Found problems: 3632
2019 AMC 10, 15
A sequence of numbers is defined recursively by $a_1 = 1$, $a_2 = \frac{3}{7}$, and
$$a_n=\frac{a_{n-2} \cdot a_{n-1}}{2a_{n-2} - a_{n-1}}$$for all $n \geq 3$ Then $a_{2019}$ can be written as $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive inegers. What is $p+q ?$
$\textbf{(A) } 2020 \qquad\textbf{(B) } 4039 \qquad\textbf{(C) } 6057 \qquad\textbf{(D) } 6061 \qquad\textbf{(E) } 8078$
1959 AMC 12/AHSME, 31
A square, with an area of $40$, is inscribed in a semicircle. The area of a square that could be inscribed in the entire circle with the same radius, is:
$ \textbf{(A)}\ 80 \qquad\textbf{(B)}\ 100\qquad\textbf{(C)}\ 120\qquad\textbf{(D)}\ 160\qquad\textbf{(E)}\ 200 $
2014 AMC 12/AHSME, 16
The product $(8)(888\ldots 8)$, where the second factor has $k$ digits, is an integer whose digits have a sum of $1000$. What is $k$?
$\textbf{(A) }901\qquad
\textbf{(B) }911\qquad
\textbf{(C) }919\qquad
\textbf{(D) }991\qquad
\textbf{(E) }999\qquad$
2025 AIME, 3
Four unit squares form a $2 \times 2$ grid. Each of the $12$ unit line segments forming the sides of the squares is colored either red or blue in such way that each square has $2$ red sides and blue sides. One example is shown below (red is solid, blue is dashed). Find the number of such colorings.
[asy]
size(4cm);
defaultpen(linewidth(1.2));
draw((0, 0) -- (2, 0) -- (2, 1));
draw((0, 1) -- (1, 1) -- (1, 2) -- (2,2));
draw((0, 0) -- (0, 1), dotted);
draw((1, 0) -- (1, 1) -- (2, 1) -- (2, 2), dotted);
draw((0, 1) -- (0, 2) -- (1, 2), dotted);
[/asy]
2006 AMC 10, 4
A digital watch displays hours and minutes with $ \text c{AM}$ and $ \text c{PM}$. What is the largest possible sum of the digits in the display?
$ \textbf{(A) } 17\qquad \textbf{(B) } 19\qquad \textbf{(C) } 21\qquad \textbf{(D) } 22\qquad \textbf{(E) } 23$
2010 AMC 10, 23
Each of 2010 boxes in a line contains a single red marble, and for $ 1 \le k \le 2010$, the box in the $ kth$ position also contains $ k$ white marbles. Isabella begins at the first box and successively draws a single marble at random from each box, in order. She stops when she first draws a red marble. Let $ P(n)$ be the probability that Isabella stops after drawing exactly $ n$ marbles. What is the smallest value of $ n$ for which $ P(n) < \frac {1}{2010}$?
$ \textbf{(A)}\ 45 \qquad
\textbf{(B)}\ 63 \qquad
\textbf{(C)}\ 64 \qquad
\textbf{(D)}\ 201 \qquad
\textbf{(E)}\ 1005$
2009 AMC 10, 2
Which of the following is equal to $ \dfrac{\frac{1}{3}\minus{}\frac{1}{4}}{\frac{1}{2}\minus{}\frac{1}{3}}$?
$ \textbf{(A)}\ \frac{1}{4} \qquad
\textbf{(B)}\ \frac{1}{3} \qquad
\textbf{(C)}\ \frac{1}{2} \qquad
\textbf{(D)}\ \frac{2}{3} \qquad
\textbf{(E)}\ \frac{3}{4}$
1996 AMC 8, 15
The remainder when the product $1492\cdot 1776\cdot 1812\cdot 1996$ is divided by $5$ is
$\text{(A)}\ 0 \qquad \text{(B)}\ 1 \qquad \text{(C)}\ 2 \qquad \text{(D)}\ 3 \qquad \text{(E)}\ 4$
1961 AMC 12/AHSME, 11
Two tangents are drawn to a circle from an exterior point $A$; they touch the circle at points $B$ and $C$ respectively. A third tangent intersects segment $AB$ in $P$ and $AC$ in $R$, and touches the circle at $Q$. If $AB=20$, then the perimeter of triangle $APR$ is
${{ \textbf{(A)}\ 42\qquad\textbf{(B)}\ 40.5 \qquad\textbf{(C)}\ 40\qquad\textbf{(D)}\ 39\frac{7}{8} }\qquad\textbf{(E)}\ \text{not determined by the given information} } $
1968 AMC 12/AHSME, 21
If $S=1!+2!+3!+ \cdots +99!$, then the units' digit in the value of $S$ is:
$\textbf{(A)}\ 9 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ 0$
2012 AMC 10, 10
How many ordered pairs of positive integers $(M,N)$ staisfy the equation $\frac{M}{6}=\frac{6}{N}$?
