Found problems: 25
2020 MIG, 9
Lily has an unfair coin that has $\tfrac23$ probability of showing heads and $\tfrac13$ probability of showing tails. She flips the coin twice. What is the probability that the first flip is heads while the second is tails?
$\textbf{(A) }0\qquad\textbf{(B) }1/9\qquad\textbf{(C) }2/9\qquad\textbf{(D) }4/9\qquad\textbf{(E) }1$
2020 MIG, 18
When $171$ is written as the sum of $19$ consecutive integers, the median of those numbers is $M$. When $171$ is written as the sum of $18$ consecutive integers, the median of those numbers is $N$. Find $|M - N|$.
$\textbf{(A) }{-}1\qquad\textbf{(B) }{-}0.5\qquad\textbf{(C) }0\qquad\textbf{(D) }0.5\qquad\textbf{(E) }1$
2020 MIG, 11
The numbers $1$, $2$, $3$, $4$, $5$, $6$ are placed onto the following six spots such that the average of the leftmost two spots, middle two spots, and rightmost two spots are all equal. What is the difference between the largest and smallest possibilities of the number on the shaded spot shown below?
[asy]
size(110);
draw(Circle((0,0),0.7));
draw(Circle((2,0),0.7));label("$1$",(2,0));
filldraw(Circle((4,0),0.7),gray);
draw(Circle((6,0),0.7));
draw(Circle((8,0),0.7));
draw(Circle((10,0),0.7));label("$2$",(10,0));
[/asy]
$\textbf{(A) }1\qquad\textbf{(B) }2\qquad\textbf{(C) }3\qquad\textbf{(D) }4\qquad\textbf{(E) }5$
2020 MIG, 25
A number $N$ is defined as follows:
\[N=2+22+202+2002+20002+\cdots+2\overbrace{00\ldots000}^{19~0\text{'s}}2\]
When the value of $N$ is simplified, what is the sum of its digits?
$\textbf{(A) }42\qquad\textbf{(B) }44\qquad\textbf{(C) }46\qquad\textbf{(D) }50\qquad\textbf{(E) }52$
2020 MIG, 10
In the diagram below, for each row except the bottom row, the number in each cell is determined by
the sum of the two numbers beneath it. Find the sum of all cells denoted with a question mark.
[asy]
unitsize(2cm);
path box = (-0.5,-0.2)--(-0.5,0.2)--(0.5,0.2)--(0.5,-0.2)--cycle;
draw(box); label("$2$",(0,0));
draw(shift(1,0)*box); label("$?$",(1,0));
draw(shift(2,0)*box); label("$?$",(2,0));
draw(shift(3,0)*box); label("$?$",(3,0));
draw(shift(0.5,0.4)*box); label("$4$",(0.5,0.4));
draw(shift(1.5,0.4)*box); label("$?$",(1.5,0.4));
draw(shift(2.5,0.4)*box); label("$?$",(2.5,0.4));
draw(shift(1,0.8)*box); label("$5$",(1,0.8));
draw(shift(2,0.8)*box); label("$?$",(2,0.8));
draw(shift(1.5,1.2)*box); label("$9$",(1.5,1.2));
[/asy]
$\textbf{(A) }6\qquad\textbf{(B) }8\qquad\textbf{(C) }12\qquad\textbf{(D) }13\qquad\textbf{(E) }14$
2020 MIG, 7
John's digital clock is broken. It scrambles the digits of the time and displays them in a random order. For example, if the current time is $4:21$, it could display $4:12$, $2:14$, or any other reordering of $4$, $1$, and $2$. If his clock reads $6:71$ one morning, how many possibilities are there for the correct time?
$\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }4\qquad\textbf{(E) }6$
2020 MIG, 19
In the diagram below, $AB$ is a diameter of circle $O$. Point C is drawn such that $\overline{BC}$ is tangent to circle $O$, and $AB = BC$. A point $F$ is selected on line $AB$ and a point $D$ is selected on circle $O$ such that $\angle CDF = 90^\circ$. Line $\overline{BD}$ is then extended to point $E$ such that $AE$ is tangent to circle $O$. Given that $AE = 5$, calculate the length of $\overline{AF}$. (Diagram not to scale)
[asy]
size(120);
pair A,O,F,B,D,EE,C;
A=(-5,0);
O=(0,0);
B=(5,0);
EE=(-5,6);
F=(3.8,0);
D=(-2.5,4.33);
C=(5,10);
dot(A^^O^^B^^EE^^F^^D^^C);
draw(circle(O,5));
draw(A--EE--F--cycle);
draw(D--B--C--cycle);
draw(A--B);
label("$A$",A,W);
label("$O$",O,S);
label("$B$",B,E);
label("$F$",F,S);
label("$E$",EE,N);
label("$D$",D,N);
label("$C$",C,N);
[/asy]
$\textbf{(A) }\dfrac92\qquad\textbf{(B) }5\qquad\textbf{(C) }3\sqrt3\qquad\textbf{(D) }7\qquad\textbf{(E) }\text{impossible to determine}$
2020 MIG, 22
Jane's uncle gives her a "$4$-balance." The $4$-balance acts like a normal balance scale, but it compares four masses instead of two, tilting towards the weight that is heaviest (if all four are equal, it stays balanced). He then gives her $25$ coins, one of which is a counterfeit heavier than the rest. What is the minimum number of uses of the $4$-balance needed to ensure she identifies the counterfeit?
