Found problems: 3632
2018 AMC 10, 4
How many ways can a student schedule 3 mathematics courses -- algebra, geometry, and number theory -- in a 6-period day if no two mathematics courses can be taken in consecutive periods? (What courses the student takes during the other 3 periods is of no concern here.)
$\textbf{(A) }3\qquad\textbf{(B) }6\qquad\textbf{(C) }12\qquad\textbf{(D) }18\qquad\textbf{(E) }24$
2012 AMC 10, 17
Let $a$ and $b$ be relatively prime integers with $a>b>0$ and $\tfrac{a^3-b^3}{(a-b)^3}=\tfrac{73}{3}$. What is $a-b$?
$ \textbf{(A)}\ 1
\qquad\textbf{(B)}\ 2
\qquad\textbf{(C)}\ 3
\qquad\textbf{(D)}\ 4
\qquad\textbf{(E)}\ 5
$
2024 AMC 12/AHSME, 18
On top of a rectangular card with sides of length $1$ and $2+\sqrt{3}$, an identical card is placed so that two of their diagonals line up, as shown ($\overline{AC}$, in this case). [asy]
defaultpen(fontsize(12)+0.85); size(150);
real h=2.25;
pair C=origin,B=(0,h),A=(1,h),D=(1,0),Dp=reflect(A,C)*D,Bp=reflect(A,C)*B;
pair L=extension(A,Dp,B,C),R=extension(Bp,C,A,D);
draw(L--B--A--Dp--C--Bp--A);
draw(C--D--R);
draw(L--C^^R--A,dashed+0.6);
draw(A--C,black+0.6);
dot("$C$",C,2*dir(C-R)); dot("$A$",A,1.5*dir(A-L)); dot("$B$",B,dir(B-R));
[/asy] Continue the process, adding a third card to the second, and so on, lining up successive diagonals after rotating clockwise. In total, how many cards must be used until a vertex of a new card lands exactly on the vertex labeled $B$ in the figure?
$\textbf{(A) }6\qquad\textbf{(B) }8\qquad\textbf{(C) }10\qquad\textbf{(D) }12\qquad\textbf{(E) }\text{No new vertex will land on }B.$
2012 AMC 10, 7
For a science project, Sammy observed a chipmunk and a squirrel stashing acorns in holes. The chipmunk hid $3$ acorns in each of the holes it dug. The squirrel hid $4$ acorns in each of the holes it dug. They each hid the same
number of acorns, although the squirrel needed $4$ fewer holes. How many acorns did the chipmunk hide?
${{ \textbf{(A)}\ 30\qquad\textbf{(B)}\ 36\qquad\textbf{(C)}\ 42\qquad\textbf{(D)}\ 48}\qquad\textbf{(E)}\ 54} $
2011 AMC 12/AHSME, 16
Rhombus $ABCD$ has side length $2$ and $\angle B = 120 ^\circ$. Region $R$ consists of all points inside the rhombus that are closer to vertex $B$ than any of the other three vertices. What is the area of $R$?
$ \textbf{(A)}\ \frac{\sqrt{3}}{3} \qquad
\textbf{(B)}\ \frac{\sqrt{3}}{2} \qquad
\textbf{(C)}\ \frac{2\sqrt{3}}{3} \qquad
\textbf{(D)}\ 1+\frac{\sqrt{3}}{3} \qquad
\textbf{(E)}\ 2 $
2002 USAMO, 5
Let $a,b$ be integers greater than 2. Prove that there exists a positive integer $k$ and a finite sequence $n_1, n_2, \dots, n_k$ of positive integers such that $n_1 = a$, $n_k = b$, and $n_i n_{i+1}$ is divisible by $n_i + n_{i+1}$ for each $i$ ($1 \leq i < k$).
