Found problems: 38
2017 AMC 10, 23
How many triangles with positive area have all their vertices at points $(i,j)$ in the coordinate plane, where $i$ and $j$ are integers between $1$ and $5$, inclusive?
$\textbf{(A)}\ 2128 \qquad\textbf{(B)}\ 2148 \qquad\textbf{(C)}\ 2160 \qquad\textbf{(D)}\ 2200 \qquad\textbf{(E)}\ 2300$
2017 AMC 10, 11
The region consisting of all points in three-dimensional space within $3$ units of line segment $\overline{AB}$ has volume $216\pi$. What is the length $AB$?
$\textbf{(A)}\ 6\qquad\textbf{(B)}\ 12\qquad\textbf{(C)}\ 18\qquad\textbf{(D)}\ 20\qquad\textbf{(E)}\ 24$
2017 AMC 12/AHSME, 19
A square with side length $x$ is inscribed in a right triangle with sides of length $3$, $4$, and $5$ so that one vertex of the square coincides with the right-angle vertex of the triangle. A square with side length $y$ is inscribed so that one side of the square lies on the hypotenuse of the triangle. What is $\frac{x}{y}$?
$\textbf{(A)}\ \frac{12}{13}\qquad\textbf{(B)}\ \frac{35}{37}\qquad\textbf{(C)}\ 1\qquad\textbf{(D)}\ \frac{37}{35}\qquad\textbf{(E)}\ \frac{13}{12}$
2017 AMC 12/AHSME, 9
Let $S$ be the set of points $(x,y)$ in the coordinate plane such that two of the three quantities $3$, $x+2$, and $y-4$ are equal and the third of the three quantities is no greater than this common value. Which of the following is a correct description of $S$?
$\textbf{(A) } \text{a single point} \qquad \textbf{(B) } \text{two intersecting lines} \\ \\ \textbf{(C) } \text{three lines whose pairwise intersections are three distinct points} \\ \\ \textbf{(D) } \text{a triangle} \qquad \textbf{(E) } \text{three rays with a common endpoint}$
2017 AMC 10, 2
Pablo buys popsicles for his friends. The store sells single popsicles for $\$1$ each, 3-popsicle boxes for $\$2$, and 5-popsicle boxes for $\$3$. What is the greatest number of popsicles that Pablo can buy with $\$8$?
$\textbf{(A)}\ 8\qquad\textbf{(B)}\ 11\qquad\textbf{(C)}\ 12\qquad\textbf{(D)}\ 13\qquad\textbf{(E)}\ 15$
2017 AMC 10, 7
Jerry and Silvia wanted to go from the southwest corner of a square field to the northeast corner. Jerry walked due east and then due north to reach the goal, but Silvia headed northeast and reached the goal walking in a straight line. Which of the following is closest to how much shorter Silvia's trip was, compared to Jerry's trip?
$\textbf{(A)}\ 30 \%\qquad\textbf{(B)}\ 40 \%\qquad\textbf{(C)}\ 50 \%\qquad\textbf{(D)}\ 60 \%\qquad\textbf{(E)}\ 70 \%$
2017 AMC 10, 20
Let $S(n)$ equal the sum of the digits of positive integer $n$. For example, $S(1507) = 13$. For a particular positive integer $n$, $S(n) = 1274$. Which of the following could be the value of $S(n+1)$?
$\textbf{(A)}\ 1 \qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 12\qquad\textbf{(D)}\ 1239\qquad\textbf{(E)}\ 1265$
2017 AMC 10, 13
Define a sequence recursively by $F_0 = 0$, $F_1 = 1$, and $F_n = $ the remainder when $F_{n-1} + F_{n-2}$ is divided by $3$, for all $n \ge 2$. Thus the sequence starts $0,1,1,2,0,2 \ldots$. What is $F_{2017} + F_{2018} + F_{2019} + F_{2020} + F_{2021} + F_{2022} + F_{2023} + F_{2024}$?
$\textbf{(A)}\ 6\qquad\textbf{(B)}\ 7\qquad\textbf{(C)}\ 8\qquad\textbf{(D)}\ 9\qquad\textbf{(E)}\ 10$
2017 AMC 10, 25
How many integers between $100$ and $999$, inclusive, have the property that some permutation of its digits is a multiple of $11$ between $100$ and $999$? For example, both $121$ and $211$ have this property.
