Found problems: 260
2014 AMC 8, 10
The first AMC $8$ was given in $1985$ and it has been given annually since that time. Samantha turned $12$ years old the year that she took the seventh AMC $8$. In what year was Samantha born?
$\textbf{(A) }1979\qquad\textbf{(B) }1980\qquad\textbf{(C) }1981\qquad\textbf{(D) }1982\qquad \textbf{(E) }1983$
2019 AMC 8, 24
In triangle $ABC$, point $D$ divides side $\overline{AC}$ so that $AD:DC=1:2$. Let $E$ be the midpoint of $\overline{BD}$ and let $F$ be the point of intersection of line $BC$ and line $AE$. Given that the area of $\triangle ABC$ is $360$, what is the area of $\triangle EBF$?
[asy]
unitsize(1.5cm);
pair A,B,C,DD,EE,FF;
B = (0,0); C = (3,0);
A = (1.2,1.7);
DD = (2/3)*A+(1/3)*C;
EE = (B+DD)/2;
FF = intersectionpoint(B--C,A--A+2*(EE-A));
draw(A--B--C--cycle);
draw(A--FF);
draw(B--DD);dot(A);
label("$A$",A,N);
dot(B);
label("$B$",
B,SW);dot(C);
label("$C$",C,SE);
dot(DD);
label("$D$",DD,NE);
dot(EE);
label("$E$",EE,NW);
dot(FF);
label("$F$",FF,S);
[/asy]
$\textbf{(A) }24\qquad\textbf{(B) }30\qquad\textbf{(C) }32\qquad\textbf{(D) }36\qquad\textbf{(E) }40$
2017 AMC 8, 22
In the right triangle $ABC$, $AC=12$, $BC=5$, and angle $C$ is a right angle. A semicircle is inscribed in the triangle as shown. What is the radius of the semicircle?
[asy] draw((0,0)--(12,0)--(12,5)--(0,0)); draw(arc((8.67,0),(12,0),(5.33,0))); label("$A$", (0,0), W); label("$C$", (12,0), E); label("$B$", (12,5), NE); label("$12$", (6, 0), S); label("$5$", (12, 2.5), E);[/asy]
$\textbf{(A) }\frac{7}{6}\qquad\textbf{(B) }\frac{13}{5}\qquad\textbf{(C) }\frac{59}{18}\qquad\textbf{(D) }\frac{10}{3}\qquad\textbf{(E) }\frac{60}{13}$
2025 AMC 8, 11
A [i]tetromino[/i] consists of four squares connected along their edges. There are five possible tetromino shapes, I, O, L, T, S, shown below, which can be rotated or flipped over. Three tetrominos are used to completely cover a $3\times 4$ rectangle. At least one of the titles is an S tile. What are the other two tiles?
[img]https://i.imgur.com/9Nxq4y6.png[/img]
$\textbf{(A) } \text{I and L} \qquad\textbf{(B) }\text{I and T} \qquad\textbf{(C) }\text{L and L}\qquad\textbf{(D) }\text{L and S} \qquad\textbf{(E) }\text{O and T}$\\
1994 AMC 8, 12
Each of the three large squares shown below is the same size. Segments that intersect the sides of the squares intersect at the midpoints of the sides. How do the shaded areas of these squares compare?
