Found problems: 649
2022 AMC 12/AHSME, 16
A [i]triangular number[/i] is a positive integer that can be expressed in the form $t_n = 1 + 2 + 3 +\cdots + n$, for some positive integer $n$. The three smallest triangular numbers that are also perfect squares are $t_1 = 1 = 1^2$, $t_8 = 36 = 6^2$, and $t_{49} = 1225 = 35^2$. What is the sum of the digits of the fourth smallest triangular number that is also a perfect square?
$\textbf{(A)} ~6 \qquad\textbf{(B)} ~9 \qquad\textbf{(C)} ~12 \qquad\textbf{(D)} ~18 \qquad\textbf{(E)} ~27 $
2024 AMC 12/AHSME, 16
A group of $16$ people will be partitioned into $4$ indistinguishable $4$-person committees. Each committee will have one chairperson and one secretary. The number of different ways to make these assignments can be written as $3^{r}M$, where $r$ and $M$ are positive integers and $M$ is not divisible by $3$. What is $r$?
$
\textbf{(A) }5 \qquad
\textbf{(B) }6 \qquad
\textbf{(C) }7 \qquad
\textbf{(D) }8 \qquad
\textbf{(E) }9 \qquad
$
2023 AMC 10, 3
A $3-4-5$ right triangle is inscribed in circle $A$, and a $5-12-13$ right triangle is inscribed in circle $B$. What is the ratio of the area of circle $A$ to the area of circle $B$?
$\textbf{(A)}~\frac{9}{25}\qquad\textbf{(B)}~\frac{1}{9}\qquad\textbf{(C)}~\frac{1}{5}\qquad\textbf{(D)}~\frac{25}{169}\qquad\textbf{(E)}~\frac{4}{25}$
2018 AMC 12/AHSME, 6
For positive integers $m$ and $n$ such that $m+10<n+1$, both the mean and the median of the set $\{m, m+4, m+10, n+1, n+2, 2n\}$ are equal to $n$. What is $m+n$?
$\textbf{(A) }20\qquad\textbf{(B) }21\qquad\textbf{(C) }22\qquad\textbf{(D) }23\qquad\textbf{(E) }24$
2023 AMC 12/AHSME, 19
What is the product of all the solutions to the equation $$\log_{7x}2023 \cdot \log_{289x} 2023 = \log_{2023x} 2023?$$
$\textbf{(A) }(\log_{2023}7 \cdot \log_{2023}289)^2 \qquad\textbf{(B) }\log_{2023}7 \cdot \log_{2023}289\qquad\textbf{(C) }1\qquad\textbf{(D) }\log_{7}2023 \cdot \log_{289}2023\qquad\textbf{(E) }(\log_{7}2023 \cdot \log_{289}2023)^2$
2023 AMC 12/AHSME, 9
A square of area $2$ is inscribed in a square of area $3$, creating four congruent triangles, as shown below. What is the ratio of the shorter leg to the longer leg in the shaded right triangle?
[asy]
size(200);
defaultpen(linewidth(0.6pt)+fontsize(10pt));
real y = sqrt(3);
pair A,B,C,D,E,F,G,H;
A = (0,0);
B = (0,y);
C = (y,y);
D = (y,0);
E = ((y + 1)/2,y);
F = (y, (y - 1)/2);
G = ((y - 1)/2, 0);
H = (0,(y + 1)/2);
fill(H--B--E--cycle, gray);
draw(A--B--C--D--cycle);
draw(E--F--G--H--cycle);
[/asy]
$\textbf{(A) }\frac15\qquad\textbf{(B) }\frac14\qquad\textbf{(C) }2-\sqrt3\qquad\textbf{(D) }\sqrt3-\sqrt2\qquad\textbf{(E) }\sqrt2-1$
2020 AMC 10, 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 $b$?
$\textbf{(A) }2\qquad\textbf{(B) }3\qquad\textbf{(C) }5\qquad\textbf{(D) }7\qquad\textbf{(E) }11$
2024 AMC 12/AHSME, 4
What is the least value of $n$ such that $n!$ is a multiple of $2024$?
