Found problems: 25
2018 MIG, 14
How many integers between $80$ and $100$ are prime?
$\textbf{(A) } 3\qquad\textbf{(B) } 4\qquad\textbf{(C) } 5\qquad\textbf{(D) } 6\qquad\textbf{(E) } 7$
2018 MIG, 12
A unit cube is sliced by a plane passing through two of its vertices and the midpoints of the edges it passes through. What is the area of the rhombus formed by this intersection?
[center][img]https://cdn.artofproblemsolving.com/attachments/3/5/3ed19fa0b4d454a3afc16c6bcf9d69403f6b2c.png[/img][/center]
$\textbf{(A) } \dfrac{\sqrt6}{2}\qquad\textbf{(B) }\sqrt2\qquad\textbf{(C) }\sqrt3\qquad\textbf{(D) }\sqrt6\qquad\textbf{(E) }2\sqrt6$
2018 MIG, 11
Square $ABCD$ and triangle $ABE$ have equal area. Square $ABCD$ has sidelength $4$, while triangle $ABE$ has height $h$ and base $4$. Find the value of $h$.
[center][img]https://cdn.artofproblemsolving.com/attachments/7/c/efe7bed4232f9d2440f5719d6f2ddae0ef7d05.png[/img][/center]
$\textbf{(A) }\dfrac43\qquad\textbf{(B) }2\qquad\textbf{(C) }4\qquad\textbf{(D) }6\qquad\textbf{(E) }8$
2018 MIG, 5
Some of the values produced by two functions, $f(x)$ and $g(x)$, are shown below. Find $f(g(3))$
\begin{tabular}{c||c|c|c|c|c}
$x$ & 1 & 3 & 5 & 7 & 9 \\ \hline\hline
$f(x)$ & 3 & 7 & 9 & 13 & 17 \\ \hline
$g(x)$ & 54 & 9 & 25 & 19 & 44
\end{tabular}
$\textbf{(A) }3\qquad\textbf{(B) }7\qquad\textbf{(C) }8\qquad\textbf{(D) }13\qquad\textbf{(E) }17$
2018 MIG, 25
The figure below contains two squares which share an edge, one with side length $200$ units and the other with side length $289$ units. The figure is divided into a whole number of regions, each with an equal whole number area but not necessarily of the same shape. Given that there is more than one region and each region has an area greater than $1$, find the sum of the number of regions and the area of each region.
[asy]
size(4cm);
draw((0,0)--(200,0)--(200,200)--(0,200)--cycle);
label("$200$",(0,0)--(200,0));
label("$289$",(200,0)--(489,0));
draw((200,0)--(489,0)--(489,289)--(200,289)--cycle);
[/asy]
$\textbf{(A) } 704\qquad\textbf{(B) } 874\qquad\textbf{(C) } 924\qquad\textbf{(D) } 978\qquad\textbf{(E) } 1028$
2018 MIG, 8
The set of natural numbers are arranged as so:
$$\begin{array}{ccccccccc}
& & & & 1 & & & &\\
& & & 2 & 3 & 4 & & &\\
& & 5 & 6 & 7 & 8 & 9 &\\
& 10 & 11 & 12 & 13 & 14 & 15 & 16 &\\
17 & 18 & 19 & 20 & 21 & 22 & 23 & 24 & 25\\
& & & & \vdots & & & &
\end{array}$$
so that each row has $2$ more numbers in it, and the rows are centered. What is the number under $49$?
$\textbf{(A) }60\qquad\textbf{(B) }61\qquad\textbf{(C) }62\qquad\textbf{(D) }63\qquad\textbf{(E) }64$
2018 MIG, 15
Gordon has the least number of coins (half-dollars, quarters, dimes, nickels, pennies) needed to make $99\cent$. He randomly chooses one. What is the probability that it is a penny?