[asy]
unitsize(36);
fill((0,0)--(1,0)--(1,1)--cycle,gray);
fill((1,1)--(1,2)--(2,2)--cycle,gray);
draw((0,0)--(2,0)--(2,2)--(0,2)--cycle);
draw((1,0)--(1,2));
draw((0,0)--(2,2));
fill((3,1)--(4,1)--(4,2)--(3,2)--cycle,gray);
draw((3,0)--(5,0)--(5,2)--(3,2)--cycle);
draw((4,0)--(4,2));
draw((3,1)--(5,1));
fill((6,1)--(6.5,0.5)--(7,1)--(7.5,0.5)--(8,1)--(7.5,1.5)--(7,1)--(6.5,1.5)--cycle,gray);
draw((6,0)--(8,0)--(8,2)--(6,2)--cycle);
draw((6,0)--(8,2));
draw((6,2)--(8,0));
draw((7,0)--(6,1)--(7,2)--(8,1)--cycle);
label("$I$",(1,2),N);
label("$II$",(4,2),N);
label("$III$",(7,2),N); [/asy]
$\text{(A)}\ \text{The shaded areas in all three are equal.}$
$\text{(B)}\ \text{Only the shaded areas of }I\text{ and }II\text{ are equal.}$
$\text{(C)}\ \text{Only the shaded areas of }I\text{ and }III\text{ are equal.}$
$\text{(D)}\ \text{Only the shaded areas of }II\text{ and }III\text{ are equal.}$
$\text{(E)}\ \text{The shaded areas of }I, II\text{ and }III\text{ are all different.}$
2015 AMC 8, 18
An arithmetic sequence is a sequence in which each term after the first is obtained by adding a constant to the previous term. For example, $2,5,8,11,14$ is an arithmetic sequence with five terms, in which the first term is $2$ and the constant added is $3$. Each row and each column in this $5\times5$ array is an arithmetic sequence with five terms. What is the value of $X$?
$\textbf{(A) }21\qquad\textbf{(B) }31\qquad\textbf{(C) }36\qquad\textbf{(D) }40\qquad \textbf{(E) }42$
[asy]
size(3.85cm);
label("$X$",(2.5,2.1),N);
for (int i=0; i<=5; ++i)
draw((i,0)--(i,5), linewidth(.5));
for (int j=0; j<=5; ++j)
draw((0,j)--(5,j), linewidth(.5));
void draw_num(pair ll_corner, int num)
{
label(string(num), ll_corner + (0.5, 0.5), p = fontsize(19pt));
}
draw_num((0,0), 17);
draw_num((4, 0), 81);
draw_num((0, 4), 1);
draw_num((4,4), 25);
void foo(int x, int y, string n)
{
label(n, (x+0.5,y+0.5), p = fontsize(19pt));
}
foo(2, 4, " ");
foo(3, 4, " ");
foo(0, 3, " ");
foo(2, 3, " ");
foo(1, 2, " ");
foo(3, 2, " ");
foo(1, 1, " ");
foo(2, 1, " ");
foo(3, 1, " ");
foo(4, 1, " ");
foo(2, 0, " ");
foo(3, 0, " ");
foo(0, 1, " ");
foo(0, 2, " ");
foo(1, 0, " ");
foo(1, 3, " ");
foo(1, 4, " ");
foo(3, 3, " ");
foo(4, 2, " ");
foo(4, 3, " ");
[/asy]
1972 AMC 12/AHSME, 9
Ann and Sue bought identical boxes of stationery. Ann used hers to write 1-sheet letters and Sue used hers to write 3-sheet letters. Ann used all the envelopes and had 50 sheets of paper left, while Sue used all of the sheets of paper and had 50 envelopes left. The number of sheets of paper in each box was
\[ \begin{array}{rlrlrlrlrlrl} \hbox {(A)}& 150 \qquad & \hbox {(B)}& 125 \qquad & \hbox {(C)}& 120 \qquad & \hbox {(D)}& 100 \qquad & \hbox {(E)}& 80 & \end{array} \]
2015 AMC 8, 17
Jeremy's father drives him to school in rush hour traffic in 20 minutes. One day there is no traffic, so his father can drive him 18 miles per hour faster and gets him to school in 12 minutes. How far in miles is it to school?
$
\textbf{(A) } 4 \qquad
\textbf{(B) } 6 \qquad
\textbf{(C) } 8 \qquad
\textbf{(D) } 9 \qquad
\textbf{(E) } 12
$
2016 AMC 8, 23
Two congruent circles centered at points $A$ and $B$ each pass through the other circle's center. The line containing both $A$ and $B$ is extended to intersect the circles at points $C$ and $D$. The circles intersect at two points, one of which is $E$. What is the degree measure of $\angle CED$?
$\textbf{(A) }90\qquad\textbf{(B) }105\qquad\textbf{(C) }120\qquad\textbf{(D) }135\qquad \textbf{(E) }150$
2024 AMC 8 -, 8
On Monday Taye has \$2. Everyday he either gains \$3 or doubles the amount of money he had on the previous day. How many different dollar amounts could Taye have on Thursday, 3 days later?
