Found problems: 3632
2006 AMC 10, 8
A parabola with equation $ y \equal{} x^2 \plus{} bx \plus{} c$ passes through the points $ (2,3)$ and $ (4,3)$. What is $ c$?
$ \textbf{(A) } 2 \qquad \textbf{(B) } 5 \qquad \textbf{(C) } 7 \qquad \textbf{(D) } 10 \qquad \textbf{(E) } 11$
1967 AMC 12/AHSME, 16
Let the product $(12)(15)(16)$, each factor written in base $b$, equal $3146$ in base $b$. Let $s=12+15+16$, each term expressed in base $b$. Then $s$, in base $b$, is
$\textbf{(A)}\ 43\qquad
\textbf{(B)}\ 44\qquad
\textbf{(C)}\ 45\qquad
\textbf{(D)}\ 46\qquad
\textbf{(E)}\ 47$
2015 AMC 10, 12
For how many integers $x$ is the point $(x,-x)$ inside or on the circle of radius $10$ centered at $(5,5)$?
$\textbf{(A) } 11
\qquad\textbf{(B) } 12
\qquad\textbf{(C) } 13
\qquad\textbf{(D) } 14
\qquad\textbf{(E) } 15
$
1968 AMC 12/AHSME, 27
Let $S_n=1-2+3-4+\cdots +(-1)^{n-1}n,\ n=1, 2, \cdots$. Then $S_{17}+S_{33}+S_{50}$ equals:
$\textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ -1 \qquad\textbf{(E)}\ -2$
2012 AMC 8, 2
In the country of East Westmore, statisticians estimate there is a baby born every 8 hours and a death every day. To the nearest hundred, how many people are added to the population of East Westmore each year?
$\textbf{(A)}\hspace{.05in}600 \qquad \textbf{(B)}\hspace{.05in}700 \qquad \textbf{(C)}\hspace{.05in}800 \qquad \textbf{(D)}\hspace{.05in}900 \qquad \textbf{(E)}\hspace{.05in}1000 $
2012 AMC 10, 23
Adam, Benin, Chiang, Deshawn, Esther, and Fiona have internet accounts. Some, but not all, of them are internet friends with each other, and none of them has an internet friend outside this group. Each of them has the same number of internet friends. In how many different ways can this happen?
$ \textbf{(A)}\ 60
\qquad\textbf{(B)}\ 170
\qquad\textbf{(C)}\ 290
\qquad\textbf{(D)}\ 320
\qquad\textbf{(E)}\ 660
$
1959 AMC 12/AHSME, 23
The set of solutions of the equation $\log_{10}\left( a^2-15a\right)=2$ consists of
$ \textbf{(A)}\ \text{two integers } \qquad\textbf{(B)}\ \text{one integer and one fraction}\qquad$ $\textbf{(C)}\ \text{two irrational numbers }\qquad\textbf{(D)}\ \text{two non-real numbers} \qquad\textbf{(E)}\ \text{no numbers, that is, the empty set} $
2006 AMC 10, 14
Let $ a$ and $ b$ be the roots of the equation $ x^2 \minus{} mx \plus{} 2 \equal{} 0$. Suppose that $ a \plus{} (1/b)$ and $ b \plus{} (1/a)$ are the roots of the equation $ x^2 \minus{} px \plus{} q \equal{} 0$. What is $ q$?
$ \textbf{(A) } \frac 52 \qquad \textbf{(B) } \frac 72 \qquad \textbf{(C) } 4 \qquad \textbf{(D) } \frac 92 \qquad \textbf{(E) } 8$
2023 AMC 10, 18
A rhombic dodecahedron is a solid with $12$ congruent rhombus faces. At every vertex, $3$ or $4$ edges meet, depending on the vertex. How many vertices have exactly $3$ edges meet?
