Found problems: 13
2018 AMC 10, 1
What is the value of \[\bigg(\Big((2+1)^{-1}+1\Big)^{-1}+1\bigg)^{-1}+1?\]
$\textbf{(A) } \frac{5}{8} \qquad\textbf{(B) } \frac{11}{7} \qquad\textbf{(C) } \frac{8}{5} \qquad\textbf{(D) } \frac{18}{11} \qquad\textbf{(E) } \frac{15}{8}$
2018 AMC 10, 24
Triangle $ABC$ with $AB=50$ and $AC=10$ has area $120$. Let $D$ be the midpoint of $\overline{AB}$, and let $E$ be the midpoint of $\overline{AC}$. The angle bisector of $\angle BAC$ intersects $\overline{DE}$ and $\overline{BC}$ at $F$ and $G$, respectively. What is the area of quadrilateral $FDBG$?
$
\textbf{(A) }60 \qquad
\textbf{(B) }65 \qquad
\textbf{(C) }70 \qquad
\textbf{(D) }75 \qquad
\textbf{(E) }80 \qquad
$
2018 AMC 10, 22
Let $a, b, c,$ and $d$ be positive integers such that $\gcd(a, b)=24$, $\gcd(b, c)=36$, $\gcd(c, d)=54$, and $70<\gcd(d, a)<100$. Which of the following must be a divisor of $a$?
$\textbf{(A)} \text{ 5} \qquad \textbf{(B)} \text{ 7} \qquad \textbf{(C)} \text{ 11} \qquad \textbf{(D)} \text{ 13} \qquad \textbf{(E)} \text{ 17}$
2018 AMC 10, 4
How many ways can a student schedule 3 mathematics courses -- algebra, geometry, and number theory -- in a 6-period day if no two mathematics courses can be taken in consecutive periods? (What courses the student takes during the other 3 periods is of no concern here.)
$\textbf{(A) }3\qquad\textbf{(B) }6\qquad\textbf{(C) }12\qquad\textbf{(D) }18\qquad\textbf{(E) }24$
2018 AMC 12/AHSME, 18
Triangle $ABC$ with $AB=50$ and $AC=10$ has area $120$. Let $D$ be the midpoint of $\overline{AB}$, and let $E$ be the midpoint of $\overline{AC}$. The angle bisector of $\angle BAC$ intersects $\overline{DE}$ and $\overline{BC}$ at $F$ and $G$, respectively. What is the area of quadrilateral $FDBG$?
$
\textbf{(A) }60 \qquad
\textbf{(B) }65 \qquad
\textbf{(C) }70 \qquad
\textbf{(D) }75 \qquad
\textbf{(E) }80 \qquad
$
2018 AMC 12/AHSME, 3
How many ways can a student schedule 3 mathematics courses -- algebra, geometry, and number theory -- in a 6-period day if no two mathematics courses can be taken in consecutive periods? (What courses the student takes during the other 3 periods is of no concern here.)
$\textbf{(A) }3\qquad\textbf{(B) }6\qquad\textbf{(C) }12\qquad\textbf{(D) }18\qquad\textbf{(E) }24$
2018 AMC 12/AHSME, 16
Which of the following describes the set of values of $a$ for which the curves $x^2+y^2=a^2$ and $y=x^2-a$ in the real $xy$-plane intersect at exactly $3$ points?
$
\textbf{(A) }a=\frac14 \qquad
\textbf{(B) }\frac14 < a < \frac12 \qquad
\textbf{(C) }a>\frac14 \qquad
\textbf{(D) }a=\frac12 \qquad
\textbf{(E) }a>\frac12 \qquad
$
2018 AMC 10, 17
Let $S$ be a set of 6 integers taken from $\{1,2,\dots,12\}$ with the property that if $a$ and $b$ are elements of $S$ with $a<b$, then $b$ is not a multiple of $a$. What is the least possible value of an element in $S$?
$\textbf{(A)}\ 2\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}\ 5\qquad\textbf{(E)}\ 7$
2018 AMC 10, 21
Which of the following describes the set of values of $a$ for which the curves $x^2+y^2=a^2$ and $y=x^2-a$ in the real $xy$-plane intersect at exactly $3$ points?
$
\textbf{(A) }a=\frac14 \qquad
\textbf{(B) }\frac14 < a < \frac12 \qquad
\textbf{(C) }a>\frac14 \qquad
\textbf{(D) }a=\frac12 \qquad
\textbf{(E) }a>\frac12 \qquad
$
2018 AMC 10, 13
A paper triangle with sides of lengths 3, 4, and 5 inches, as shown, is folded so that point $A$ falls on point $B$. What is the length in inches of the crease?
[asy]
draw((0,0)--(4,0)--(4,3)--(0,0));
label("$A$", (0,0), SW);
label("$B$", (4,3), NE);
label("$C$", (4,0), SE);
label("$4$", (2,0), S);
label("$3$", (4,1.5), E);
label("$5$", (2,1.5), NW);
fill(origin--(0,0)--(4,3)--(4,0)--cycle, gray(0.9));
[/asy]
$\textbf{(A) } 1+\frac12 \sqrt2 \qquad \textbf{(B) } \sqrt3 \qquad \textbf{(C) } \frac74 \qquad \textbf{(D) } \frac{15}{8} \qquad \textbf{(E) } 2 $
2018 AMC 10, 15
Two circles of radius 5 are externally tangent to each other and are internally tangent to a circle of radius 13 at points $A$ and $B$, as shown in the diagram. The distance $AB$ can be written in the form $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$?
[asy]
draw(circle((0,0),13));
draw(circle((5,-6.2),5));
draw(circle((-5,-6.2),5));
label("$B$", (9.5,-9.5), S);
label("$A$", (-9.5,-9.5), S);
[/asy]
$\textbf{(A) } 21 \qquad \textbf{(B) } 29 \qquad \textbf{(C) } 58 \qquad \textbf{(D) } 69 \qquad \textbf{(E) } 93 $
2018 AMC 12/AHSME, 12
Let $S$ be a set of 6 integers taken from $\{1,2,\dots,12\}$ with the property that if $a$ and $b$ are elements of $S$ with $a<b$, then $b$ is not a multiple of $a$. What is the least possible value of an element in $S$?
$\textbf{(A)}\ 2\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}\ 5\qquad\textbf{(E)}\ 7$
2018 AMC 12/AHSME, 11
A paper triangle with sides of lengths 3, 4, and 5 inches, as shown, is folded so that point $A$ falls on point $B$. What is the length in inches of the crease?
[asy]
draw((0,0)--(4,0)--(4,3)--(0,0));
label("$A$", (0,0), SW);
label("$B$", (4,3), NE);
label("$C$", (4,0), SE);
label("$4$", (2,0), S);
label("$3$", (4,1.5), E);
label("$5$", (2,1.5), NW);
fill(origin--(0,0)--(4,3)--(4,0)--cycle, gray(0.9));
[/asy]
$\textbf{(A) } 1+\frac12 \sqrt2 \qquad \textbf{(B) } \sqrt3 \qquad \textbf{(C) } \frac74 \qquad \textbf{(D) } \frac{15}{8} \qquad \textbf{(E) } 2 $