$\textbf{(A) }1\qquad\textbf{(B) }2\qquad\textbf{(C) }3\qquad\textbf{(D) }4\qquad\textbf{(E) }5$
2020 MIG, 5
What is the side length, in meters, of a square with area $49 \text{ m}^2$?
$\textbf{(A) }3\qquad\textbf{(B) }4\qquad\textbf{(C) }5\qquad\textbf{(D) }6\qquad\textbf{(E) }7$
2020 MIG, 15
In the city of Urbextorto, the sales tax is $25\%$. A certain clothing store in the city is currently giving an $n\%$ discount on all items, and $n$ is special in that, after both the sales tax and discount are applied, a $\$20$ shirt ends up costing $\$20$. Find the value of $n$.
$\textbf{(A) }5\qquad\textbf{(B) }10\qquad\textbf{(C) }20\qquad\textbf{(D) }25\qquad\textbf{(E) }50$
2020 MIG, 16
Two $1$ inch by $1$ inch squares are cutout from opposite corners of a $7$ inch by $5$ inch piece of paper to form an octagon. What is the distance, in inches, between the two dotted points, both of which lie on corners of the octagon?
[asy]
size(120);
draw((0,1)--(0,5)--(6,5));
draw((1,0)--(7,0)--(7,4));
draw((0,1)--(0,0)--(1,0),dashed);
draw((6,5)--(7,5)--(7,4),dashed);
draw((0,1)--(1,1)--(1,0));
draw((6,5)--(6,4)--(7,4));
draw((1,1)--(6,4),dashed);
dot((1,1),linewidth(5));
dot((6,4),linewidth(5));
label("$?$",(1,1)--(6,4),N);
[/asy]
$\textbf{(A) }5\qquad\textbf{(B) }\sqrt{34}\qquad\textbf{(C) }5\sqrt2\qquad\textbf{(D) }8\qquad\textbf{(E) }\sqrt{74}$
2020 MIG, 21
Consider the following $2 \times 3$ arrangement of pegs on a board. Jane places three rubber bands on the
pegs on the board such that the following conditions are satisfied:
$~$
[center]
(I) No two rubber bands cross each other.
(II) Each peg has a rubber band wrapped around it
[/center]$~$
How many distinct arrangements could Jane create exist? One acceptable arrangement is shown below.
[asy]
size(100);
filldraw(circle((0,0),0.2),black);
filldraw(circle((1,0),0.2),black);
filldraw(circle((2,0),0.2),black);
filldraw(circle((0,1),0.2),black);
filldraw(circle((1,1),0.2),black);
filldraw(circle((2,1),0.2),black);
draw((0,1.2)--(1,1.2));
draw((0,0.8)--(1,0.8));
draw((1,0.2)--(2,0.2));
draw((1,-0.2)--(2,-0.2));
draw((0,0.2)--(2,1.2));
draw((0,-0.2)--(2,0.8));
[/asy]
$\textbf{(A) }2\qquad\textbf{(B) }3\qquad\textbf{(C) }5\qquad\textbf{(D) }6\qquad\textbf{(E) }8$
2020 MIG, 14
Given that $x$ satisfies $2^{4x} \cdot 2^{4x} \cdot 8^{4x} = 16^5$, find the value of $x$.
$\textbf{(A) }1\qquad\textbf{(B) }2\qquad\textbf{(C) }4\qquad\textbf{(D) }5\qquad\textbf{(E) }10$
2020 MIG, 24
[asy]
size(140);
import geometry;
dot((0,0));label("$(0,0)$",(0,0),SW);
dot((4,3));
dot((5,4));label("$(5,4)$",(5,4),NE);
draw((0,0)--(7,0), EndArrow);
draw((0,0)--(0,6), EndArrow);
add(grid(5,4));
[/asy]
A leprechaun wishes to travel from the origin to a pot of gold located at the coordinate point $(5,4)$. If she can only move upwards and rightwards along the unit grid, must pass a checkpoint at $(1,2)$, and must avoid an evil thief at $(4,3)$, how many distinct paths can she take?
