Found problems: 730
2017 AMC 10, 14
An integer $N$ is selected at random in the range $1\le N \le 2020.$ What is the probability that the remainder when $N^{16}$ is divided by $5$ is $1$?
$\textbf{(A)} \text{ }\frac{1}{5} \qquad \textbf{(B)} \text{ }\frac{2}{5} \qquad \textbf{(C)} \text{ }\frac{3}{5} \qquad \textbf{(D)} \text{ }\frac{4}{5} \qquad \textbf{(E)} \text{ 1}$
2020 AMC 10, 17
Define $$P(x) =(x-1^2)(x-2^2)\cdots(x-100^2).$$
How many integers $n$ are there such that $P(n)\leq 0$?
$\textbf{(A) } 4900 \qquad \textbf{(B) } 4950\qquad \textbf{(C) } 5000\qquad \textbf{(D) } 5050 \qquad \textbf{(E) } 5100$
2017 AMC 12/AHSME, 6
Joy has $30$ thin rods, one each of every integer length from $1$ cm through $30$ cm. She places the rods with lengths $3$ cm, $7$ cm, and $15$ cm on a table. She then wants to choose a fourth rod that she can put with these three to form a quadrilateral with positive area. How many of the remaining rods can she choose as the fourth rod?
$\textbf{(A) }16\qquad\textbf{(B) }17\qquad\textbf{(C) }18\qquad\textbf{(D) }19\qquad\textbf{(E) }20$
2017 AMC 10, 11
At Typico High School, $60\%$ of the students like dancing, and the rest dislike it. Of those who like dancing, $80\%$ say that they like it, and the rest say that they dislike it. Of those who dislike dancing, $90\%$ say that they dislike it, and the rest say that they like it. What fraction of students who say they dislike dancing actually like it?
$\textbf{(A) } 10\%\qquad \textbf{(B) } 12\%\qquad \textbf{(C) } 20\%\qquad \textbf{(D) } 25\%\qquad \textbf{(E) } 33\frac{1}{3}\%$
2012 AMC 12/AHSME, 21
Let $a,b,$ and $c$ be positive integers with $a\ge b\ge c$ such that
\begin{align*} a^2-b^2-c^2+ab&=2011\text{ and}\\
a^2+3b^2+3c^2-3ab-2ac-2bc&=-1997\end{align*}
What is $a$?
$ \textbf{(A)}\ 249
\qquad\textbf{(B)}\ 250
\qquad\textbf{(C)}\ 251
\qquad\textbf{(D)}\ 252
\qquad\textbf{(E)}\ 253
$
2020 AMC 10, 5
How many distinguishable arrangements are there of $1$ brown tile, $1$ purple tile, $2$ green tiles, and $3$ yellow tiles in a row from left to right? (Tiles of the same color are indistinguishable)
$\textbf{(A)}\ 210\qquad\textbf{(B)}\ 420\qquad\textbf{(C)}\ 630\qquad\textbf{(D)}\ 840\qquad\textbf{(E)}\ 1050$
2014 AMC 10, 25
In a small pond there are eleven lily pads in a row labeled $0$ through $10$. A frog is sitting on pad $1$. When the frog is on pad $N$, $0<N<10$, it will jump to pad $N-1$ with probability $\frac{N}{10}$ and to pad $N+1$ with probability $1-\frac{N}{10}$. Each jump is independent of the previous jumps. If the frog reaches pad $0$ it will be eaten by a patiently waiting snake. If the frog reaches pad $10$ it will exit the pond, never to return. What is the probability that the frog will escape being eaten by the snake?
$ \textbf {(A) } \frac{32}{79} \qquad \textbf {(B) } \frac{161}{384} \qquad \textbf {(C) } \frac{63}{146} \qquad \textbf {(D) } \frac{7}{16} \qquad \textbf {(E) } \frac{1}{2} $
2013 AMC 12/AHSME, 3
A flower bouquet contains pink roses, red roses, pink carnations, and red carnations. One third of the pink flowers are roses, three fourths of the red flowers are carnations, and six tenths of the flowers are pink. What percent of the flowers are carnations?
$ \textbf{(A)}\ 15\qquad\textbf{(B)}\ 30\qquad\textbf{(C)}\ 40\qquad\textbf{(D)}\ 60\qquad\textbf{(E)}\ 70 $
2023 AMC 10, 19
The line segment formed by $A(1, 2)$ and $B(3, 3)$ is rotated to the line segment formed by $A'(3, 1)$ and $B'(4, 3)$ about the point $P(r, s)$. What is $|r-s|$?
