Found problems: 3632
1986 AMC 12/AHSME, 12
John scores $93$ on this year's AHSME. Had the old scoring system still been in effect, he would score only $84$ for the same answers. How many questions does he leave unanswered? (In the new scoring system one receives 5 points for correct answers, 0 points for wrong answers, and 2 points for unanswered questions. In the old system, one started with 30 points, received 4 more for each correct answer, lost one point for each wrong answer, and neither gained nor lost points for unanswered questions. There are 30 questions in the 1986 AHSME.)
$ \textbf{(A)}\ 6\qquad\textbf{(B)}\ 9\qquad\textbf{(C)}\ 11\qquad\textbf{(D)}\ 14\qquad\textbf{(E)}\ \text{Not uniquely determined} $
2010 AMC 12/AHSME, 8
Every high school in the city of Euclid sent a team of 3 students to a math contest. Each participant in the contest received a different score. Andrea's score was the median among all students, and hers was the highest score on her team. Andrea's teammates Beth and Carla placed 37th and 64th, respectively. How many schools are in the city?
$ \textbf{(A)}\ 22\qquad\textbf{(B)}\ 23\qquad\textbf{(C)}\ 24\qquad\textbf{(D)}\ 25\qquad\textbf{(E)}\ 26$
2021 AMC 12/AHSME Spring, 14
What is the value of $$\left(\sum_{k=1}^{20} \log_{5^k} 3^{k^2}\right)\cdot\left(\sum_{k=1}^{100} \log_{9^k} 25^k\right)?$$
$\textbf{(A) }21 \qquad \textbf{(B) }100\log_5 3 \qquad \textbf{(C) }200\log_3 5 \qquad \textbf{(D) }2,200\qquad \textbf{(E) }21,000$
2001 AMC 10, 10
If $ x$, $ y$, and $ z$ are positive with $ xy \equal{} 24$, $ xz \equal{} 48$, and $ yz \equal{} 72$, then $ x \plus{} y \plus{} z$ is
$ \textbf{(A) }18\qquad\textbf{(B) }19\qquad\textbf{(C) }20\qquad\textbf{(D) }22\qquad\textbf{(E) }24$
1966 AMC 12/AHSME, 33
If $ab\ne0$ and $|a|\ne|b|$ the number of distinct values of $x$ satisfying the equation
\[\dfrac{x-a}{b}+\dfrac{x-b}{a}=\dfrac{b}{x-a}+\dfrac{a}{x-b}\]
is:
$\text{(A)}\ \text{zero}\qquad
\text{(B)}\ \text{one}\qquad
\text{(C)}\ \text{two}\qquad
\text{(D)}\ \text{three}\qquad
\text{(E)}\ \text{four}$
1972 AMC 12/AHSME, 11
The value(s) of $y$ for which the following pair of equations \[x^2+y^2-16=0\text{ and }x^2-3y+12=0\] may have a real common solution, are
$\textbf{(A) }4\text{ only}\qquad\textbf{(B) }-7,~4\qquad\textbf{(C) }0,~4\qquad\textbf{(D) }\text{no }y\qquad \textbf{(E) }\text{all }y$
1963 AMC 12/AHSME, 22
Acute-angled triangle $ABC$ is inscribed in a circle with center at $O$; $\stackrel \frown {AB} = 120$ and $\stackrel \frown {BC} = 72$. A point $E$ is taken in minor arc $AC$ such that $OE$ is perpendicular to $AC$. Then the ratio of the magnitudes of angles $OBE$ and $BAC$ is:
$\textbf{(A)}\ \dfrac{5}{18} \qquad
\textbf{(B)}\ \dfrac{2}{9} \qquad
\textbf{(C)}\ \dfrac{1}{4} \qquad
\textbf{(D)}\ \dfrac{1}{3} \qquad
\textbf{(E)}\ \dfrac{4}{9}$
1967 AMC 12/AHSME, 13
A triangle $ABC$ is to be constructed given a side $a$ (oppisite angle $A$). angle $B$, and $h_c$, the altitude from $C$. If $N$ is the number of noncongruent solutions, then $N$
$\textbf{(A)}\ \text{is} \; 1\qquad
\textbf{(B)}\ \text{is} \; 2\qquad
\textbf{(C)}\ \text{must be zero}\qquad
\textbf{(D)}\ \text{must be infinite}\qquad
\textbf{(E)}\ \text{must be zero or infinite}$
2015 USA Team Selection Test, 3
A physicist encounters $2015$ atoms called usamons. Each usamon either has one electron or zero electrons, and the physicist can't tell the difference. The physicist's only tool is a diode. The physicist may connect the diode from any usamon $A$ to any other usamon $B$. (This connection is directed.) When she does so, if usamon $A$ has an electron and usamon $B$ does not, then the electron jumps from $A$ to $B$. In any other case, nothing happens. In addition, the physicist cannot tell whether an electron jumps during any given step. The physicist's goal is to isolate two usamons that she is sure are currently in the same state. Is there any series of diode usage that makes this possible?