$\textbf{(A) } \dfrac15\qquad\textbf{(B) } \dfrac13\qquad\textbf{(C) } \dfrac12\qquad\textbf{(D) } \dfrac23\qquad\textbf{(E) } \dfrac34$
2018 MIG, 21
Find the sum: \[11 \times \dbinom20 + 10 \times \dbinom31 + 9 \times \dbinom42 + \cdots + 2 \times \dbinom{11}9 + \dbinom{12}{10}\] Where $\tbinom{n}{r}$ is combination function given by $\tfrac{n!}{r!(n-r)!}$
$\textbf{(A) } 351\qquad\textbf{(B) } 841\qquad\textbf{(C) } 901\qquad\textbf{(D) } 991\qquad\textbf{(E) } 1001$
2018 MIG, 6
How many more hours are in $10$ years than seconds in $1$ day?
$\textbf{(A) }1000\qquad\textbf{(B) }1100\qquad\textbf{(C) }1150\qquad\textbf{(D) }1200\qquad\textbf{(E) }1300$
2018 MIG, 1
Evaluate $1 + 2 + 4 + 7$
$\textbf{(A) }14\qquad\textbf{(B) }15\qquad\textbf{(C) }16\qquad\textbf{(D) }17\qquad\textbf{(E) }18$
2018 MIG, 22
Country $A$ uses a currency known as the shell. The nation uses only two coins, each worth a whole number of shells. The largest amount of shell not obtainable using a combination of these two coins is $215$. Find the number of possible pairs of values these two coins could have. (a value of $15$ and $4$ is the same as having a $4$ and $15$)
$\textbf{(A) } 6\qquad\textbf{(B) } 7\qquad\textbf{(C) } 8\qquad\textbf{(D) } 9\qquad\textbf{(E) } 10$
2018 MIG, 17
Two standard six sided dice labeled with the numbers $1$-$6$ are rolled, and the numbers that come up are multiplied. What is the probability that their product is a multiple of five?
$\textbf{(A) } \dfrac14\qquad\textbf{(B) } \dfrac5{18}\qquad\textbf{(C) } \dfrac{11}{36}\qquad\textbf{(D) } \dfrac13\qquad\textbf{(E) } \dfrac49$
2018 MIG, 3
Solve for $x$ if $4x + 1 = 37$.
$\textbf{(A) }4\qquad\textbf{(B) }5\qquad\textbf{(C) }7\qquad\textbf{(D) }9\qquad\textbf{(E) }10$
2018 MIG, 20
Point $O$ is selected in equilateral $\triangle ABC$ such that the sum of the distances from $O$ to each side of $ABC$ is $15$. Compute the area of $ABC$.
[center][img]https://cdn.artofproblemsolving.com/attachments/4/0/dd573985a7c98f23fd05d11e95c4b908eaa895.png[/img][/center]
$\textbf{(A) } 15\sqrt3\qquad\textbf{(B) } 30\sqrt3\qquad\textbf{(C) } 50\sqrt3\qquad\textbf{(D) } 75\sqrt3\qquad\textbf{(E) } 225\sqrt3$
2018 MIG, 2
Edward is trying to spell the word "CAT". He has an equal chance of spelling the word in any order of letters (i.e. TAC or TCA). What is the probability that he spells "CAT" incorrectly?
$\textbf{(A) }\dfrac16\qquad\textbf{(B) }\dfrac13\qquad\textbf{(C) }\dfrac12\qquad\textbf{(D) }\dfrac23\qquad\textbf{(E) }\dfrac56$
2018 MIG, 16
A triangle with area $60\text{ units}^2$ has vertices with coordinates of $(-15,x)$, $(0,x)$, and $(25,0)$. Find the largest possible value of $x$.
$\textbf{(A) } {-}8\qquad\textbf{(B) } {-}4\qquad\textbf{(C) } 4\qquad\textbf{(D) } 8\qquad\textbf{(E) } 16$
2018 MIG, 18
How many paths are there from $A$ to $B$ in the following diagram if only moves downward are allowed?