$\text{A) } \frac{1}{4} \qquad \text{B) } \frac{1}{2} \qquad \text{C) } \frac{3}{4} \qquad \text{D) } \frac{2}{3} \qquad \text{E) } 1$
2010 AMC 12/AHSME, 23
The number obtained from the last two nonzero digits of $ 90!$ is equal to $ n$. What is $ n$?
$ \textbf{(A)}\ 12 \qquad
\textbf{(B)}\ 32 \qquad
\textbf{(C)}\ 48 \qquad
\textbf{(D)}\ 52 \qquad
\textbf{(E)}\ 68$
2018 AMC 10, 2
Sam drove $96$ miles in $90$ minutes. His average speed during the first $30$ minutes was $60$ mph (miles per hour), and his average speed during the second $30$ minutes was $65$ mph. What was his average speed, in mph, during the last $30$ minutes?
$\textbf{(A) } 64 \qquad \textbf{(B) } 65 \qquad \textbf{(C) } 66 \qquad \textbf{(D) } 67 \qquad \textbf{(E) } 68$
2023 AMC 10, 6
Let $L_1=1$, $L_2=3$, and $L_{n+2}=L_{n+1}+L_n$ for $n\geq1$. How many terms in the sequence $L_1, L_2, L_3, \dots, L_{2023}$ are even?
$\textbf{(A) }673\qquad\textbf{(B) }1011\qquad\textbf{(C) }675\qquad\textbf{(D) }1010\qquad\textbf{(E) }674$
2003 AMC 10, 1
What is the difference between the sum of the first $ 2003$ even counting numbers and the sum of the first $ 2003$ odd counting numbers?
$ \textbf{(A)}\ 0 \qquad
\textbf{(B)}\ 1 \qquad
\textbf{(C)}\ 2 \qquad
\textbf{(D)}\ 2003 \qquad
\textbf{(E)}\ 4006$
2016 AMC 10, 19
In rectangle $ABCD$, $AB=6$ and $BC=3$. Point $E$ between $B$ and $C$, and point $F$ between $E$ and $C$ are such that $BE=EF=FC$. Segments $\overline{AE}$ and $\overline{AF}$ intersect $\overline{BD}$ at $P$ and $Q$, respectively. The ratio $BP:PQ:QD$ can be written as $r:s:t$, where the greatest common factor of $r,s$ and $t$ is $1$. What is $r+s+t$?
$\textbf{(A) } 7
\qquad \textbf{(B) } 9
\qquad \textbf{(C) } 12
\qquad \textbf{(D) } 15
\qquad \textbf{(E) } 20$
2020 AMC 10, 24
How many positive integers $n$ satisfy$$\dfrac{n+1000}{70} = \lfloor \sqrt{n} \rfloor?$$(Recall that $\lfloor x\rfloor$ is the greatest integer not exceeding $x$.)
$\textbf{(A) } 2 \qquad\textbf{(B) } 4 \qquad\textbf{(C) } 6 \qquad\textbf{(D) } 30 \qquad\textbf{(E) } 32$
2013 AMC 8, 25
A ball with diameter 4 inches starts at point A to roll along the track shown. The track is comprised of 3 semicircular arcs whose radii are $R_1 = 100$ inches, $R_2 = 60$ inches, and $R_3 = 80$ inches, respectively. The ball always remains in contact with the track and does not slip. What is the distance the center of the ball travels over the course from A to B?
