Found problems: 649
2019 AMC 10, 19
Let $S$ be the set of all positive integer divisors of $100,000.$ How many numbers are the product of two distinct elements of $S?$
$\textbf{(A) }98\qquad\textbf{(B) }100\qquad\textbf{(C) }117\qquad\textbf{(D) }119\qquad\textbf{(E) }121$
2020 AMC 12/AHSME, 20
Two different cubes of the same size are to be painted, with the color of each face being chosen independently and at random to be either black or white. What is the probability that after they are painted, the cubes can be rotated to be identical in appearance?
$\textbf{(A)}\ \frac{9}{64} \qquad\textbf{(B)}\ \frac{289}{2048} \qquad\textbf{(C)}\ \frac{73}{512} \qquad\textbf{(D)}\ \frac{147}{1024} \qquad\textbf{(E)}\ \frac{589}{4096}$
2019 AMC 12/AHSME, 11
How many unordered pairs of edges of a given cube determine a plane?
$\textbf{(A) } 21 \qquad\textbf{(B) } 28 \qquad\textbf{(C) } 36 \qquad\textbf{(D) } 42 \qquad\textbf{(E) } 66$
2010 AMC 12/AHSME, 6
A [i]palindrome[/i], such as $ 83438$, is a number that remains the same when its digits are reversed. The numbers $ x$ and $ x \plus{} 32$ are three-digit and four-digit palindromes, respectively. What is the sum of the digits of x?
$ \textbf{(A)}\ 20\qquad \textbf{(B)}\ 21\qquad \textbf{(C)}\ 22\qquad \textbf{(D)}\ 23\qquad \textbf{(E)}\ 24$
2019 AMC 12/AHSME, 22
Circles $\omega$ and $\gamma$, both centered at $O$, have radii $20$ and $17$, respectively. Equilateral triangle $ABC$, whose interior lies in the interior of $\omega$ but in the exterior of $\gamma$, has vertex $A$ on $\omega$, and the line containing side $\overline{BC}$ is tangent to $\gamma$. Segments $\overline{AO}$ and $\overline{BC}$ intersect at $P$, and $\dfrac{BP}{CP} = 3$. Then $AB$ can be written in the form $\dfrac{m}{\sqrt{n}} - \dfrac{p}{\sqrt{q}}$ for positive integers $m$, $n$, $p$, $q$ with $\gcd(m,n) = \gcd(p,q) = 1$. What is $m+n+p+q$?
$\phantom{}$
$\textbf{(A) } 42 \qquad \textbf{(B) }86 \qquad \textbf{(C) } 92 \qquad \textbf{(D) } 114 \qquad \textbf{(E) } 130$
2023 AMC 12/AHSME, 17
Flora the frog starts at $0$ on the number line and makes a sequence of jumps to the right. In any one jump, independent of previous jumps, Flora leaps a positive integer distance $m$ with probability $\frac{1}{2^m}$. What is the probability that Flora will eventually land at $10$?
$\textbf{(A) } \frac{5}{512} \qquad \textbf{(B) } \frac{45}{1024} \qquad \textbf{(C) } \frac{127}{1024} \qquad \textbf{(D) } \frac{511}{1024} \qquad \textbf{(E) } \frac{1}{2}$
2013 AMC 12/AHSME, 22
A palindrome is a nonnegative integer number that reads the same forwards and backwards when written in base 10 with no leading zeros. A 6-digit palindrome $n$ is chosen uniformly at random. What is the probability that $\frac{n}{11}$ is also a palindrome?
$ \textbf{(A)} \ \frac{8}{25} \qquad \textbf{(B)} \ \frac{33}{100} \qquad \textbf{(C)} \ \frac{7}{20} \qquad \textbf{(D)} \ \frac{9}{25} \qquad \textbf{(E)} \ \frac{11}{30}$
2017 AMC 10, 18
In the figure below, $3$ of the $6$ disks are to be painted blue, $2$ are to be painted red, and $1$ is to be painted green. Two paintings that can be obtained from one another by a rotation or a reflection of the entire figure are considered the same. How many different paintings are possible?
