Found problems: 649
2022 AMC 12/AHSME, 3
Five rectangles, $A$, $B$, $C$, $D$, and $E$, are arranged in a square as shown below. These rectangles have dimensions $1\times6$, $2\times4$, $5\times6$, $2\times7$, and $2\times3$, respectively. (The figure is not drawn to scale.) Which of the five rectangles is the shaded one in the middle?
[asy]
fill((3,2.5)--(3,4.5)--(5.3,4.5)--(5.3,2.5)--cycle,mediumgray);
draw((0,0)--(7,0)--(7,7)--(0,7)--(0,0));
draw((3,0)--(3,4.5));
draw((0,4.5)--(5.3,4.5));
draw((5.3,7)--(5.3,2.5));
draw((7,2.5)--(3,2.5));
[/asy]
$\textbf{(A) }A\qquad\textbf{(B) }B \qquad\textbf{(C) }C \qquad\textbf{(D) }D\qquad\textbf{(E) }E$
2006 AMC 10, 1
What is $ ( \minus{} 1)^1 \plus{} ( \minus{} 1)^2 \plus{} \cdots \plus{} ( \minus{} 1)^{2006}$?
$ \textbf{(A) } \minus{} 2006 \qquad \textbf{(B) } \minus{} 1 \qquad \textbf{(C) } 0 \qquad \textbf{(D) } 1 \qquad \textbf{(E) } 2006$
2024 AMC 10, 23
Integers $a$, $b$, and $c$ satisfy $ab + c = 100$, $bc + a = 87$, and $ca + b = 60$. What is $ab + bc + ca$?
$
\textbf{(A) }212 \qquad
\textbf{(B) }247 \qquad
\textbf{(C) }258 \qquad
\textbf{(D) }276 \qquad
\textbf{(E) }284 \qquad
$
2022 AMC 12/AHSME, 7
Camila writes down five positive integers. The unique mode of these integers is $2$ greater than their median, and the median is $2$ greater than their arithmetic mean. What is the least possible value for the mode?
$\textbf{(A) }5\qquad\textbf{(B) }7\qquad\textbf{(C) }9\qquad\textbf{(D) }11\qquad\textbf{(E) }13$
2021 AMC 10 Fall, 1
What is the value of $1234+2341+3412+4123$?
$\textbf{(A) } 10,000 \qquad \textbf{(B) }10,010 \qquad \textbf{(C) }10,110 \qquad \textbf{(D) }11,000 \qquad \textbf{(E) }11,110$
2022 AMC 12/AHSME, 22
Ant Amelia starts on the number line at $0$ and crawls in the following manner. For $n=1,2,3,$ Amelia chooses a time duration $t_n$ and an increment $x_n$ independently and uniformly at random from the interval $(0,1).$ During the $n$th step of the process, Amelia moves $x_n$ units in the positive direction, using up $t_n$ minutes. If the total elapsed time has exceeded $1$ minute during the $n$th step, she stops at the end of that step; otherwise, she continues with the next step, taking at most $3$ steps in all. What is the probability that Amelia’s position when she stops will be greater than $1$?
$\textbf{(A) }\frac{1}{3} \qquad \textbf{(B) }\frac{1}{2} \qquad \textbf{(C) }\frac{2}{3} \qquad \textbf{(D) }\frac{3}{4} \qquad \textbf{(E) }\frac{5}{6}$
2020 AMC 12/AHSME, 4
The acute angles of a right triangle are $a^{\circ}$ and $b^{\circ}$, where $a>b$ and both $a$ and $b$ are prime numbers. What is the least possible value of $b$?
