Found problems: 3632
1987 AMC 12/AHSME, 9
The first four terms of an arithmetic sequence are $a, x, b, 2x$. The ratio of $a$ to $b$ is
$ \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)}\ 2 $
2007 AIME Problems, 1
How many positive perfect squares less than $10^{6}$ are multiples of $24$?
1992 AMC 12/AHSME, 26
Semicircle $\stackrel{\frown}{AB}$ has center $C$ and radius $1$. Point $D$ is on $\stackrel{\frown}{AB}$ and $\overline{CD} \perp \overline{AB}$. Extend $\overline{BD}$ and $\overline{AD}$ to $E$ and $F$, respectively, so that circular arcs $\stackrel{\frown}{AE}$ and $\stackrel{\frown}{BF}$ have $B$ and $A$ as their respective centers. Circular arc $\stackrel{\frown}{EF}$ has center $D$. The area of the shaded "smile", $AEFBDA$, is
[asy]
size(200);
defaultpen(fontsize(10pt)+linewidth(.8pt));
pair A=(-1,0), B=(1,0), D=(0,-1), C=(0,0), E=(1-sqrt(2),-sqrt(2)), F=(-1+sqrt(2),-sqrt(2));
fill(Arc((1,0),2,180,225)--Arc((0,-1),(2-sqrt(2)),225,315)--Arc((-1,0),2,315,360)--Arc((0,0),1,360,180)--cycle,mediumgray);
draw(A--B^^C--D^^A--F^^B--E);
draw(Arc((1,0),2,180,225)--Arc((0,-1),(2-sqrt(2)),225,315)--Arc((-1,0),2,315,360)--Arc((0,0),1,360,180));
label("$A$",A,N);
label("$B$",B,N);
label("$C$",C,N);
label("$D$",(-0.1,-.7));
label("$E$",E,SW);
label("$F$",F,SE);
[/asy]
$ \textbf{(A)}\ (2 - \sqrt{2})\pi\qquad\textbf{(B)}\ 2\pi - \pi\sqrt{2} - 1\qquad\textbf{(C)}\ \left(1 - \frac{\sqrt{2}}{2}\right)\pi\qquad\textbf{(D)}\ \frac{5\pi}{2} - \pi\sqrt{2} - 1\qquad\textbf{(E)}\ (3 - 2\sqrt{2})\pi $
2021 AMC 10 Spring, 12
Let $N=34 \cdot 34 \cdot 63 \cdot 270$. What is the ratio of the sum of the odd divisors of $N$ to the sum of the even divisors of $N$?
$\textbf{(A) }1:16 \qquad \textbf{(B) }1:15 \qquad \textbf{(C) }1:14 \qquad \textbf{(D) }1:8 \qquad \textbf{(E) }1:3$
2021 AMC 10 Fall, 1
What is the value of $1234+2341+3412+4123$?
$\textbf{(A) } 10,000 \qquad \textbf{(B) }10,010 \qquad \textbf{(C) }10,110 \qquad \textbf{(D) }11,000 \qquad \textbf{(E) }11,110$
1969 AMC 12/AHSME, 17
The equation $2^{2x}-8\cdot 2^x+12=0$ is satisfied by:
$\textbf{(A) }\log3\qquad
\textbf{(B) }\tfrac12\log6\qquad
\textbf{(C) }1+\log\tfrac34\qquad$
$\textbf{(D) }1+\tfrac{\log3}{\log2}\qquad
\textbf{(E) }\text{none of these}$
2012 AMC 8, 10
How many 4-digit numbers greater than 1000 are there that use the four digits of 2012?
$\textbf{(A)}\hspace{.05in}6 \qquad \textbf{(B)}\hspace{.05in}7 \qquad \textbf{(C)}\hspace{.05in}8 \qquad \textbf{(D)}\hspace{.05in}9 \qquad \textbf{(E)}\hspace{.05in}12 $
2015 AIME Problems, 7
In the diagram below, $ABCD$ is a square. Point $E$ is the midpoint of $\overline{AD}$. Points $F$ and $G$ lie on $\overline{CE}$, and $H$ and $J$ lie on $\overline{AB}$ and $\overline{BC}$, respectively, so that $FGHJ$ is a square. Points $K$ and $L$ lie on $\overline{GH}$, and $M$ and $N$ lie on $\overline{AD}$ and $\overline{AB}$, respectively, so that $KLMN$ is a square. The area of $KLMN$ is 99. Find the area of $FGHJ$.