$\textbf{(A) } 3\qquad\textbf{(B) } 4\qquad\textbf{(C) } 5\qquad\textbf{(D) } 6\qquad\textbf{(E) } 7$
2022 AMC 8 -, 18
The midpoints of the four sides of a rectangle are $(-3, 0), (2, 0), (5, 4)$ and $(0, 4)$. What is the area of the rectangle?
$\textbf{(A)} ~20\qquad\textbf{(B)} ~25\qquad\textbf{(C)} ~40\qquad\textbf{(D)} ~50\qquad\textbf{(E)} ~80\qquad$
2024 AMC 8 -, 4
When Yunji added all the integers from $1$ to $9$, she mistakenly left out a number. Her incorrect sum turned out to be a square number. Which number did Yunji leave out?
$\textbf{(A) } 5\qquad\textbf{(B) } 6\qquad\textbf{(C) } 7\qquad\textbf{(D) } 8\qquad\textbf{(E) } 9$
2015 AMC 8, 1
How many square yards of carpet are required to cover a rectangular floor that is $12$ feet long and $9$ feet wide? (There are 3 feet in a yard.)
$\textbf{(A) }12\qquad\textbf{(B) }36\qquad\textbf{(C) }108\qquad\textbf{(D) }324\qquad \textbf{(E) }972$
2023 AMC 8, 1
What is the value of $(8 \times 4 + 2) - (8 + 4 \times 2)?$
$\textbf{(A)}~0\qquad\textbf{(B)}~6\qquad\textbf{(C)}~10\qquad\textbf{(D)}~18\qquad\textbf{(E)}~24$
2024 AMC 8 -, 19
Jordan owns 15 pairs of sneakers. Three fifths of the pairs are red and the rest are white. Two thirds of the pairs are high-top and the rest are low-top. The red high-top sneakers make up a fraction of the collection. What is the least possible value of this fraction?
[img]https://wiki-images.artofproblemsolving.com//thumb/a/a2/2024_AMC_8_-19.png/1200px-2024_AMC_8_-19.png[/img]
$\textbf{(A) } 0\qquad\textbf{(B) } \dfrac{1}{5} \qquad\textbf{(C) } \dfrac{4}{15} \qquad\textbf{(D) } \dfrac{1}{3} \qquad\textbf{(E) } \dfrac{2}{5}$
2017 AMC 8, 19
For any positive integer $M$, the notation $M!$ denotes the product of the integers $1$ through $M$. What is the largest integer $n$ for which $5^n$ is a factor of the sum $98!+99!+100!$ ?
$\textbf{(A) }23\qquad\textbf{(B) }24\qquad\textbf{(C) }25\qquad\textbf{(D) }26\qquad\textbf{(E) }27$
2017 AMC 8, 12
The smallest positive integer greater than 1 that leaves a remainder of 1 when divided by 4, 5, and 6 lies between which of the following pairs of numbers?
$\textbf{(A) }2\text{ and }19\qquad\textbf{(B) }20\text{ and }39\qquad\textbf{(C) }40\text{ and }59\qquad\textbf{(D) }60\text{ and }79\qquad\textbf{(E) }80\text{ and }124$
2010 Contests, 2
If $a @ b = \frac{a\times b}{a+b}$, for $a,b$ positive integers, then what is $5 @10$?
$\textbf{(A)}\ \frac{3}{10} \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ \frac{10}{3} \qquad\textbf{(E)}\ 50$
2002 AMC 8, 22
Six cubes, each an inch on an edge, are fastened together, as shown. Find the total surface area in square inches. Include the top, bottom and sides.