$\textbf{(A) }2\qquad\textbf{(B) }3\qquad\textbf{(C) }5\qquad\textbf{(D) }7\qquad\textbf{(E) }11$
1959 AMC 12/AHSME, 36
The base of a triangle is $80$, and one side of the base angle is $60^\circ$. The sum of the lengths of the other two sides is $90$. The shortest side is:
$ \textbf{(A)}\ 45 \qquad\textbf{(B)}\ 40\qquad\textbf{(C)}\ 36\qquad\textbf{(D)}\ 17\qquad\textbf{(E)}\ 12 $
2016 AMC 10, 22
For some positive integer $n$, the number $110n^3$ has $110$ positive integer divisors, including $1$ and the number $110n^3$. How many positive integer divisors does the number $81n^4$ have?
$\textbf{(A) }110 \qquad \textbf{(B) } 191 \qquad \textbf{(C) } 261 \qquad \textbf{(D) } 325 \qquad \textbf{(E) } 425$
2021 AMC 10 Fall, 4
At noon on a certain day, Minneapolis is $N$ degrees warmer than St. Louis. At $4{:}00$ the temperature in Minneapolis has fallen by $5$ degrees while the temperature in St. Louis has risen by $3$ degrees, at which time the temperatures in the two cities differ by $2$ degrees. What is the product of all possible values of $N?$
$(\textbf{A})\: 10\qquad(\textbf{B}) \: 30\qquad(\textbf{C}) \: 60\qquad(\textbf{D}) \: 100\qquad(\textbf{E}) \: 120$
2019 AMC 12/AHSME, 12
Right triangle $ACD$ with right angle at $C$ is constructed outwards on the hypotenuse $\overline{AC}$ of isosceles right triangle $ABC$ with leg length $1$, as shown, so that the two triangles have equal perimeters. What is $\sin(2\angle BAD)$?
[asy]
/* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki go to User:Azjps/geogebra */
import graph; size(8.016233639805293cm);
real labelscalefactor = 0.5; /* changes label-to-point distance */
pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */
pen dotstyle = black; /* point style */
real xmin = -4.001920114613276, xmax = 4.014313525192017, ymin = -2.552570341575814, ymax = 5.6249093771911145; /* image dimensions */
draw((-1.6742337260757447,-1.)--(-1.6742337260757445,-0.6742337260757447)--(-2.,-0.6742337260757447)--(-2.,-1.)--cycle, linewidth(2.));
draw((-1.7696484586262846,2.7696484586262846)--(-1.5392969172525692,3.)--(-1.7696484586262846,3.2303515413737154)--(-2.,3.)--cycle, linewidth(2.));
/* draw figures */
draw((-2.,3.)--(-2.,-1.), linewidth(2.));
draw((-2.,-1.)--(2.,-1.), linewidth(2.));
draw((2.,-1.)--(-2.,3.), linewidth(2.));
draw((-0.6404058554606791,4.3595941445393205)--(-2.,3.), linewidth(2.));
draw((-0.6404058554606791,4.3595941445393205)--(2.,-1.), linewidth(2.));
label("$D$",(-0.9382446143428628,4.887784444795223),SE*labelscalefactor,fontsize(14));
label("$A$",(1.9411496528285788,-1.0783204767840298),SE*labelscalefactor,fontsize(14));
label("$B$",(-2.5046350956841272,-0.9861798602345433),SE*labelscalefactor,fontsize(14));
label("$C$",(-2.5737405580962416,3.5747806589650395),SE*labelscalefactor,fontsize(14));
label("$1$",(-2.665881174645728,1.2712652452278765),SE*labelscalefactor,fontsize(14));
label("$1$",(-0.3393306067712029,-1.3547423264324894),SE*labelscalefactor,fontsize(14));
/* dots and labels */
dot((-2.,3.),linewidth(4.pt) + dotstyle);
