(C)} \ 33 \qquad \textbf{(D)} \ 61 \qquad \textbf{(E)} \ 72 $
2015 AMC 10, 18
Johann has $64$ fair coins. He flips all the coins. Any coin that lands on tails is tossed again. Coins that land on tails on the second toss are tossed a third time. What is the expected number of coins that are now heads?
$\textbf{(A) } 32
\qquad\textbf{(B) } 40
\qquad\textbf{(C) } 48
\qquad\textbf{(D) } 56
\qquad\textbf{(E) } 64
$
2000 AIME Problems, 6
One base of a trapezoid is 100 units longer than the other base. The segment that joins the midpoints of the legs divides the trapezoid into two regions whose areas are in the ratio $2: 3.$ Let $x$ be the length of the segment joining the legs of the trapezoid that is parallel to the bases and that divides the trapezoid into two regions of equal area. Find the greatest integer that does not exceed $x^2/100.$
1972 AMC 12/AHSME, 16
There are two positive numbers that may be inserted between $3$ and $9$ such that the first three are in geometric progression while the last three are in arithmetic progression. The sum of those two positive numbers is
$\textbf{(A) }13\textstyle\frac{1}{2}\qquad\textbf{(B) }11\frac{1}{4}\qquad\textbf{(C) }10\frac{1}{2}\qquad\textbf{(D) }10\qquad \textbf{(E) }9\frac{1}{2}$
2022 AMC 12/AHSME, 16
A [i]triangular number[/i] is a positive integer that can be expressed in the form $t_n = 1 + 2 + 3 +\cdots + n$, for some positive integer $n$. The three smallest triangular numbers that are also perfect squares are $t_1 = 1 = 1^2$, $t_8 = 36 = 6^2$, and $t_{49} = 1225 = 35^2$. What is the sum of the digits of the fourth smallest triangular number that is also a perfect square?
$\textbf{(A)} ~6 \qquad\textbf{(B)} ~9 \qquad\textbf{(C)} ~12 \qquad\textbf{(D)} ~18 \qquad\textbf{(E)} ~27 $