$
\textbf{(A) }11 \qquad
\textbf{(B) }21 \qquad
\textbf{(C) }22 \qquad
\textbf{(D) }23 \qquad
\textbf{(E) }253 \qquad
$
2023 AMC 10, 25
A regular pentagon with area $\sqrt{5}+1$ is printed on paper and cut out. The five vertices of the pentagon are folded into the center of the pentagon, creating a smaller pentagon. What is the area of the new pentagon?
$\textbf{(A)}~4-\sqrt{5}\qquad\textbf{(B)}~\sqrt{5}-1\qquad\textbf{(C)}~8-3\sqrt{5}\qquad\textbf{(D)}~\frac{\sqrt{5}+1}{2}\qquad\textbf{(E)}~\frac{2+\sqrt{5}}{3}$
2017 AMC 12/AHSME, 16
In the figure below, semicircles with centers at $A$ and $B$ and with radii $2$ and $1$, respectively, are drawn in the interior of, and sharing bases with, a semicircle with diameter $\overline{JK}$. The two smaller semicircles are externally tangent to each other and internally tangent to the largest semicircle. A circle centered at $P$ is drawn externally tangent to the two smaller semicircles and internally tangent to the largest semicircle. What is the radius of the circle centered at $P$?
[asy]
size(8cm);
draw(arc((0,0),3,0,180));
draw(arc((2,0),1,0,180));
draw(arc((-1,0),2,0,180));
draw((-3,0)--(3,0));
pair P = (-1,0)+(2+6/7)*dir(36.86989);
draw(circle(P,6/7));
dot((-1,0)); dot((2,0)); dot((-3,0)); dot((3,0)); dot(P);
label("$J$",(-3,0),W);
label("$A$",(-1,0),NW);
label("$B$",(2,0),NE);
label("$K$",(3,0),E);
label("$P$",P,NW);
[/asy]
$ \textbf{(A)}\ \frac{3}{4}
\qquad \textbf{(B)}\ \frac{6}{7}
\qquad\textbf{(C)}\ \frac{1}{2}\sqrt{3}
\qquad\textbf{(D)}\ \frac{5}{8}\sqrt{2}
\qquad\textbf{(E)}\ \frac{11}{12} $
2024 AMC 12/AHSME, 22
The figure below shows a dotted grid $8$ cells wide and $3$ cells tall consisting of $1''\times1''$ squares. Carl places $1$-inch toothpicks along some of the sides of the squares to create a closed loop that does not intersect itself. The numbers in the cells indicate the number of sides of that square that are to be covered by toothpicks, and any number of toothpicks are allowed if no number is written. In how many ways can Carl place the toothpicks? [asy]
size(6cm);
for (int i=0; i<9; ++i) {
draw((i,0)--(i,3),dotted);
}
for (int i=0; i<4; ++i){
draw((0,i)--(8,i),dotted);
}
for (int i=0; i<8; ++i) {
for (int j=0; j<3; ++j) {
if (j==1) {
label("1",(i+0.5,1.5));
}}}
[/asy] $\textbf{(A) }130\qquad\textbf{(B) }144\qquad\textbf{(C) }146\qquad\textbf{(D) }162\qquad\textbf{(E) }196$
2015 AMC 10, 25
A rectangular box measures $a \times b \times c$, where $a,$ $b,$ and $c$ are integers and $1 \leq a \leq b \leq c$. The volume and surface area of the box are numerically equal. How many ordered triples $(a,b,c)$ are possible?
$ \textbf{(A) }4\qquad\textbf{(B) }10\qquad\textbf{(C) }12\qquad\textbf{(D) }21\qquad\textbf{(E) }26 $
2021 AMC 12/AHSME Fall, 25
For $n$ a positive integer, let $R(n)$ be the sum of the remainders when $n$ is divided by $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, and $10$. For example, $R(15) = 1+0+3+0+3+1+7+6+5=26$. How many two-digit positive integers $n$ satisfy $R(n) = R(n+1)\,?$
$\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }3\qquad\textbf{(E) }4$
2021 AMC 12/AHSME Fall, 4
Let $n = 8^{2022}$. Which of the following is equal to $\frac{n}{4}$?
