Found problems: 3632
1969 AMC 12/AHSME, 9
The arithmetic mean (ordinary average) of the fifty-two successive positive integers beginning with $2$ is:
$\textbf{(A) }27\qquad
\textbf{(B) }27\tfrac14\qquad
\textbf{(C) }27\tfrac12\qquad
\textbf{(D) }28\qquad
\textbf{(E) }28\tfrac12$
1971 AMC 12/AHSME, 19
If the line $y=mx+1$ intersects the ellipse $x^2+4y^2=1$ exactly once, then the value of $m^2$ is
$\textbf{(A) }\textstyle\frac{1}{2}\qquad\textbf{(B) }\frac{2}{3}\qquad\textbf{(C) }\frac{3}{4}\qquad\textbf{(D) }\frac{4}{5}\qquad \textbf{(E) }\frac{5}{6}$
1994 AMC 12/AHSME, 24
A sample consisting of five observations has an arithmetic mean of $10$ and a median of $12$. The smallest value that the range (largest observation minus smallest) can assume for such a sample is
$ \textbf{(A)}\ 2 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}\ 7 \qquad\textbf{(E)}\ 10 $
1990 AIME Problems, 1
The increasing sequence $2,3,5,6,7,10,11,\ldots$ consists of all positive integers that are neither the square nor the cube of a positive integer. Find the 500th term of this sequence.
2016 AMC 10, 16
The sum of an infinite geometric series is a positive number $S$, and the second term in the series is $1$. What is the smallest possible value of $S?$
$\textbf{(A)}\ \frac{1+\sqrt{5}}{2} \qquad
\textbf{(B)}\ 2 \qquad
\textbf{(C)}\ \sqrt{5} \qquad
\textbf{(D)}\ 3 \qquad
\textbf{(E)}\ 4$
1970 AMC 12/AHSME, 32
$A$ and $B$ travel around a circular track at uniform speeds in opposite directions, starting from diametrically opposite points. If they start at the same time, meet first after $B$ has travelled $100$ yards, and meet a second time $60$ yards before $A$ completes one lap, then the circumference of the track in yards is
$\textbf{(A) }400\qquad\textbf{(B) }440\qquad\textbf{(C) }480\qquad\textbf{(D) }560\qquad \textbf{(E) }880$
2020 AMC 12/AHSME, 1
What is the value in simplest form of the following expression? \[\sqrt{1} + \sqrt{1+3} + \sqrt{1+3+5} + \sqrt{1+3+5+7}\]
$\textbf{(A) }5 \qquad \textbf{(B) }4 + \sqrt{7} + \sqrt{10} \qquad \textbf{(C) } 10 \qquad \textbf{(D) } 15 \qquad \textbf{(E) } 4 + 3\sqrt{3} + 2\sqrt{5} + \sqrt{7}$
1993 Brazil National Olympiad, 1
The sequence $(a_n)_{n \in\mathbb{N}}$ is defined by $a_1 = 8, a_2 = 18, a_{n+2} = a_{n+1}a_{n}$. Find all terms which are perfect squares.
2004 AIME Problems, 7
Let $C$ be the coefficient of $x^2$ in the expansion of the product \[(1 - x)(1 + 2x)(1 - 3x)\cdots(1 + 14x)(1 - 15x).\] Find $|C|$.
1969 AMC 12/AHSME, 10
The number of points equidistant from a circle and two parallel tangents to the circle is:
$\textbf{(A) }0\qquad
\textbf{(B) }2\qquad
\textbf{(C) }3\qquad
\textbf{(D) }4\qquad
\textbf{(E) }\text{infinite}$
1961 AMC 12/AHSME, 38
Triangle $ABC$ is inscribed in a semicircle of radius $r$ so that its base $AB$ coincides with diameter $AB$. Point $C$ does not coincide with either $A$ or $B$. Let $s=AC+BC$. Then, for all permissible positions of $C$:
$ \textbf{(A)}\ s^2\le8r^2$
$\qquad\textbf{(B)}\ s^2=8r^2$
$\qquad\textbf{(C)}\ s^2 \ge 8r^2$
${\qquad\textbf{(D)}\ s^2\le4r^2 }$
${\qquad\textbf{(E)}\ x^2=4r^2 } $
1983 USAMO, 3
Each set of a finite family of subsets of a line is a union of two closed intervals. Moreover, any three of the sets of the family have a point in common. Prove that there is a point which is common to at least half the sets of the family.
2010 AMC 10, 6
For positive numbers $ x$ and $ y$ the operation $ \spadesuit(x,y)$ is defined as
\[ \spadesuit(x,y)\equal{}x\minus{}\frac1y\]What is $ \spadesuit(2,\spadesuit(2,2))$?
$ \textbf{(A)}\ \frac23 \qquad
\textbf{(B)}\ 1 \qquad
\textbf{(C)}\ \frac43 \qquad
\textbf{(D)}\ \frac53 \qquad
\textbf{(E)}\ 2$
1960 AMC 12/AHSME, 31
For $x^2+2x+5$ to be a factor of $x^4+px^2+q$, the values of $p$ and $q$ must be, respectively:
$ \textbf{(A)}\ -2, 5\qquad\textbf{(B)}\ 5, 25\qquad\textbf{(C)}\ 10, 20\qquad\textbf{(D)}\ 6, 25\qquad\textbf{(E)}\ 14, 25 $
2008 AMC 10, 8
A class collects $ \$50$ to buy flowers for a classmate who is in the hospital. Roses cost $ \$3$ each, and carnations cost $ \$2$ each. No other flowers are to be used. How many different bouquets could be purchased for exactly $ \$50$?