$\textbf{(A) }5\qquad\textbf{(B) }6\qquad\textbf{(C) }7\qquad\textbf{(D) }8\qquad\textbf{(E) }9$
2021 AMC 12/AHSME Spring, 24
Semicircle $\Gamma$ has diameter $\overline{AB}$ of length $14$. Circle $\Omega$ lies tangent to $\overline{AB}$ at a point $P$ and intersects $\Gamma$ at points $Q$ and $R$. If $QR=3\sqrt3$ and $\angle QPR=60^\circ$, then the area of $\triangle PQR$ is $\frac{a\sqrt{b}}{c}$, where $a$ and $c$ are relatively prime positive integers, and $b$ is a positive integer not divisible by the square of any prime. What is $a+b+c$?
$\textbf{(A) }110 \qquad \textbf{(B) }114 \qquad \textbf{(C) }118 \qquad \textbf{(D) }122\qquad \textbf{(E) }126$
1986 AMC 12/AHSME, 11
In $\triangle ABC$, $AB = 13$, $BC = 14$ and $CA = 15$. Also, $M$ is the midpoint of side $AB$ and $H$ is the foot of the altitude from $A$ to $BC$. The length of $HM$ is
[asy]
size(200);
defaultpen(linewidth(0.7)+fontsize(10));
pair H=origin, A=(0,6), B=(-4,0), C=(5,0), M=B+3.6*dir(B--A);
draw(B--C--A--B^^M--H--A^^rightanglemark(A,H,C));
label("$A$", A, NE);
label("$B$", B, W);
label("$C$", C, E);
label("$H$", H, S);
label("$M$", M, dir(M));
[/asy]
$ \textbf{(A)}\ 6\qquad\textbf{(B)}\ 6.5\qquad\textbf{(C)}\ 7\qquad\textbf{(D)}\ 7.5\qquad\textbf{(E)}\ 8 $
2000 AMC 8, 2
Which of these numbers is less than its reciprocal?
$\textbf{(A)}\ -2\qquad
\textbf{(B)}\ -1\qquad
\textbf{(C)}\ 0\qquad
\textbf{(D)}\ 1\qquad
\textbf{(E)}\ 2$
2016 AMC 10, 6
Ximena lists the whole numbers $1$ through $30$ once. Emilio copies Ximena's numbers, replacing each occurrence of the digit $2$ by the digit $1$. Ximena adds her numbers and Emilio adds his numbers. How much larger is Ximena's sum than Emilio's?
$\textbf{(A)}\ 13\qquad\textbf{(B)}\ 26\qquad\textbf{(C)}\ 102\qquad\textbf{(D)}\ 103\qquad\textbf{(E)}\ 110$
1996 AIME Problems, 7
Two of the squares of a $ 7\times 7$ checkerboard are painted yellow, and the rest are painted green. Two color schemes are equivalent if one can be obtained from the other by applying a rotation in the plane of the board. How many inequivalent color schemes are possible?
2020 CHMMC Winter (2020-21), 2
Caltech's 900 students are evenly spaced along the circumference of a circle. How many equilateral triangles can be formed with at least two Caltech students as vertices?
2010 AMC 12/AHSME, 15
A coin is altered so that the probability that it lands on heads is less than $ \frac {1}{2}$ and when the coin is flipped four times, the probability of an equal number of heads and tails is $ \frac {1}{6}$. What is the probability that the coin lands on heads?
$ \textbf{(A)}\ \frac {\sqrt {15} \minus{} 3}{6}\qquad
\textbf{(B)}\ \frac {6 \minus{} \sqrt {6\sqrt {6} \plus{} 2}}{12}\qquad
\textbf{(C)}\ \frac {\sqrt {2} \minus{} 1}{2}\qquad
\textbf{(D)}\ \frac {3 \minus{} \sqrt {3}}{6}\qquad
\textbf{(E)}\ \frac {\sqrt {3} \minus{} 1}{2}$
2019 AMC 10, 14
For a set of four distinct lines in a plane, there are exactly $N$ distinct points that lie on two or more of the lines. What is the sum of all possible values of $N$?
$\textbf{(A) } 14 \qquad \textbf{(B) } 16 \qquad \textbf{(C) } 18 \qquad \textbf{(D) } 19 \qquad \textbf{(E) } 21$
2012 AIME Problems, 2
The terms of an arithmetic sequence add to $715$. The first term of the sequence is increased by $1$, the second term is increased by $3$, the third term is increased by $5$, and in general, the $k$th term is increased by the $k$th odd positive integer. The terms of the new sequence add to $836$. Find the sum of the first, last, and middle terms of the original sequence.