[center][img]https://cdn.artofproblemsolving.com/attachments/f/d/62a14f7959cc0461543b0f76bba51be9786847.png[/img][/center]
$\textbf{(A) } 65\qquad\textbf{(B) } 67\qquad\textbf{(C) } 70\qquad\textbf{(D) } 74\qquad\textbf{(E) } 75$
2018 MIG, 7
How many perfect squares are greater than $0$ but less than or equal to $100$?
$\textbf{(A) }6\qquad\textbf{(B) }7\qquad\textbf{(C) }8\qquad\textbf{(D) }9\qquad\textbf{(E) }10$
2018 MIG, 9
Define $f(x) = x^2 + 5$. Find the product of all $x$ such that $f(x) = 14$.
$\textbf{(A) }{-}9\qquad\textbf{(B) }{-}3\qquad\textbf{(C) }0\qquad\textbf{(D) }3\qquad\textbf{(E) }9$
2018 MIG, 10
A survey was taken in Ms. Susan's class to see what grades the class received:
[center][img width=35]https://cdn.artofproblemsolving.com/attachments/5/c/e96cb42de6d5e1b100f37bbb71768d399842cb.png[/img][/center]
What percent of the class received an "A"?
$\textbf{(A) }3\%\qquad\textbf{(B) }5\%\qquad\textbf{(C) }10\%\qquad\textbf{(D) }15\%\qquad\textbf{(E) }27\%$
2018 MIG, 13
Find the sum of the $2$ smallest prime factors of $2^{1024} - 1$.
$\textbf{(A) } 4\qquad\textbf{(B) } 6\qquad\textbf{(C) } 8\qquad\textbf{(D) } 10\qquad\textbf{(E) } 12$
2018 MIG, 23
Diagonal $AC$ is drawn in rectangle $ABCD$. Points $E$ and $F$ are placed on $BC$ such that $CE:EF:FB=2:1:1$. Let $G$ be the intersection of $DF$ with $AC$ and $H$ the intersection of $DE$ with $AC$. Given that $AD=4$ and $AB=8$, find the length of $GH$. Express your answer as a common fraction in simplest radical form.
[center][img]https://cdn.artofproblemsolving.com/attachments/4/c/b69d79cd47bcb945e7a489533eb9761ccc7ccd.png[/img][/center]
$\textbf{(A) } \dfrac{4\sqrt5}{21}\qquad\textbf{(B) } \dfrac{8\sqrt5}{21}\qquad\textbf{(C) } \dfrac{10\sqrt5}{21}\qquad\textbf{(D) } \dfrac{4\sqrt5}{5}\qquad\textbf{(E) } \sqrt5$
2018 MIG, 4
What is the positive difference between the sum of the first $5$ positive even integers and the first $5$ positive odd integers?
$\textbf{(A) }2\qquad\textbf{(B) }3\qquad\textbf{(C) }4\qquad\textbf{(D) }5\qquad\textbf{(E) }6$
2018 MIG, 24
The sides of $\triangle ABC$ form an arithmetic sequence of integers. Incircle $I$ is tangent to $AB$, $BC$, and $CA$ at $D$, $E$, and $F$, respectively. Given that $DB = \tfrac32$, $FA = \tfrac12$, find the radius of $I$.
$\textbf{(A) } \dfrac12\qquad\textbf{(B) } \dfrac{\sqrt{15}}7\qquad\textbf{(C) } \dfrac{\sqrt{15}}6\qquad\textbf{(D) } \dfrac{2\sqrt{15}}{9}\qquad\textbf{(E) } \dfrac{\sqrt{15}}{4}$
2018 MIG, 19
Rectangle $ABCD$, with integer side lengths, has equal area and perimeter. What is the positive difference between the two possible areas of $ABCD$?
$\textbf{(A) } 0\qquad\textbf{(B) } 2\qquad\textbf{(C) } 4\qquad\textbf{(D) } 5\qquad\textbf{(E) } 6$