[asy]/* AMC8 2002 #22 Problem */
draw((0,0)--(0,1)--(1,1)--(1,0)--cycle);
draw((0,1)--(0.5,1.5)--(1.5,1.5)--(1,1));
draw((1,0)--(1.5,0.5)--(1.5,1.5));
draw((0.5,1.5)--(1,2)--(1.5,2));
draw((1.5,1.5)--(1.5,3.5)--(2,4)--(3,4)--(2.5,3.5)--(2.5,0.5)--(1.5,.5));
draw((1.5,3.5)--(2.5,3.5));
draw((1.5,1.5)--(3.5,1.5)--(3.5,2.5)--(1.5,2.5));
draw((3,4)--(3,3)--(2.5,2.5));
draw((3,3)--(4,3)--(4,2)--(3.5,1.5));
draw((4,3)--(3.5,2.5));
draw((2.5,.5)--(3,1)--(3,1.5));[/asy]
$ \textbf{(A)}\ 18\qquad\textbf{(B)}\ 24\qquad\textbf{(C)}\ 26\qquad\textbf{(D)}\ 30\qquad\textbf{(E)}\ 36$
2016 AMC 8, 7
Which of the following numbers is [b]not[/b] a perfect square?
$\textbf{(A) }1^{2016}\qquad\textbf{(B) }2^{2017}\qquad\textbf{(C) }3^{2018}\qquad\textbf{(D) }4^{2019}\qquad \textbf{(E) }5^{2020}$
2020 AMC 8 -, 19
A number is called [i]flippy[/i] if its digits alternate between two distinct digits. For example, $2020$ and $37373$ are flippy, but $3883$ and $123123$ are not. How many five-digit flippy numbers are divisible by $15$?
$\textbf{(A)}\ 3 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 6 \qquad \textbf{(E)}\ 8$
2018 AMC 8, 25
How many perfect cubes lie between $2^8+1$ and $2^{18}+1$, inclusive?
$\textbf{(A) }4\qquad\textbf{(B) }9\qquad\textbf{(C) }10\qquad\textbf{(D) }57\qquad \textbf{(E) }58$
2022 AMC 8 -, 8
What is the value of \[\displaystyle\frac{1}{3}\cdot\displaystyle\frac{2}{4}\cdot\displaystyle\frac{3}{5}\cdots\displaystyle\frac{18}{20}\cdot\displaystyle\frac{19}{21}\cdot\displaystyle\frac{20}{22}?\]
$\textbf{(A)} ~\displaystyle\frac{1}{462}\qquad\textbf{(B)} ~\displaystyle\frac{1}{231}\qquad\textbf{(C)} ~\displaystyle\frac{1}{132}\qquad\textbf{(D)} ~\displaystyle\frac{2}{213}\qquad\textbf{(E)} ~\displaystyle\frac{1}{22}\qquad$
2015 AMC 8, 16
In a middle-school mentoring program, a number of the sixth graders are paired with a ninth-grade student as a buddy. No ninth grader is assigned more than one sixth-grade buddy. If $\tfrac{1}{3}$ of all the ninth graders are paired with $\tfrac{2}{5}$ of all the sixth graders, what fraction of the total number of sixth and ninth graders have a buddy?
$
\textbf{(A) } \frac{2}{15} \qquad
\textbf{(B) } \frac{4}{11} \qquad
\textbf{(C) } \frac{11}{30} \qquad
\textbf{(D) } \frac{3}{8} \qquad
\textbf{(E) } \frac{11}{15}
$
2003 AMC 8, 21
The area of trapezoid $ ABCD$ is $ 164 \text{cm}^2$. The altitude is $ 8 \text{cm}$, $ AB$ is $ 10 \text{cm}$, and $ CD$ is $ 17 \text{cm}$. What is $ BC$, in centimeters?
[asy]/* AMC8 2003 #21 Problem */
size(4inch,2inch);
draw((0,0)--(31,0)--(16,8)--(6,8)--cycle);
draw((11,8)--(11,0), linetype("8 4"));
draw((11,1)--(12,1)--(12,0));
label("$A$", (0,0), SW);
label("$D$", (31,0), SE);
label("$B$", (6,8), NW);
label("$C$", (16,8), NE);
label("10", (3,5), W);
label("8", (11,4), E);
label("17", (22.5,5), E);[/asy]
$ \textbf{(A)}\ 9\qquad\textbf{(B)}\ 10\qquad\textbf{(C)}\ 12\qquad\textbf{(D)}\ 15\qquad\textbf{(E)}\ 20$