[i]Proposed by Linus Hamilton[/i]
1969 AMC 12/AHSME, 12
Let $F=\dfrac{6x^2+16x+3m}6$ be the square of an expression which is linear in $x$. Then $m$ has a particular value between:
$\textbf{(A) }3\text{ and }4\qquad
\textbf{(B) }4\text{ and }5\qquad
\textbf{(C) }5\text{ and }6\qquad$
$\textbf{(D) }-4\text{ and }-3\qquad
\textbf{(E) }-6\text{ and }-5$
1964 AMC 12/AHSME, 6
If $x, 2x+2, 3x+3, \dots$ are in geometric progression, the fourth term is:
${{ \textbf{(A)}\ -27 \qquad\textbf{(B)}\ -13\frac{1}{2} \qquad\textbf{(C)}\ 12\qquad\textbf{(D)}\ 13\frac{1}{2} }\qquad\textbf{(E)}\ 27 } $
2001 AIME Problems, 8
Call a positive integer $N$ a $\textit{7-10 double}$ if the digits of the base-7 representation of $N$ form a base-10 number that is twice $N.$ For example, 51 is a 7-10 double because its base-7 representation is 102. What is the largest 7-10 double?
2010 AMC 10, 13
Angelina drove at an average rate of $ 80$ kph and then stopped $ 20$ minutes for gas. After the stop, she drove at an average rate of $ 100$ kph. Altogether she drove $ 250$ km in a total trip time of $ 3$ hours including the stop. Which equation could be used to solve for the time $ t$ in hours that she drove before her stop?
$ \textbf{(A)}\ 80t\plus{}100(8/3\minus{}t)\equal{}250 \qquad
\textbf{(B)}\ 80t\equal{}250 \qquad
\textbf{(C)}\ 100t\equal{}250 \\
\textbf{(D)}\ 90t\equal{}250 \qquad
\textbf{(E)}\ 80(8/3\minus{}t)\plus{}100t\equal{}250$
2012 AMC 10, 21
Four distinct points are arranged in a plane so that the segments connecting them has lengths $a,a,a,a,2a,$ and $b$. What is the ratio of $b$ to $a$?
$ \textbf{(A)}\ \sqrt{3}\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ \sqrt{5}\qquad\textbf{(D)}\ 3\qquad\textbf{(E)}\ \pi $
2013 AMC 10, 18
The number $2013$ has the property that its units digit is the sum of its other digits, that is $2+0+1=3$. How many integers less than $2013$ but greater than $1000$ share this property?
${ \textbf{(A)}\ 33 \qquad\textbf{(B)}\ 34 \qquad\textbf{(C)}\ 45 \qquad\textbf{(D)}}\ 46\qquad\textbf{(E)}\ 58 $
2022 AMC 10, 23
Isosceles trapezoid $ABCD$ has parallel sides $\overline{AD}$ and $\overline{BC},$ with $BC < AD$ and $AB = CD.$ There is a point $P$ in the plane such that $PA=1, PB=2, PC=3,$ and $PD=4.$ What is $\tfrac{BC}{AD}?$
$\textbf{(A) }\frac{1}{4}\qquad\textbf{(B) }\frac{1}{3}\qquad\textbf{(C) }\frac{1}{2}\qquad\textbf{(D) }\frac{2}{3}\qquad\textbf{(E) }\frac{3}{4}$
2013 AMC 12/AHSME, 10
Alex has $75$ red tokens and $75$ blue tokens. There is a booth where Alex can give two red tokens and receive in return a silver token and a blue token, and another booth where Alex can give three blue tokens and receive in return a silver token and a red token. Alex continues to exchange tokens until no more exchanges are possible. How many silver tokens will Alex have at the end?