$\textbf{(A) }4^{1010}\qquad\textbf{(B) }2^{2022}\qquad\textbf{(C) }8^{2018}\qquad\textbf{(D) }4^{3031}\qquad\textbf{(E) }4^{3032}$
2019 AMC 10, 21
A sphere with center $O$ has radius $6$. A triangle with sides of length $15, 15,$ and $24$ is situated in space so that each of its sides is tangent to the sphere. What is the distance between $O$ and the plane determined by the triangle?
$
\textbf{(A) }2\sqrt{3}\qquad
\textbf{(B) }4\qquad
\textbf{(C) }3\sqrt{2}\qquad
\textbf{(D) }2\sqrt{5}\qquad
\textbf{(E) }5\qquad
$
2006 AMC 12/AHSME, 16
Circles with centers $ A$ and $ B$ have radii 3 and 8, respectively. A common internal tangent intersects the circles at $ C$ and $ D$, respectively. Lines $ AB$ and $ CD$ intersect at $ E$, and $ AE \equal{} 5$. What is $ CD$?
[asy]unitsize(2.5mm);
defaultpen(fontsize(10pt)+linewidth(.8pt));
dotfactor=3;
pair A=(0,0), Ep=(5,0), B=(5+40/3,0);
pair M=midpoint(A--Ep);
pair C=intersectionpoints(Circle(M,2.5),Circle(A,3))[1];
pair D=B+8*dir(180+degrees(C));
dot(A);
dot(C);
dot(B);
dot(D);
draw(C--D);
draw(A--B);
draw(Circle(A,3));
draw(Circle(B,8));
label("$A$",A,W);
label("$B$",B,E);
label("$C$",C,SE);
label("$E$",Ep,SSE);
label("$D$",D,NW);[/asy]$ \textbf{(A) } 13\qquad \textbf{(B) } \frac {44}{3}\qquad \textbf{(C) } \sqrt {221}\qquad \textbf{(D) } \sqrt {255}\qquad \textbf{(E) } \frac {55}{3}$
2019 AMC 12/AHSME, 6
The figure below shows line $\ell$ with a regular, infinite, recurring pattern of squares and line segments.
[asy]
size(300);
defaultpen(linewidth(0.8));
real r = 0.35;
path P = (0,0)--(0,1)--(1,1)--(1,0), Q = (1,1)--(1+r,1+r);
path Pp = (0,0)--(0,-1)--(1,-1)--(1,0), Qp = (-1,-1)--(-1-r,-1-r);
for(int i=0;i <= 4;i=i+1)
{
draw(shift((4*i,0)) * P);
draw(shift((4*i,0)) * Q);
}
for(int i=1;i <= 4;i=i+1)
{
draw(shift((4*i-2,0)) * Pp);
draw(shift((4*i-1,0)) * Qp);
}
draw((-1,0)--(18.5,0),Arrows(TeXHead));
[/asy]
How many of the following four kinds of rigid motion transformations of the plane in which this figure is drawn, other than the identity transformation, will transform this figure into itself?
[list]
[*] some rotation around a point of line $\ell$
[*] some translation in the direction parallel to line $\ell$
[*] the reflection across line $\ell$
[*] some reflection across a line perpendicular to line $\ell$
[/list]
$\textbf{(A) } 0 \qquad\textbf{(B) } 1 \qquad\textbf{(C) } 2 \qquad\textbf{(D) } 3 \qquad\textbf{(E) } 4$
2016 AMC 12/AHSME, 25
Let $k$ be a positive integer. Bernardo and Silvia take turns writing and erasing numbers on a blackboard as follows. Bernardo starts by writing the smallest perfect square with $k+1$ digits. Every time Bernardo writes a number, Silvia erases the last $k$ digits of it. Bernardo then writes the next perfect square, Silvia erases the last $k$ digits of it, and this process continues until the last two numbers that remain on the board differ by at least $2$. Let $f(k)$ be the smallest positive integer not written on the board. For example, if $k = 1$, then the numbers that Bernardo writes are $16$, $25$, $36$, $49$, and $64$, and the numbers showing on the board after Silvia erases are $1$, $2$, $3$, $4$, and $6$, and thus $f(1) = 5$. What is the sum of the digits of $f(2) + f(4) + f(6) + \cdots + f(2016)$?