[asy]
pair A,B,C,D,E,F,G,H,J,K,L,M,N;
B=(0,0);
real m=7*sqrt(55)/5;
J=(m,0);
C=(7*m/2,0);
A=(0,7*m/2);
D=(7*m/2,7*m/2);
E=(A+D)/2;
H=(0,2m);
N=(0,2m+3*sqrt(55)/2);
G=foot(H,E,C);
F=foot(J,E,C);
draw(A--B--C--D--cycle);
draw(C--E);
draw(G--H--J--F);
pair X=foot(N,E,C);
M=extension(N,X,A,D);
K=foot(N,H,G);
L=foot(M,H,G);
draw(K--N--M--L);
label("$A$",A,NW);
label("$B$",B,SW);
label("$C$",C,SE);
label("$D$",D,NE);
label("$E$",E,dir(90));
label("$F$",F,NE);
label("$G$",G,NE);
label("$H$",H,W);
label("$J$",J,S);
label("$K$",K,SE);
label("$L$",L,SE);
label("$M$",M,dir(90));
label("$N$",N,dir(180));
[/asy]
2014 JHMMC 7 Contest, 26
Alex is training to make $\text{MOP}$. Currently he will score a $0$ on $\text{the AMC,}\text{ the AIME,}\text{and the USAMO}$. He can expend $3$ units of effort to gain $6$ points on the $\text{AMC}$, $7$ units of effort to gain $10$ points on the $\text{AIME}$, and $10$ units of effort to gain $1$ point on the $\text{USAMO}$. He will need to get at least $200$ points on $\text{the AMC}$ and $\text{AIME}$ combined and get at least $21$ points on $\text{the USAMO}$ to make $\text{MOP}$. What is the minimum amount of effort he can expend to make $\text{MOP}$?
2017 AIME Problems, 2
Teams $T_1$, $T_2$, $T_3$, and $T_4$ are in the playoffs. In the semifinal matches, $T_1$ plays $T_4$ and $T_2$ plays $T_3$. The winners of those two matches will play each other in the final match to determine the champion. When $T_i$ plays $T_j$, the probability that $T_i$ wins is $\frac{i}{i+j}$, and the outcomes of all the matches are independent. The probability that $T_4$ will be the champion is $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.
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 $1$?
$\textbf{(A) }\frac{1}{3} \qquad \textbf{(B) }\frac{1}{2} \qquad \textbf{(C) }\frac{2}{3} \qquad \textbf{(D) }\frac{3}{4} \qquad \textbf{(E) }\frac{5}{6}$
2008 AIME Problems, 10
Let $ ABCD$ be an isosceles trapezoid with $ \overline{AD}\parallel{}\overline{BC}$ whose angle at the longer base $ \overline{AD}$ is $ \dfrac{\pi}{3}$. The diagonals have length $ 10\sqrt {21}$, and point $ E$ is at distances $ 10\sqrt {7}$ and $ 30\sqrt {7}$ from vertices $ A$ and $ D$, respectively. Let $ F$ be the foot of the altitude from $ C$ to $ \overline{AD}$. The distance $ EF$ can be expressed in the form $ m\sqrt {n}$, where $ m$ and $ n$ are positive integers and $ n$ is not divisible by the square of any prime. Find $ m \plus{} n$.
2008 AMC 10, 13
Doug can paint a room in $ 5$ hours. Dave can paint the same room in $ 7$ hours. Doug and Dave paint the room together and take a one-hour break for lunch. Let $ t$ be the total time, in hours, required for them to complete the job working together, including lunch. Which of the following equations is satisfied by $ t$?
$ \textbf{(A)}\ \left(\frac{1}{5}\plus{}\frac{1}{7}\right)(t\plus{}1)\equal{}1 \qquad
\textbf{(B)}\ \left(\frac{1}{5}\plus{}\frac{1}{7}\right)t\plus{}1\equal{}1 \qquad
\textbf{(C)}\ \left(\frac{1}{5}\plus{}\frac{1}{7}\right)t\equal{}1 \\
\textbf{(D)}\ \left(\frac{1}{5}\plus{}\frac{1}{7}\right)(t\minus{}1)\equal{}1 \qquad
\textbf{(E)}\ (5\plus{}7)t\equal{}1$
2015 AMC 12/AHSME, 12
Let $a$, $b$, and $c$ be three distinct one-digit numbers. What is the maximum value of the sum of the roots of the equation $(x-a)(x-b)+(x-b)(x-c)=0$?