[asy]
size(100);
pair A, B, C, D, E, F;
A = (0,0);
B = (1,0);
C = (2,0);
D = rotate(60, A)*B;
E = B + D;
F = rotate(60, A)*C;
draw(Circle(A, 0.5));
draw(Circle(B, 0.5));
draw(Circle(C, 0.5));
draw(Circle(D, 0.5));
draw(Circle(E, 0.5));
draw(Circle(F, 0.5));
[/asy]
$\textbf{(A) } 6 \qquad \textbf{(B) } 8 \qquad \textbf{(C) } 9 \qquad \textbf{(D) } 12 \qquad \textbf{(E) } 15$
1959 AMC 12/AHSME, 2
Through a point $P$ inside the triangle $ABC$ a line is drawn parallel to the base $AB$, dividing the triangle into two equal areas. If the altitude to $AB$ has a length of $1$, then the distance from $P$ to $AB$ is:
$ \textbf{(A)}\ \frac12 \qquad\textbf{(B)}\ \frac14\qquad\textbf{(C)}\ 2-\sqrt2\qquad\textbf{(D)}\ \frac{2-\sqrt2}{2}\qquad\textbf{(E)}\ \frac{2+\sqrt2}{8} $
2019 AMC 10, 17
A red ball and a green ball are randomly and independently tossed into bins numbered with positive integers so that for each ball, the probability that it is tossed into bin $k$ is $2^{-k}$ for $k=1,2,3,\ldots.$ What is the probability that the red ball is tossed into a higher-numbered bin than the green ball?
$\textbf{(A) } \frac{1}{4} \qquad\textbf{(B) } \frac{2}{7} \qquad\textbf{(C) } \frac{1}{3} \qquad\textbf{(D) } \frac{3}{8} \qquad\textbf{(E) } \frac{3}{7}$
2023 AMC 12/AHSME, 7
For how many integers $n$ does the expression \[\sqrt{\frac{\log (n^2) - (\log n)^2}{\log n - 3}} \] represent a real number, where log denotes the base $10$ logarithm?
$
\textbf{(A) }900 \qquad \textbf{(B) }2\qquad \textbf{(C) }902 \qquad \textbf{(D) } 2 \qquad \textbf{(E) }901$
2023 AMC 12/AHSME, 9
What is the area of the region in the coordinate plane defined by the inequality \[\left||x|-1\right|+\left||y|-1\right|\leq 1?\]
$\textbf{(A)}~4\qquad\textbf{(B)}~8\qquad\textbf{(C)}~10\qquad\textbf{(D)}~12\qquad\textbf{(E)}~15$
2023 AMC 12/AHSME, 21
If $A$ and $B$ are vertices of a polyhedron, define the [i]distance[/i] $d(A, B)$ to be the minimum number of edges of the polyhedron one must traverse in order to connect $A$ and $B$. For example, if $\overline{AB}$ is an edge of the polyhedron, then $d(A, B) = 1$, but if $\overline{AC}$ and $\overline{CB}$ are edges and $\overline{AB}$ is not an edge, then $d(A, B) = 2$. Let $Q$, $R$, and $S$ be randomly chosen distinct vertices of a regular icosahedron (regular polyhedron made up of 20 equilateral triangles). What is the probability that $d(Q, R) > d(R, S)$?
$\textbf{(A)}~\frac{7}{22}\qquad\textbf{(B)}~\frac13\qquad\textbf{(C)}~\frac38\qquad\textbf{(D)}~\frac5{12}\qquad\textbf{(E)}~\frac12$
1959 AMC 12/AHSME, 5
The value of $\left(256\right)^{.16}\left(256\right)^{.09}$ is:
$ \textbf{(A)}\ 4 \qquad\textbf{(B)}\ 16\qquad\textbf{(C)}\ 64\qquad\textbf{(D)}\ 256.25\qquad\textbf{(E)}\ -16$
1959 AMC 12/AHSME, 11
The logarithm of $.0625$ to the base $2$ is:
$ \textbf{(A)}\ .025 \qquad\textbf{(B)}\ .25\qquad\textbf{(C)}\ 5\qquad\textbf{(D)}\ -4\qquad\textbf{(E)}\ -2 $
2019 AMC 12/AHSME, 14
Let $S$ be the set of all positive integer divisors of $100,000.$ How many numbers are the product of two distinct elements of $S?$
$\textbf{(A) }98\qquad\textbf{(B) }100\qquad\textbf{(C) }117\qquad\textbf{(D) }119\qquad\textbf{(E) }121$
2017 AMC 12/AHSME, 24
Quadrilateral $ABCD$ is inscribed in circle $O$ and has sides $AB = 3$, $BC = 2$, $CD = 6$, and $DA = 8$. Let $X$ and $Y$ be points on $\overline{BD}$ such that
\[\frac{DX}{BD} = \frac{1}{4} \quad \text{and} \quad \frac{BY}{BD} = \frac{11}{36}.\]
Let $E$ be the intersection of intersection of line $AX$ and the line through $Y$ parallel to $\overline{AD}$. Let $F$ be the intersection of line $CX$ and the line through $E$ parallel to $\overline{AC}$. Let $G$ be the point on circle $O$ other than $C$ that lies on line $CX$. What is $XF \cdot XG$?