$ \textbf{(A) }226\qquad \textbf{(B) } 243 \qquad \textbf{(C) } 270 \qquad \textbf{(D) }469\qquad \textbf{(E) } 486$
2017 AMC 10, 3
Tamara has three rows of two 6-feet by 2-feet flower beds in her garden. The beds are separated and also surrounded by 1-foot-wide walkways, as shown on the diagram. What is the total area of the walkways, in square feet?
[asy]
unitsize(0.7cm);
path p1 = (0,0)--(15,0)--(15,10)--(0,10)--cycle;
fill(p1,lightgray);
draw(p1);
for (int i = 1; i <= 8; i += 7) {
for (int j = 1; j <= 7; j += 3 ) {
path p2 = (i,j)--(i+6,j)--(i+6,j+2)--(i,j+2)--cycle;
draw(p2);
fill(p2,white);
}
}
draw((0,8)--(1,8),Arrows);
label("1",(0.5,8),S);
draw((7,8)--(8,8),Arrows);
label("1",(7.5,8),S);
draw((14,8)--(15,8),Arrows);
label("1",(14.5,8),S);
draw((11,0)--(11,1),Arrows);
label("1",(11,0.5),W);
draw((11,3)--(11,4),Arrows);
label("1",(11,3.5),W);
draw((11,6)--(11,7),Arrows);
label("1",(11,6.5),W);
draw((11,9)--(11,10),Arrows);
label("1",(11,9.5),W);
label("6",(4,1),N);
label("2",(1,2),E);
[/asy]
$\textbf{(A) }72 \qquad
\textbf{(B) }78 \qquad
\textbf{(C) }90 \qquad
\textbf{(D) }120 \qquad
\textbf{(E) }150 $
2017 AMC 12/AHSME, 4
Jerry and Silvia wanted to go from the southwest corner of a square field to the northeast corner. Jerry walked due east and then due north to reach the goal, but Silvia headed northeast and reached the goal walking in a straight line. Which of the following is closest to how much shorter Silvia's trip was, compared to Jerry's trip?
$\textbf{(A)}\ 30 \%\qquad\textbf{(B)}\ 40 \%\qquad\textbf{(C)}\ 50 \%\qquad\textbf{(D)}\ 60 \%\qquad\textbf{(E)}\ 70 \%$
2017 AMC 12/AHSME, 1
Pablo buys popsicles for his friends. The store sells single popsicles for $\$1$ each, 3-popsicle boxes for $\$2$, and 5-popsicle boxes for $\$3$. What is the greatest number of popsicles that Pablo can buy with $\$8$?
$\textbf{(A)}\ 8\qquad\textbf{(B)}\ 11\qquad\textbf{(C)}\ 12\qquad\textbf{(D)}\ 13\qquad\textbf{(E)}\ 15$
2017 AMC 10, 10
The lines with equations $ax-2y=c$ and $2x+by=-c$ are perpendicular and intersect at $(1, -5)$. What is $c$?
$\textbf{(A) } -13\qquad \textbf{(B) } -8\qquad \textbf{(C) } 2\qquad \textbf{(D) } 8\qquad \textbf{(E) } 13$
2017 AMC 10, 14
Every week Roger pays for a movie ticket and a soda out of his allowance. Last week, Roger's allowance was $A$ dollars. The cost of his movie ticket was $20\%$ of the difference between $A$ and the cost of his soda, while the cost of his soda was $5\%$ of the difference between $A$ and the cost of his movie ticket. To the nearest whole percent, what fraction of $A$ did Roger pay for his movie ticket and soda?
$ \textbf{(A) }9\%\qquad \textbf{(B) } 19\%\qquad \textbf{(C) } 22\%\qquad \textbf{(D) } 23\%\qquad \textbf{(E) }25\%$
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, 4
Mia is “helping” her mom pick up $30$ toys that are strewn on the floor. Mia’s mom manages to put $3$ toys into the toy box every $30$ seconds, but each time immediately after those $30$ seconds have elapsed, Mia takes $2$ toys out of the box. How much time, in minutes, will it take Mia and her mom to put all $30$ toys into the box for the first time?
$\textbf{(A)}\ 13.5\qquad\textbf{(B)}\ 14\qquad\textbf{(C)}\ 14.5\qquad\textbf{(D)}\ 15\qquad\textbf{(E)}\ 15.5$
2017 AMC 10, 1
What is the value of $2(2(2(2(2(2+1)+1)+1)+1)+1)+1$?