dot((-2.,-1.),linewidth(4.pt) + dotstyle);
dot((2.,-1.),linewidth(4.pt) + dotstyle);
dot((-0.6404058554606791,4.3595941445393205),linewidth(4.pt) + dotstyle);
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);
/* end of picture */
[/asy]
$\textbf{(A) } \dfrac{1}{3} \qquad\textbf{(B) } \dfrac{\sqrt{2}}{2} \qquad\textbf{(C) } \dfrac{3}{4} \qquad\textbf{(D) } \dfrac{7}{9} \qquad\textbf{(E) } \dfrac{\sqrt{3}}{2}$
2021 AMC 12/AHSME Fall, 13
Let $c = \frac{2\pi}{11}.$ What is the value of
$$\frac{\sin 3c \cdot \sin 6c \cdot \sin 9c \cdot \sin 12c \cdot \sin 15c}{\sin c \cdot \sin 2c \cdot \sin 3c \cdot \sin 4c \cdot \sin 5c}?$$
$\textbf{(A)}\ -1 \qquad\textbf{(B)}\ \frac{\sqrt{-11}}{5} \qquad\textbf{(C)}\ \frac{\sqrt{11}}{5} \qquad\textbf{(D)}\
\frac{10}{11} \qquad\textbf{(E)}\ 1$
2020 AMC 12/AHSME, 15
There are 10 people standing equally spaced around a circle. Each person knows exactly 3 of the other 9 people: the 2 people standing next to her or him, as well as the person directly across the circle. How many ways are there for the 10 people to split up into 5 pairs so that the members of each pair know each other?
$\textbf{(A) } 11 \qquad \textbf{(B) } 12 \qquad \textbf{(C) } 13 \qquad \textbf{(D) } 14 \qquad \textbf{(E) } 15$
2017 AMC 12/AHSME, 14
Alice refuses to sit next to either Bob or Carla. Derek refuses to sit next to Eric. How many ways are there for the five of them to sit in a row of $5$ chairs under these conditions?
$\textbf{(A)}\ 12\qquad\textbf{(B)}\ 16\qquad\textbf{(C)}\ 28\qquad\textbf{(D)}\ 32\qquad\textbf{(E)}\ 40$
2023 AMC 12/AHSME, 23
When $n$ standard six-sided dice are rolled, the product of the numbers rolled can be any of $936$ possible values. What is $n$?
$\textbf{(A)}~6\qquad\textbf{(B)}~8\qquad\textbf{(C)}~9\qquad\textbf{(D)}~10\qquad\textbf{(E)}~11$
2017 AMC 12/AHSME, 7
Define a function on the positive integers recursively by $f(1) = 2$, $f(n) = f(n-1) + 1$ if $n$ is even, and $f(n) = f(n-2) + 2$ if $n$ is odd and greater than $1$. What is $f(2017)$?
$\textbf{(A) } 2017 \qquad \textbf{(B) } 2018 \qquad \textbf{(C) } 4034 \qquad \textbf{(D) } 4035 \qquad \textbf{(E) } 4036$
2024 AMC 10, 23
The Fibonacci numbers are defined by $F_1=1,$ $F_2=1,$ and $F_n=F_{n-1}+F_{n-2}$ for $n\geq 3.$ What is $$\dfrac{F_2}{F_1}+\dfrac{F_4}{F_2}+\dfrac{F_6}{F_3}+\cdots+\dfrac{F_{20}}{F_{10}}?$$
$\textbf{(A) }318 \qquad\textbf{(B) }319\qquad\textbf{(C) }320\qquad\textbf{(D) }321\qquad\textbf{(E) }322$
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} $
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 $a+b+c$?
$\textbf{(A) }110 \qquad \textbf{(B) }114 \qquad \textbf{(C) }118 \qquad \textbf{(D) }122\qquad \textbf{(E) }126$
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 $N$?
$\textbf{(A) } 14 \qquad \textbf{(B) } 16 \qquad \textbf{(C) } 18 \qquad \textbf{(D) } 19 \qquad \textbf{(E) } 21$
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 $N$?
$\textbf{(A)}\ 6\qquad\textbf{(B)}\ 7\qquad\textbf{(C)}\ 8\qquad\textbf{(D)}\ 9\qquad\textbf{(E)}\ 10$