$ \textbf {(A) } 15 \qquad \textbf {(B) } 15.5 \qquad \textbf {(C) } 16 \qquad \textbf {(D) } 16.5 \qquad \textbf {(E) } 17 $
2017 AMC 10, 2
Pablo buys popsicles for his friends. The store sells single popsicles for $\$1$ each, 3-popsicle boxes for $\$2$, and 5-popsicle boxes for $\$3$. What is the greatest number of popsicles that Pablo can buy with $\$8$?
$\textbf{(A)}\ 8\qquad\textbf{(B)}\ 11\qquad\textbf{(C)}\ 12\qquad\textbf{(D)}\ 13\qquad\textbf{(E)}\ 15$
1971 AMC 12/AHSME, 27
A box contains chips, each of which is red, white, or blue. The number of blue chips is at least half the number of white chips, and at most one third the number of red chips. The number which are white or blue is at least $55$. The minimum number of red chips is
$\textbf{(A) }24\qquad\textbf{(B) }33\qquad\textbf{(C) }45\qquad\textbf{(D) }54\qquad \textbf{(E) }57$
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 $
1991 AMC 12/AHSME, 23
If $ABCD$ is a $2\ X\ 2$ square, $E$ is the midpoint of $\overline{AB}$, $F$ is the midpoint of $\overline{BC}$, $\overline{AF}$ and $\overline{DE}$ intersect at $I$, and $\overline{BD}$ and $\overline{AF}$ intersect at $H$, then the area of quadrilateral $BEIH$ is
[asy]
size(200);
defaultpen(linewidth(0.7)+fontsize(10));
pair B=origin, A=(0,2), C=(2,0), D=(2,2), E=(0,1), F=(1,0);
draw(A--E--B--F--C--D--A--F^^E--D--B);
label("A", A, NW);
label("B", B, SW);
label("C", C, SE);
label("D", D, NE);
label("E", E, W);
label("F", F, S);
label("H", (.8,0.6));
label("I", (0.4,1.4));
[/asy]
$ \textbf{(A)}\ \frac{1}{3}\qquad\textbf{(B)}\ \frac{2}{5}\qquad\textbf{(C)}\ \frac{7}{15}\qquad\textbf{(D)}\ \frac{8}{15}\qquad\textbf{(E)}\ \frac{3}{5} $
2010 Stanford Mathematics Tournament, 8
Let $P(x)$ be a polynomial of degree $n$ such that $P(x)=3^k$ for $0\le k \le n$. Find $P(n+1)$
2012 AMC 10, 19
In rectangle $ABCD$, $AB=6$, $AD=30$, and $G$ is the midpoint of $\overline{AD}$. Segment $AB$ is extended $2$ units beyond $B$ to point $E$, and $F$ is the intersection of $\overline{ED}$ and $\overline{BC}$. What is the area of $BFDG$?
$ \textbf{(A)}\ \frac{133}{2}\qquad\textbf{(B)}\ 67\qquad\textbf{(C)}\ \frac{135}{2}\qquad\textbf{(D)}\ 68\qquad\textbf{(E)}\ \frac{137}{2}$
1979 AMC 12/AHSME, 22
Find the number of pairs $(m, n)$ of integers which satisfy the equation $m^3 + 6m^2 + 5m = 27n^3 + 9n^2 + 9n + 1$.
$\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }3\qquad\textbf{(D) }9\qquad\textbf{(E) }\text{infinitely many}$
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 $N?$
$(\textbf{A})\: 10\qquad(\textbf{B}) \: 30\qquad(\textbf{C}) \: 60\qquad(\textbf{D}) \: 100\qquad(\textbf{E}) \: 120$
2013 AMC 12/AHSME, 6
Real numbers $x$ and $y$ satisfy the equation $x^2+y^2=10x-6y-34$. What is $x+y$?
$ \textbf{(A) }1\qquad\textbf{(B) }2\qquad\textbf{(C) }3\qquad\textbf{(D) }6\qquad\textbf{(E) }8 $
2007 AIME Problems, 14
Let a sequence be defined as follows: $a_{1}= 3$, $a_{2}= 3$, and for $n \ge 2$, $a_{n+1}a_{n-1}= a_{n}^{2}+2007$. Find the largest integer less than or equal to $\frac{a_{2007}^{2}+a_{2006}^{2}}{a_{2007}a_{2006}}$.