$\textbf{(A) } 70 \qquad
\textbf{(B) } 97 \qquad
\textbf{(C) } 127 \qquad
\textbf{(D) } 159 \qquad
\textbf{(E) } 729 $
2017 AMC 12/AHSME, 8
The region consisting of all points in three-dimensional space within $3$ units of line segment $\overline{AB}$ has volume $216\pi$. What is the length $AB$?
$\textbf{(A)}\ 6\qquad\textbf{(B)}\ 12\qquad\textbf{(C)}\ 18\qquad\textbf{(D)}\ 20\qquad\textbf{(E)}\ 24$
2017 AMC 12/AHSME, 2
The sum of two nonzero real numbers is $4$ times their product. What is the sum of the reciprocals of the two numbers?
$\textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}\ 8\qquad\textbf{(E)}\ 12$
2017 AMC 10, 24
For certain real numbers $a$, $b$, and $c$, the polynomial \[g(x) = x^3 + ax^2 + x + 10\] has three distinct roots, and each root of $g(x)$ is also a root of the polynomial \[f(x) = x^4 + x^3 + bx^2 + 100x + c.\] What is $f(1)$?
$\textbf{(A)}\ -9009 \qquad\textbf{(B)}\ -8008 \qquad\textbf{(C)}\ -7007 \qquad\textbf{(D)}\ -6006 \qquad\textbf{(E)}\ -5005$
2017 AMC 10, 12
Let $S$ be the set of points $(x,y)$ in the coordinate plane such that two of the three quantities $3$, $x+2$, and $y-4$ are equal and the third of the three quantities is no greater than this common value. Which of the following is a correct description of $S$?
$\textbf{(A) } \text{a single point} \qquad \textbf{(B) } \text{two intersecting lines} \\ \\ \textbf{(C) } \text{three lines whose pairwise intersections are three distinct points} \\ \\ \textbf{(D) } \text{a triangle} \qquad \textbf{(E) } \text{three rays with a common endpoint}$
2017 AMC 10, 18
Amelia has a coin that lands heads with probability $\frac{1}{3}$, and Blaine has a coin that lands on heads with probability $\frac{2}{5}$. Amelia and Blaine alternately toss their coins until someone gets a head; the first one to get a head wins. All coin tosses are independent. Amelia goes first. The probability that Amelia wins is $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. What is $q-p$?
$\textbf{(A) }1\qquad\textbf{(B) }2\qquad\textbf{(C) }3\qquad\textbf{(D) }4\qquad\textbf{(E) }5$
2017 AMC 10, 21
A square with side length $x$ is inscribed in a right triangle with sides of length $3$, $4$, and $5$ so that one vertex of the square coincides with the right-angle vertex of the triangle. A square with side length $y$ is inscribed so that one side of the square lies on the hypotenuse of the triangle. What is $\frac{x}{y}$?
$\textbf{(A)}\ \frac{12}{13}\qquad\textbf{(B)}\ \frac{35}{37}\qquad\textbf{(C)}\ 1\qquad\textbf{(D)}\ \frac{37}{35}\qquad\textbf{(E)}\ \frac{13}{12}$
2017 AMC 10, 6
Ms. Carroll promised that anyone who got all the multiple choice questions right on the upcoming exam would receive an A on the exam. Which of these statements necessarily follows logically?
$\textbf{(A)}$ If Lewis did not receive an A, then he got all of the multiple choice questions wrong. \\
$\textbf{(B)}$ If Lewis did not receive an A, then he got at least one of the multiple choice questions wrong. \\
$\textbf{(C)}$ If Lewis got at least one of the multiple choice questions wrong, then he did not receive an A. \\
$\textbf{(D)}$ If Lewis received an A, then he got all of the multiple choice questions right. \\
$\textbf{(E)}$ If Lewis received an A, then he got at least one of the multiple choice questions right.
2017 AMC 10, 17
Distinct points $P$, $Q$, $R$, $S$ lie on the circle $x^2+y^2=25$ and have integer coordinates. The distances $PQ$ and $RS$ are irrational numbers. What is the greatest possible value of the ratio $\frac{PQ}{RS }$?
$\textbf{(A)}\ 3\qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\ 3\sqrt{5}\qquad\textbf{(D)}\ 7\qquad\textbf{(E)}\ 5\sqrt{2}$