1959 AMC 12/AHSME, 26
The base of an isosceles triangle is $\sqrt 2$. The medians to the leg intersect each other at right angles. The area of the triangle is:
$ \textbf{(A)}\ 1.5 \qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 2.5\qquad\textbf{(D)}\ 3.5\qquad\textbf{(E)}\ 4 $
2023 AMC 12/AHSME, 18
Last academic year Yolanda and Zelda took different courses that did not necessarily administer the same number of quizzes during each of the two semesters. Yolanda's average on all the quizzes she took during the first semester was 3 points higher than Zelda's average on all the quizzes she took during the first semester. Yolanda's average on all the quizzes she took during the second semester was 18 points higher than her average for the first semester and was again 3 points higher than Zelda's average on all the quizzes Zelda took during her second semester. Which one of the following statements cannot possibly be true?
(A) Yolanda's quiz average for the academic year was 22 points higher than Zelda's.
(B) Zelda's quiz average for the academic year was higher than Yolanda's.
(C) Yolanda's quiz average for the academic year was 3 points higher than Zelda's.
(D) Zelda's quiz average for the academic year equaled Yolanda's.
(E) If Zelda had scored 3 points higher on each quiz she took, then she would have had the same average for the academic year as Yolanda.
2017 AMC 10, 7
Jerry and Silvia wanted to go from the southwest corner of a square field to the northeast corner. Jerry walked due east and then due north to reach the goal, but Silvia headed northeast and reached the goal walking in a straight line. Which of the following is closest to how much shorter Silvia's trip was, compared to Jerry's trip?
$\textbf{(A)}\ 30 \%\qquad\textbf{(B)}\ 40 \%\qquad\textbf{(C)}\ 50 \%\qquad\textbf{(D)}\ 60 \%\qquad\textbf{(E)}\ 70 \%$
2016 AMC 10, 9
A triangular array of $2016$ coins has $1$ coin in the first row, $2$ coins in the second row, $3$ coins in the third row, and so on up to $N$ coins in the $N$th row. What is the sum of the digits of $N$?
$\textbf{(A)}\ 6\qquad\textbf{(B)}\ 7\qquad\textbf{(C)}\ 8\qquad\textbf{(D)}\ 9\qquad\textbf{(E)}\ 10$
2014 NIMO Problems, 8
Three of the below entries, with labels $a$, $b$, $c$, are blatantly incorrect (in the United States).
What is $a^2+b^2+c^2$?
041. The Gentleman's Alliance Cross
042. Glutamine (an amino acid)
051. Grant Nelson and Norris Windross
052. A compact region at the center of a galaxy
061. The value of \verb+'wat'-1+. (See \url{https://www.destroyallsoftware.com/talks/wat}.)
062. Threonine (an amino acid)
071. Nintendo Gamecube
072. Methane and other gases are compressed
081. A prank or trick
082. Three carbons
091. Australia's second largest local government area
092. Angoon Seaplane Base
101. A compressed archive file format
102. Momordica cochinchinensis
111. Gentaro Takahashi
112. Nat Geo
121. Ante Christum Natum
122. The supreme Siberian god of death
131. Gnu C Compiler
132. My TeX Shortcut for $\angle$.
2021 AMC 10 Spring, 9
What is the least possible value of $(xy-1)^2+(x+y)^2$ for real numbers $x$ and $y$?
$\textbf{(A)}\ 0 \qquad\textbf{(B)}\ \frac14 \qquad\textbf{(C)}\ \frac12 \qquad\textbf{(D)}\ 1 \qquad\textbf{(E)}\ 2$
1970 AMC 12/AHSME, 33
Find the sum of the digits of all numerals in the sequence $1,2,3,4,\cdots ,10000$.
$\textbf{(A) }180,001\qquad\textbf{(B) }154,756\qquad\textbf{(C) }45,001\qquad\textbf{(D) }154,755\qquad \textbf{(E) }270,001$