$ \textbf{(A)}\ 6\qquad\textbf{(B)}\ 7\qquad\textbf{(C)}\ 8\qquad\textbf{(D)}\ 9\qquad\textbf{(E)}\ 10 $
1987 AIME Problems, 10
Al walks down to the bottom of an escalator that is moving up and he counts 150 steps. His friend, Bob, walks up to the top of the escalator and counts 75 steps. If Al's speed of walking (in steps per unit time) is three times Bob's walking speed, how many steps are visible on the escalator at a given time? (Assume that this value is constant.)
2021 AIME Problems, 6
Segments $\overline{AB}, \overline{AC},$ and $\overline{AD}$ are edges of a cube and $\overline{AG}$ is a diagonal through the center of the cube. Point $P$ satisfies $BP=60\sqrt{10}$, $CP=60\sqrt{5}$, $DP=120\sqrt{2}$, and $GP=36\sqrt{7}$. Find $AP.$
2009 AIME Problems, 12
In right $ \triangle ABC$ with hypotenuse $ \overline{AB}$, $ AC \equal{} 12$, $ BC \equal{} 35$, and $ \overline{CD}$ is the altitude to $ \overline{AB}$. Let $ \omega$ be the circle having $ \overline{CD}$ as a diameter. Let $ I$ be a point outside $ \triangle ABC$ such that $ \overline{AI}$ and $ \overline{BI}$ are both tangent to circle $ \omega$. The ratio of the perimeter of $ \triangle ABI$ to the length $ AB$ can be expressed in the form $ \displaystyle\frac{m}{n}$, where $ m$ and $ n$ are relatively prime positive integers. Find $ m\plus{}n$.
2014 AMC 12/AHSME, 22
The number $5^{867}$ is between $2^{2013}$ and $2^{2014}$. How many pairs of integers $(m,n)$ are there such that $1\leq m\leq 2012$ and \[5^n<2^m<2^{m+2}<5^{n+1}?\]
$\textbf{(A) }278\qquad
\textbf{(B) }279\qquad
\textbf{(C) }280\qquad
\textbf{(D) }281\qquad
\textbf{(E) }282\qquad$
2014 India IMO Training Camp, 2
Let $n$ be a natural number.A triangulation of a convex n-gon is a division of the polygon into $n-2$ triangles by drawing $n-3$ diagonals no two of which intersect at an interior point of the polygon.Let $f(n)$ denote the number of triangulations of a regular n-gon such that each of the triangles formed is isosceles.Determine $f(n)$ in terms of $n$.
1996 AMC 8, 11
Let $x$ be the number
\[0.\underbrace{0000...0000}_{1996\text{ zeros}}1,\]
where there are 1996 zeros after the decimal point. Which of the following expressions represents the largest number?
$\text{(A)}\ 3+x \qquad \text{(B)}\ 3-x \qquad \text{(C)}\ 3\cdot x \qquad \text{(D)}\ 3/x \qquad \text{(E)}\ x/3$
1967 AMC 12/AHSME, 12
If the (convex) area bounded by the x-axis and the lines $y=mx+4$, $x=1$, and $x=4$ is $7$, then $m$ equals:
$\textbf{(A)}\ -\frac{1}{2}\qquad
\textbf{(B)}\ -\frac{2}{3}\qquad
\textbf{(C)}\ -\frac{3}{2} \qquad
\textbf{(D)}\ -2 \qquad
\textbf{(E)}\ \text{none of these}$
2019 AMC 12/AHSME, 2
Consider the statement, "If $n$ is not prime, then $n-2$ is prime." Which of the following values of $n$ is a counterexample to this statement?