2017 AMC 12/AHSME, 22
Abby, Bernardo, Carl, and Debra play a game in which each of them starts with four coins. The game consists of four rounds. In each round, four balls are placed in an urn - one green, one red, and two white. The players each draw a ball at random without replacement. Whoever gets the green ball gives one coin to whoever gets the red ball. What is the probability that, at the end of the fourth round, each of the players has four coins?
$\textbf{(A)} \dfrac{7}{576} \qquad \textbf{(B)} \dfrac{5}{192} \qquad \textbf{(C)} \dfrac{1}{36} \qquad \textbf{(D)} \dfrac{5}{144} \qquad \textbf{(E)}\dfrac{7}{48}$
2012 China Girls Math Olympiad, 8
Find the number of integers $k$ in the set $\{0, 1, 2, \dots, 2012\}$ such that $\binom{2012}{k}$ is a multiple of $2012$.
2023 AMC 10, 22
Circle $C_1$ and $C_2$ each have radius $1$, and the distance between their centers is $\frac{1}{2}$. Circle $C_3$ is the largest circle internally tangent to both $C_1$ and $C_2$. Circle $C_4$ is internally tangent to both $C_1$ and $C_2$ and externally tangent to $C_3$. What is the radius of $C_4$?
[asy]
import olympiad;
size(10cm);
draw(circle((0,0),0.75));
draw(circle((-0.25,0),1));
draw(circle((0.25,0),1));
draw(circle((0,6/7),3/28));
pair A = (0,0), B = (-0.25,0), C = (0.25,0), D = (0,6/7), E = (-0.95710678118, 0.70710678118), F = (0.95710678118, -0.70710678118);
dot(B^^C);
draw(B--E, dashed);
draw(C--F, dashed);
draw(B--C);
label("$C_4$", D);
label("$C_1$", (-1.375, 0));
label("$C_2$", (1.375,0));
label("$\frac{1}{2}$", (0, -.125));
label("$C_3$", (-0.4, -0.4));
label("$1$", (-.85, 0.70));
label("$1$", (.85, -.7));
import olympiad;
markscalefactor=0.005;
[/asy]
$\textbf{(A) } \frac{1}{14} \qquad \textbf{(B) } \frac{1}{12} \qquad \textbf{(C) } \frac{1}{10} \qquad \textbf{(D) } \frac{3}{28} \qquad \textbf{(E) } \frac{1}{9}$
1995 USAMO, 3
Given a nonisosceles, nonright triangle ABC, let O denote the center of its circumscribed circle, and let $A_1$, $B_1$, and $C_1$ be the midpoints of sides BC, CA, and AB, respectively. Point $A_2$ is located on the ray $OA_1$ so that $OAA_1$ is similar to $OA_2A$. Points $B_2$ and $C_2$ on rays $OB_1$ and $OC_1$, respectively, are defined similarly. Prove that lines $AA_2$, $BB_2$, and $CC_2$ are concurrent, i.e. these three lines intersect at a point.
2017 AMC 10, 19
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?
$\textbf{(A)}\ 12\qquad\textbf{(B)}\ 16\qquad\textbf{(C)}\ 28\qquad\textbf{(D)}\ 32\qquad\textbf{(E)}\ 40$
2014 AMC 10, 15
In rectangle $ABCD$, $DC = 2CB$ and points $E$ and $F$ lie on $\overline{AB}$ so that $\overline{ED}$ and $\overline{FD}$ trisect $\angle ADC$ as shown. What is the ratio of the area of $\triangle DEF$ to the area of rectangle $ABCD$?