$\textbf{(A) }17\qquad\textbf{(B) }\frac{59 - 5\sqrt{2}}{3}\qquad\textbf{(C) }\frac{91 - 12\sqrt{3}}{4}\qquad\textbf{(D) }\frac{67 - 10\sqrt{2}}{3}\qquad\textbf{(E) }18$
2022 AMC 12/AHSME, 12
Let $M$ be the midpoint of $\overline{AB}$ in regular tetrahedron $ABCD$. What is $\cos({\angle CMD})$?
$\textbf{(A)} ~\frac{1}{4} \qquad\textbf{(B)} ~\frac{1}{3} \qquad\textbf{(C)} ~\frac{2}{5} \qquad\textbf{(D)} ~\frac{1}{2} \qquad\textbf{(E)} ~\frac{\sqrt{3}}{2} $
2019 AMC 10, 5
What is the greatest number of consecutive integers whose sum is $45 ?$
$\textbf{(A) } 9 \qquad\textbf{(B) } 25 \qquad\textbf{(C) } 45 \qquad\textbf{(D) } 90 \qquad\textbf{(E) } 120$
2018 AMC 12/AHSME, 14
Joey and Chloe and their daughter Zoe all have the same birthday. Joey is 1 year older than Chloe, and Zoe is exactly 1 year old today. Today is the first of the 9 birthdays on which Chloe's age will be an integral multiple of Zoe's age. What will be the sum of the two digits of Joey's age the next time his age is a multiple of Zoe's age?
$\textbf{(A) } 7 \qquad \textbf{(B) } 8 \qquad \textbf{(C) } 9 \qquad \textbf{(D) } 10 \qquad \textbf{(E) } 11 $
2022 AMC 12/AHSME, 23
Let $x_{0}$, $x_{1}$, $x_{2}$, $\cdots$ be a sequence of numbers, where each $x_{k}$ is either $0$ or $1$. For each positive integer $n$, define
\[S_{n} = \displaystyle\sum^{n-1}_{k=0}{x_{k}2^{k}}\]
Suppose $7S_{n} \equiv 1\pmod {2^{n}}$ for all $n\geq 1$. What is the value of the sum
\[x_{2019}+2x_{2020}+4x_{2021}+8x_{2022}?\]
$ \textbf{(A)}\ 6 \qquad
\textbf{(B)}\ 7 \qquad
\textbf{(C)}\ 12 \qquad
\textbf{(D)}\ 14 \qquad
\textbf{(E)}\ 15$
2019 AMC 12/AHSME, 24
For how many integers $n$ between $1$ and $50$, inclusive, is
\[
\frac{(n^2-1)!}{(n!)^n}
\]an integer? (Recall that $0! = 1$.)
$\textbf{(A) } 31 \qquad \textbf{(B) } 32 \qquad \textbf{(C) } 33 \qquad \textbf{(D) } 34 \qquad \textbf{(E) } 35$
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?
$\textbf{(A) }6\qquad\textbf{(B) }8\qquad\textbf{(C) }10\qquad\textbf{(D) }12\qquad\textbf{(E) }\text{No new vertex will land on }B.$
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?
$\textbf{(A)} \dfrac{7}{576} \qquad \textbf{(B)} \dfrac{5}{192} \qquad \textbf{(C)} \dfrac{1}{36} \qquad \textbf{(D)} \dfrac{5}{144} \qquad \textbf{(E)}\dfrac{7}{48}$
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]
$\textbf{(A) } \frac{1}{14} \qquad \textbf{(B) } \frac{1}{12} \qquad \textbf{(C) } \frac{1}{10} \qquad \textbf{(D) } \frac{3}{28} \qquad \textbf{(E) } \frac{1}{9}$