$\textbf{(A) } 11 \qquad \textbf{(B) } 15 \qquad \textbf{(C) } 19 \qquad \textbf{(D) } 21 \qquad \textbf{(E) } 27$
2006 AMC 12/AHSME, 15
Suppose $ \cos x \equal{} 0$ and $ \cos (x \plus{} z) \equal{} 1/2$. What is the smallest possible positive value of $ z$?
$ \textbf{(A) } \frac {\pi}{6}\qquad \textbf{(B) } \frac {\pi}{3}\qquad \textbf{(C) } \frac {\pi}{2}\qquad \textbf{(D) } \frac {5\pi}{6}\qquad \textbf{(E) } \frac {7\pi}{6}$
2021 AMC 10 Fall, 13
Each of $6$ balls is randomly and independently painted either black or white with equal probability. What is the probability that every ball is different in color from more than half of the other $5$ balls?
$\textbf{(A) }\dfrac1{64}\qquad\textbf{(B) }\dfrac16\qquad\textbf{(C) }\dfrac14\qquad\textbf{(D) }\dfrac5{16}\qquad\textbf{(E) }\dfrac12$
2022 AIME Problems, 14
Given $\triangle ABC$ and a point $P$ on one of its sides, call line $\ell$ the splitting line of $\triangle ABC$ through $P$ if $\ell$ passes through $P$ and divides $\triangle ABC$ into two polygons of equal perimeter. Let $\triangle ABC$ be a triangle where $BC = 219$ and $AB$ and $AC$ are positive integers. Let $M$ and $N$ be the midpoints of $\overline{AB}$ and $\overline{AC}$, respectively, and suppose that the splitting lines of $\triangle ABC$ through $M$ and $N$ intersect at $30^{\circ}$. Find the perimeter of $\triangle ABC$.
1987 AMC 12/AHSME, 16
A cryptographer devises the following method for encoding positive integers. First, the integer is expressed in base $5$. Second, a 1-to-1 correspondence is established between the digits that appear in the expressions in base $5$ and the elements of the set $\{V, W, X, Y, Z\}$. Using this correspondence, the cryptographer finds that three consecutive integers in increasing order are coded as $VYZ$, $VYX$, $VVW$, respectively. What is the base-10 expression for the integer coded as $XYZ$?
$ \textbf{(A)}\ 48 \qquad\textbf{(B)}\ 71 \qquad\textbf{(C)}\ 82 \qquad\textbf{(D)}\ 108 \qquad\textbf{(E)}\ 113$
2018 AMC 10, 25
For a positive integer $n$ and nonzero digits $a$, $b$, and $c$, let $A_n$ be the $n$-digit integer each of whose digits is equal to $a$; let $B_n$ be the $n$-digit integer each of whose digits is equal to $b$, and let $C_n$ be the $2n$-digit (not $n$-digit) integer each of whose digits is equal to $c$. What is the greatest possible value of $a + b + c$ for which there are at least two values of $n$ such that $C_n - B_n = A_n^2$?
$\textbf{(A)} \text{ 12} \qquad \textbf{(B)} \text{ 14} \qquad \textbf{(C)} \text{ 16} \qquad \textbf{(D)} \text{ 18} \qquad \textbf{(E)} \text{ 20}$
1993 AMC 12/AHSME, 14
The convex pentagon $ABCDE$ has $\angle A=\angle B=120^{\circ}$, $EA=AB=BC=2$ and $CD=DE=4$. What is the area of $ABCDE$?
[asy]
draw((0,0)--(1,0)--(1.5,sqrt(3)/2)--(0.5,3sqrt(3)/2)--(-0.5,sqrt(3)/2)--cycle);
dot((0,0));
dot((1,0));
dot((1.5,sqrt(3)/2));
dot((0.5,3sqrt(3)/2));
dot((-0.5,sqrt(3)/2));
label("A", (0,0), SW);
label("B", (1,0), SE);
label("C", (1.5,sqrt(3)/2), E);
label("D", (0.5,3sqrt(3)/2), N);
label("E", (-.5, sqrt(3)/2), W);
[/asy]
$ \textbf{(A)}\ 10 \qquad\textbf{(B)}\ 7\sqrt{3} \qquad\textbf{(C)}\ 15 \qquad\textbf{(D)}\ 9\sqrt{3} \qquad\textbf{(E)}\ 12\sqrt{5} $