$ \textbf{(A)}\ 1 \qquad
\textbf{(B)}\ 7 \qquad
\textbf{(C)}\ 9 \qquad
\textbf{(D)}\ 16 \qquad
\textbf{(E)}\ 17$
2022 AMC 10, 13
The positive difference between a pair of primes is equal to $2$, and the positive difference between the cubes of the two primes is $31106$. What is the sum of the digits of the least prime that is greater than those two primes?
$\textbf{(A) } 8 \qquad \textbf{(B) } 10 \qquad \textbf{(C) } 11 \qquad \textbf{(D) } 13 \qquad \textbf{(E) } 16$
2004 France Team Selection Test, 1
Let $n$ be a positive integer, and $a_1,...,a_n, b_1,..., b_n$ be $2n$ positive real numbers such that
$a_1 + ... + a_n = b_1 + ... + b_n = 1$.
Find the minimal value of
$ \frac {a_1^2} {a_1 + b_1} + \frac {a_2^2} {a_2 + b_2} + ...+ \frac {a_n^2} {a_n + b_n}$.
2024 AMC 12/AHSME, 12
The first three terms of a geometric sequence are the integers $a,\,720,$ and $b,$ where $a<720<b.$ What is the sum of the digits of the least possible value of $b?$
$\textbf{(A) } 9 \qquad \textbf{(B) } 12 \qquad \textbf{(C) } 16 \qquad \textbf{(D) } 18 \qquad \textbf{(E) } 21$
2013 USAMO, 3
Let $n$ be a positive integer. There are $\tfrac{n(n+1)}{2}$ marks, each with a black side and a white side, arranged into an equilateral triangle, with the biggest row containing $n$ marks. Initially, each mark has the black side up. An [i]operation[/i] is to choose a line parallel to the sides of the triangle, and flipping all the marks on that line. A configuration is called [i]admissible [/i] if it can be obtained from the initial configuration by performing a finite number of operations. For each admissible configuration $C$, let $f(C)$ denote the smallest number of operations required to obtain $C$ from the initial configuration. Find the maximum value of $f(C)$, where $C$ varies over all admissible configurations.
2021 AMC 10 Fall, 15
Isosceles triangle $ABC$ has $AB = AC = 3\sqrt6$, and a circle with radius $5\sqrt2$ is tangent to line $AB$ at $B$ and to line $AC$ at $C$. What is the area of the circle that passes through vertices $A$, $B$, and $C?$
$\textbf{(A) }24\pi\qquad\textbf{(B) }25\pi\qquad\textbf{(C) }26\pi\qquad\textbf{(D) }27\pi\qquad\textbf{(E) }28\pi$
2024 AMC 12/AHSME, 10
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
$
2014 AMC 10, 9
For real numbers $w$ and $z$,
\[ \frac{\frac{1}{w} + \frac{1}{z}}{\frac{1}{w} - \frac{1}{z}} = 2014. \]
What is $\tfrac{w+z}{w-z}$ ?
${ \textbf{(A)}\ \ -2014\qquad\textbf{(B)}\ \frac{-1}{2014}\qquad\textbf{(C)}\ \frac{1}{2014}\qquad\textbf{(D)}}\ 1\qquad\textbf{(E)}\ 2014$
2017 AMC 10, 17
Distinct points $P$, $Q$, $R$, $S$ lie on the circle $x^2+y^2=25$ and have integer coordinates. The distances $PQ$ and $RS$ are irrational numbers. What is the greatest possible value of the ratio $\frac{PQ}{RS }$?
$\textbf{(A)}\ 3\qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\ 3\sqrt{5}\qquad\textbf{(D)}\ 7\qquad\textbf{(E)}\ 5\sqrt{2}$
1979 AMC 12/AHSME, 1
[asy]
draw((-2,1)--(2,1)--(2,-1)--(-2,-1)--cycle);
draw((0,0)--(0,-1)--(-2,-1)--(-2,0)--cycle);
label("$F$",(0,0),E);
label("$A$",(-2,1),W);
label("$B$",(2,1),E);
label("$C$", (2,-1),E);
label("$D$",(-2,-1),WSW);
label("$E$",(-2,0),W);
label("$G$",(0,-1),S);
//Credit to TheMaskedMagician for the diagram
[/asy]
If rectangle $ABCD$ has area $72$ square meters and $E$ and $G$ are the midpoints of sides $AD$ and $CD$, respectively, then the area of rectangle $DEFG$ in square meters is
$\textbf{(A) }8\qquad\textbf{(B) }9\qquad\textbf{(C) }12\qquad\textbf{(D) }18\qquad\textbf{(E) }24$
2006 AIME Problems, 11
A collection of 8 cubes consists of one cube with edge-length $k$ for each integer $k,\thinspace 1 \le k \le 8.$ A tower is to be built using all 8 cubes according to the rules:
$\bullet$ Any cube may be the bottom cube in the tower.
$\bullet$ The cube immediately on top of a cube with edge-length $k$ must have edge-length at most $k+2.$
Let $T$ be the number of different towers than can be constructed. What is the remainder when $T$ is divided by 1000?