$\textbf{(A) } 7986 \qquad\textbf{(B) } 8002 \qquad\textbf{(C) } 8030 \qquad\textbf{(D) } 8048 \qquad\textbf{(E) } 8064$
2024 AMC 12/AHSME, 25
A graph is $\textit{symmetric}$ about a line if the graph remains unchanged after reflection in that line. For how many quadruples of integers $(a,b,c,d)$, where $|a|,|b|,|c|,|d|\le5$ and $c$ and $d$ are not both $0$, is the graph of \[y=\frac{ax+b}{cx+d}\] symmetric about the line $y=x$?
$\textbf{(A) }1282\qquad\textbf{(B) }1292\qquad\textbf{(C) }1310\qquad\textbf{(D) }1320\qquad\textbf{(E) }1330$
2020 AMC 12/AHSME, 1
What is the value in simplest form of the following expression? \[\sqrt{1} + \sqrt{1+3} + \sqrt{1+3+5} + \sqrt{1+3+5+7}\]
$\textbf{(A) }5 \qquad \textbf{(B) }4 + \sqrt{7} + \sqrt{10} \qquad \textbf{(C) } 10 \qquad \textbf{(D) } 15 \qquad \textbf{(E) } 4 + 3\sqrt{3} + 2\sqrt{5} + \sqrt{7}$
2024 AMC 12/AHSME, 12
The first three terms of a geometric sequence are the integers $a,\,720,$ and $b,$ where $a<720<b.$ What is the sum of the digits of the least possible value of $b?$
$\textbf{(A) } 9 \qquad \textbf{(B) } 12 \qquad \textbf{(C) } 16 \qquad \textbf{(D) } 18 \qquad \textbf{(E) } 21$
2024 AMC 12/AHSME, 10
A list of 9 real numbers consists of $1$, $2.2 $, $3.2 $, $5.2 $, $6.2 $, $7$, as well as $x, y,z$ with $x\leq y\leq z$. The range of the list is $7$, and the mean and median are both positive integers. How many ordered triples $(x,y,z)$ are possible?
$
\textbf{(A) }1 \qquad
\textbf{(B) }2 \qquad
\textbf{(C) }3 \qquad
\textbf{(D) }4 \qquad
\textbf{(E) infinitely many}\qquad
$
1959 AMC 12/AHSME, 19
With the use of three different weights, namely 1 lb., 3 lb., and 9 lb., how many objects of different weights can be weighed, if the objects is to be weighed and the given weights may be placed in either pan of the scale?
$ \textbf{(A)}\ 15 \qquad\textbf{(B)}\ 13\qquad\textbf{(C)}\ 11\qquad\textbf{(D)}\ 9\qquad\textbf{(E)}\ 7 $
2023 AMC 10, 13
What is the area of the region in the coordinate plane defined by the inequality \[\left||x|-1\right|+\left||y|-1\right|\leq 1?\]
$\textbf{(A)}~4\qquad\textbf{(B)}~8\qquad\textbf{(C)}~10\qquad\textbf{(D)}~12\qquad\textbf{(E)}~15$
2024 AMC 12/AHSME, 24
A $\textit{disphenoid}$ is a tetrahedron whose triangular faces are congruent to one another. What is the least total surface area of a disphenoid whose faces are scalene triangles with integer side lengths?
$\textbf{(A) }\sqrt{3}\qquad\textbf{(B) }3\sqrt{15}\qquad\textbf{(C) }15\qquad\textbf{(D) }15\sqrt{7}\qquad\textbf{(E) }24\sqrt{6}$