[asy]
size(8cm);
draw((0,0)--(480,0),linetype("3 4"));
filldraw(circle((8,0),8),black);
draw((0,0)..(100,-100)..(200,0));
draw((200,0)..(260,60)..(320,0));
draw((320,0)..(400,-80)..(480,0));
draw((100,0)--(150,-50sqrt(3)),Arrow(size=4));
draw((260,0)--(290,30sqrt(3)),Arrow(size=4));
draw((400,0)--(440,-40sqrt(3)),Arrow(size=4));
label("$R_1$",(100,0)--(150,-50sqrt(3)), W, fontsize(10));
label("$R_2$",(260,0)--(290,30sqrt(3)), W, fontsize(10));
label("$R_3$",(400,0)--(440,-40sqrt(3)), W, fontsize(10));
filldraw(circle((8,0),8),black);
label("$A$",(0,0),W,fontsize(10));[/asy]
$\textbf{(A)}\ 238\pi \qquad \textbf{(B)}\ 240\pi \qquad \textbf{(C)}\ 260\pi \qquad \textbf{(D)}\ 280\pi \qquad \textbf{(E)}\ 500\pi$
2024 AMC 10, 19
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 $b?$
$\textbf{(A) } 9 \qquad \textbf{(B) } 12 \qquad \textbf{(C) } 16 \qquad \textbf{(D) } 18 \qquad \textbf{(E) } 21$
2020 AMC 10, 10
Seven cubes, whose volumes are $1$, $8$, $27$, $64$, $125$, $216$, and $343$ cubic units, are stacked vertically to form a tower in which the volumes of the cubes decrease from bottom to top. Except for the bottom cube, the bottom face of each cube lies completely on top of the cube below it. What is the total surface area of the tower (including the bottom) in square units?
$\textbf{(A) } 644 \qquad \textbf{(B) } 658 \qquad \textbf{(C) } 664 \qquad \textbf{(D) } 720 \qquad \textbf{(E) } 749$
2016 AMC 10, 11
Carl decided to fence in his rectangular garden. He bought $20$ fence posts, placed one on each of the four corners, and spaced out the rest evenly along the edges of the garden, leaving exactly $4$ yards between neighboring posts. The longer side of his garden, including the corners, has twice as many posts as the shorter side, including the corners. What is the area, in square yards, of Carl’s garden?
$\textbf{(A)}\ 256\qquad\textbf{(B)}\ 336\qquad\textbf{(C)}\ 384\qquad\textbf{(D)}\ 448\qquad\textbf{(E)}\ 512$
2021 AMC 10 Spring, 2
What is the value of $\sqrt{(3-2\sqrt{3})^2}+\sqrt{(3+2\sqrt{3})^2}$?
$\textbf{(A)}\ 0 \qquad \textbf{(B)}\ 4\sqrt{3}-6 \qquad \textbf{(C)}\ 6 \qquad \textbf{(D)}\ 4\sqrt{3} \qquad \textbf{(E)}\ 4\sqrt{3}+6$
2023 AMC 10, 25
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 $d(Q, R) > d(R, S)$?
$\textbf{(A)}~\frac{7}{22}\qquad\textbf{(B)}~\frac13\qquad\textbf{(C)}~\frac38\qquad\textbf{(D)}~\frac5{12}\qquad\textbf{(E)}~\frac12$
2023 AMC 10, 5
How many digits are in the base-ten representation of $8^5 \cdot 5^{10} \cdot 15^5$?
$\textbf{(A)}~14\qquad\textbf{(B)}~15\qquad\textbf{(C)}~16\qquad\textbf{(D)}~17\qquad\textbf{(E)}~18\qquad$
2024 AMC 10, 25
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] $\textbf{(A) }130\qquad\textbf{(B) }144\qquad\textbf{(C) }146\qquad\textbf{(D) }162\qquad\textbf{(E) }196$
2024 AMC 10, 11
How many ordered pairs of integers $(m, n)$ satisfy $\sqrt{n^2 - 49} = m$?
$
\textbf{(A) }1 \qquad
\textbf{(B) }2 \qquad
\textbf{(C) }3 \qquad
\textbf{(D) }4 \qquad
\textbf{(E) } \text{Infinitely many} \qquad
$
2024 AMC 10, 25
Each of $27$ bricks (right rectangular prisms) has dimensions $a \times b \times c$, where $a$, $b$, and $c$ are pairwise relatively prime positive integers. These bricks are arranged to form a $3 \times 3 \times 3$ block, as shown on the left below. A $28$[sup]th[/sup] brick with the same dimensions is introduced, and these bricks are reconfigured into a $2 \times 2 \times 7$ block, shown on the right. The new block is $1$ unit taller, $1$ unit wider, and $1$ unit deeper than the old one. What is $a + b + c$?
[img]https://cdn.artofproblemsolving.com/attachments/2/d/b18d3d0a9e5005c889b34e79c6dab3aaefeffd.png[/img]
$
\textbf{(A) }88 \qquad
\textbf{(B) }89 \qquad
\textbf{(C) }90 \qquad
\textbf{(D) }91 \qquad
\textbf{(E) }92 \qquad
$