${ \textbf{(A)}\ 62 \qquad\textbf{(B)}\ 82 \qquad\textbf{(C)}\ 83\qquad\textbf{(D}}\ 102\qquad\textbf{(E)}\ 103 $
1963 AMC 12/AHSME, 21
The expression $x^2-y^2-z^2+2yz+x+y-z$ has:
$\textbf{(A)}\ \text{no linear factor with integer coeficients and integer exponents} \qquad$
$
\textbf{(B)}\ \text{the factor }-x+y+z \qquad$
$
\textbf{(C)}\ \text{the factor }x-y-z+1 \qquad$
$
\textbf{(D)}\ \text{the factor }x+y-z+1 \qquad$
$
\textbf{(E)}\ \text{the factor }x-y+z+1$
1961 AMC 12/AHSME, 24
Thirty-one books are arranged from left to right in order of increasing prices. The price of each book differs by $\$2$ from that of each adjacent book. For the price of the book at the extreme right a customer can buy the middle book and the adjacent one. Then:
$ \textbf{(A)}\ \text{The adjacent book referred to is at the left of the middle book}$
$\qquad\textbf{(B)}\ \text{The middle book sells for \$36 }$
$\qquad\textbf{(C)}\ \text{The cheapest book sells for \$4 }$
$\qquad\textbf{(D)}\ \text{The most expensive book sells for \$64 }$
$\qquad\textbf{(E)}\ \text{None of these is correct } $
2004 AIME Problems, 4
How many positive integers less than 10,000 have at most two different digits?
2015 AMC 12/AHSME, 4
The sum of two positive numbers is $5$ times their difference. What is the ratio of the larger number to the smaller?
$\textbf{(A) }\dfrac54\qquad\textbf{(B) }\dfrac32\qquad\textbf{(C) }\dfrac95\qquad\textbf{(D) }2\qquad\textbf{(E) }\dfrac52$
1961 AMC 12/AHSME, 39
Any five points are taken inside or on a square with side length $1$. Let $a$ be the [i]smallest[/i] possible number with the property that it is always possible to select one pair of points from these five such that the distance between them is equal to or less than $a$. Then $a$ is:
${{ \textbf{(A)}\ \sqrt{3}/3 \qquad\textbf{(B)}\ \sqrt{2}/2 \qquad\textbf{(C)}\ 2\sqrt{2}/3 \qquad\textbf{(D)}\ 1 }\qquad\textbf{(E)}\ \sqrt{2} } $
2010 AMC 12/AHSME, 5
Lucky Larry's teacher asked him to substitute numbers for $ a$, $ b$, $ c$, $ d$, and $ e$ in the expression $ a\minus{}(b\minus{}(c\minus{}(d\plus{}e)))$ and evaluate the result. Larry ignored the parentheses but added and subtracted correctly and obtained the correct result by coincedence. The numbers Larry substituted for $ a$, $ b$, $ c$, and $ d$ were $ 1$, $ 2$, $ 3$, and $ 4$, respectively. What number did Larry substitute for $ e$?
$ \textbf{(A)}\ \minus{}5\qquad\textbf{(B)}\ \minus{}3\qquad\textbf{(C)}\ 0\qquad\textbf{(D)}\ 3\qquad\textbf{(E)}\ 5$
2012 AMC 10, 16
Three runners start running simultaneously from the same point on a $500$-meter circular track. They each run clockwise around the course maintaining constant speeds of $4.4$, $4.8$, and $5.0$ meters per second. The runners stop once they are all together again somewhere on the circular course. How many seconds do the runners run?
$ \textbf{(A)}\ 1,000
\qquad\textbf{(B)}\ 1,250
\qquad\textbf{(C)}\ 2,500
\qquad\textbf{(D)}\ 5,000
\qquad\textbf{(E)}\ 10,000
$
2019 AMC 12/AHSME, 8
Let $f(x) = x^{2}(1-x)^{2}$. What is the value of the sum
\begin{align*}
f\left(\frac{1}{2019}\right)-f\left(\frac{2}{2019}\right)+f\left(\frac{3}{2019}\right)-&f\left(\frac{4}{2019}\right)+\cdots\\
&\,+f\left(\frac{2017}{2019}\right) - f\left(\frac{2018}{2019}\right)?
\end{align*}
$\textbf{(A) }0\qquad\textbf{(B) }\frac{1}{2019^{4}}\qquad\textbf{(C) }\frac{2018^{2}}{2019^{4}}\qquad\textbf{(D) }\frac{2020^{2}}{2019^{4}}\qquad\textbf{(E) }1$