[asy]
draw((0, 0)--(0, 1)--(2, 1)--(2, 0)--cycle);
draw((0, 0)--(sqrt(3)/3, 1));
draw((0, 0)--(sqrt(3), 1));
label("A", (0, 1), N);
label("B", (2, 1), N);
label("C", (2, 0), S);
label("D", (0, 0), S);
label("E", (sqrt(3)/3, 1), N);
label("F", (sqrt(3), 1), N);
[/asy]
${ \textbf{(A)}\ \ \frac{\sqrt{3}}{6}\qquad\textbf{(B)}\ \frac{\sqrt{6}}{8}\qquad\textbf{(C)}\ \frac{3\sqrt{3}}{16}\qquad\textbf{(D)}}\ \frac{1}{3}\qquad\textbf{(E)}\ \frac{\sqrt{2}}{4}$
2018 AMC 12/AHSME, 4
Alice, Bob, and Charlie were on a hike and were wondering how far away the nearest town was. When Alice said, "We are at least 6 miles away," Bob replied, "We are at most 5 miles away." Charlie then remarked, "Actually the nearest town is at most 4 miles away." It turned out that none of the three statements were true. Let $d$ be the distance in miles to the nearest town. Which of the following intervals is the set of all possible values of $d$?
$\textbf{(A) } (0,4) \qquad \textbf{(B) } (4,5) \qquad \textbf{(C) } (4,6) \qquad \textbf{(D) } (5,6) \qquad \textbf{(E) } (5,\infty) $
2024 AMC 10, 13
Positive integers $x$ and $y$ satisfy the equation $\sqrt{x}+\sqrt{y}=\sqrt{1183}.$ What is the minimum possible value of $x+y?$
$\textbf{(A) }585 \qquad\textbf{(B) }595\qquad\textbf{(C) }623\qquad\textbf{(D) }700\qquad\textbf{(E) }791$
2008 AMC 12/AHSME, 18
A pyramid has a square base $ ABCD$ and vertex $ E$. The area of square $ ABCD$ is $ 196$, and the areas of $ \triangle{ABE}$ and $ \triangle{CDE}$ are $ 105$ and $ 91$, respectively. What is the volume of the pyramid?
$ \textbf{(A)}\ 392 \qquad
\textbf{(B)}\ 196\sqrt{6} \qquad
\textbf{(C)}\ 392\sqrt2 \qquad
\textbf{(D)}\ 392\sqrt3 \qquad
\textbf{(E)}\ 784$
2006 AMC 12/AHSME, 16
Regular hexagon $ ABCDEF$ has vertices $ A$ and $ C$ at $ (0,0)$ and $ (7,1)$, respectively. What is its area?
$ \textbf{(A) } 20\sqrt {3} \qquad \textbf{(B) } 22\sqrt {3} \qquad \textbf{(C) } 25\sqrt {3} \qquad \textbf{(D) } 27\sqrt {3} \qquad \textbf{(E) } 50$
1989 AMC 12/AHSME, 27
Let $n$ be a positive integer. If the equation $2x+2y+z=n$ has $28$ solutions in positive integers $x, y$ and $z$, then $n$ must be either
$ \textbf{(A)}\ 14\ \text{or}\ 15 \qquad\textbf{(B)}\ 15\ \text{or}\ 16 \qquad\textbf{(C)}\ 16\ \text{or}\ 17 \qquad\textbf{(D)}\ 17\ \text{or}\ 18 \qquad\textbf{(E)}\ 18\ \text{or}\ 19 $
1988 USAMO, 4
Let $I$ be the incenter of triangle $ABC$, and let $A'$, $B'$, and $C'$ be the circumcenters of triangles $IBC$, $ICA$, and $IAB$, respectively. Prove that the circumcircles of triangles $ABC$ and $A'B'C'$ are concentric.
1969 AMC 12/AHSME, 19
The number of distinct ordered pairs $(x,y)$, where $x$ and $y$ have positive integral values satisfying the equation $x^4y^4-10x^2y^2+9=0$, is:
$\textbf{(A) }0\qquad
\textbf{(B) }3\qquad
\textbf{(C) }4\qquad
\textbf{(D) }12\qquad
\textbf{(E) }\text{infinite}$
2009 AMC 12/AHSME, 20
A convex polyhedron $ Q$ has vertices $ V_1,V_2,\ldots,V_n$, and $ 100$ edges. The polyhedron is cut by planes $ P_1,P_2,\ldots,P_n$ in such a way that plane $ P_k$ cuts only those edges that meet at vertex $ V_k$. In addition, no two planes intersect inside or on $ Q$. The cuts produce $ n$ pyramids and a new polyhedron $ R$. How many edges does $ R$ have?
$ \textbf{(A)}\ 200\qquad
\textbf{(B)}\ 2n\qquad
\textbf{(C)}\ 300\qquad
\textbf{(D)}\ 400\qquad
\textbf{(E)}\ 4n$
2023 AMC 12/AHSME, 14
For how many ordered pairs $(a,b)$ of integers does the polynomial $x^3+ax^2+bx+6$ have $3$ distinct integer roots?
$\textbf{(A)}\ 5 \qquad\textbf{(B)}\ 6 \qquad\textbf{(C)}\ 8 \qquad\textbf{(D)}\ 7 \qquad\textbf{(E)}\ 4$
2018 AMC 10, 11
When 7 fair standard 6-sided dice are thrown, the probability that the sum of the numbers on the top faces is 10 can be written as $$\frac{n}{6^7},$$where $n$ is a positive integer. What is $n$?
$\textbf{(A) } 42 \qquad \textbf{(B) } 49 \qquad \textbf{(C) } 56 \qquad \textbf{(D) } 63 \qquad \textbf{(E) } 84 $
2008 AMC 12/AHSME, 9
Older television screens have an aspect ratio of $ 4: 3$. That is, the ratio of the width to the height is $ 4: 3$. The aspect ratio of many movies is not $ 4: 3$, so they are sometimes shown on a television screen by 'letterboxing' - darkening strips of equal height at the top and bottom of the screen, as shown. Suppose a movie has an aspect ratio of $ 2: 1$ and is shown on an older television screen with a $ 27$-inch diagonal. What is the height, in inches, of each darkened strip?
[asy]unitsize(1mm);
defaultpen(linewidth(.8pt));
filldraw((0,0)--(21.6,0)--(21.6,2.7)--(0,2.7)--cycle,grey,black);
filldraw((0,13.5)--(21.6,13.5)--(21.6,16.2)--(0,16.2)--cycle,grey,black);
draw((0,2.7)--(0,13.5));
draw((21.6,2.7)--(21.6,13.5));[/asy]$ \textbf{(A)}\ 2 \qquad \textbf{(B)}\ 2.25 \qquad \textbf{(C)}\ 2.5 \qquad \textbf{(D)}\ 2.7 \qquad \textbf{(E)}\ 3$
1988 AMC 12/AHSME, 25
$X$, $Y$ and $Z$ are pairwise disjoint sets of people. The average ages of people in the sets $X$, $Y$, $Z$, $X \cup Y$, $X \cup Z$ and $Y \cup Z$ are given in the table below.
\begin{tabular}{|c|c|c|c|c|c|c|} \hline
\rule{0pt}{1.1em} Set & $X$ & $Y$ & $Z$ & $X\cup Y$ & $X\cup Z$ & $Y\cup Z$\\[0.5ex] \hline \rule{0pt}{2.2em} \shortstack{Average age of \\ people in the set} & 37 & 23 & 41 & 29 & 39.5 & 33\\[1ex]\hline\end{tabular}
Find the average age of the people in set $X \cup Y \cup Z$.
$ \textbf{(A)}\ 33\qquad\textbf{(B)}\ 33.5\qquad\textbf{(C)}\ 33.6\overline{6}\qquad\textbf{(D)}\ 33.83\overline{3}\qquad\textbf{(E)}\ 34 $
1987 AMC 12/AHSME, 24
How many polynomial functions $f$ of degree $\ge 1$ satisfy
\[ f(x^2)=[f(x)]^2=f(f(x)) \ ? \]
$ \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ \text{finitely many but more than 2} \\ \qquad\textbf{(E)}\ \text{infinitely many} $