$\textbf{(A) }7\qquad\textbf{(B) }15\qquad\textbf{(C) }21\qquad\textbf{(D) }45\qquad\textbf{(E) }126$
2020 MIG, 6
The top vertex of this equilateral triangle is folded over the shown dashed line. Which of the 5 points
will the vertex lie closest to after this fold?
[asy]
size(110);
draw((0,0)--(1,0)--(0.5,sqrt(3)/2)--cycle);
dot((0.5,sqrt(3)/2));
pair A_1=(0,0);label("$A_1$",A_1,S);dot(A_1);
pair A_2=(0.25,0);label("$A_2$",A_2,S);dot(A_2);
pair A_3=(0.5,0);label("$A_3$",A_3,S);dot(A_3);
pair A_4=(0.75,0);label("$A_4$",A_4,S);dot(A_4);
pair A_5=(1,0);label("$A_5$",A_5,S);dot(A_5);
draw((0.23,0.38)--(0.86,0.22),dashed);
[/asy]
$\textbf{(A) }A_1\qquad\textbf{(B) }A_2\qquad\textbf{(C) }A_3\qquad\textbf{(D) }A_4\qquad\textbf{(E) }A_5$
2020 MIG, 2
A certain value of $x$ satisfies $1 + x + 5 - 1 = 7$. What is this value of $x$?
$\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }3\qquad\textbf{(E) }\text{impossible to determine}$
2020 MIG, 20
John can purchase pieces of gum in packs of $4$, $14$, and $20$ pieces. Given that he purchases at least one of each kind of pack, what is the positive difference between the greatest and least number of packs he can purchase to end up with exactly $86$ pieces of gum?
$\textbf{(A) }5\qquad\textbf{(B) }6\qquad\textbf{(C) }7\qquad\textbf{(D) }8\qquad\textbf{(E) }9$
2020 MIG, 1
Calculate the numerical value of $1 \times 1 + 2 \times 2 - 2$.
$\textbf{(A) }2\qquad\textbf{(B) }3\qquad\textbf{(C) }4\qquad\textbf{(D) }5\qquad\textbf{(E) }6$
2020 MIG, 13
For how many real values of $x$ is the equation $(x^2 - 7)^3 = 0$ true?
$\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }3\qquad\textbf{(E) }4$
2020 MIG, 3
What is the positive difference between the largest possible two-digit integer and the smallest possible three-digit integer?
$\textbf{(A) }1\qquad\textbf{(B) }2\qquad\textbf{(C) }3\qquad\textbf{(D) }5\qquad\textbf{(E) }9$
2020 MIG, 4
If you were to randomly select an answer to this question, what is the probability it would be correct?
$\textbf{(A) }0\%\qquad\textbf{(B) }20\%\qquad\textbf{(C) }40\%\qquad\textbf{(D) }80\%\qquad\textbf{(E) }100\%$
2020 MIG, 8
$(1 + \sqrt 3)^2$ may be written as $a + b \sqrt 3$ for certain integers $a$ and $b$. What is $a + b$?
$\textbf{(A) }1\qquad\textbf{(B) }2\qquad\textbf{(C) }4\qquad\textbf{(D) }6\qquad\textbf{(E) }7$
2020 MIG, 12
Jane's mother bakes cookies for Jane to share with her $6$ friends. When the cookies are evenly divided among the $7$ children (Jane and her $6$ friends), there is one cookie left over. Given that each child receives at least $1$ cookie, and Jane's mother baked less than $100$ cookies, how many different numbers of cookies could Jane's mother have baked? For example, she could have baked $15$ cookies, because each child receives $2$ cookies, with $1$ left over.
$\textbf{(A) }9\qquad\textbf{(B) }11\qquad\textbf{(C) }14\qquad\textbf{(D) }15\qquad\textbf{(E) }17$
2020 MIG, 23
There exists a positive integer $b$ such that the base-$10$ fraction $\tfrac{59}{48}$ can be expressed as $1.\overline{14}_b$ (or $1.141414\ldots_b$), a value in base $b$. Find $b$.
$\textbf{(A) }5\qquad\textbf{(B) }6\qquad\textbf{(C) }7\qquad\textbf{(D) }8\qquad\textbf{(E) }9$
2020 MIG, 17
A rubber band of negligible thickness encloses three pegs that lie in a perfect line, as shown. Each peg has a diameter of $4$ cm, as shown. What is the length of the rubber band used, in centimeters? All pegs shown are congruent circles.
[asy]
size(120);
draw(circle((0,0),1));draw(circle((0,2),1));draw(circle((0,4),1));
dot((0,0)^^(0,2)^^(0,4));
draw((-1,0)--(-1,4)--arc((0,4),1,180,0)--(1,4)--(1,0)--arc((0,0),1,360,180),linewidth(2));
draw((-1,0)--(1,0),dotted);
label("$4$ cm", (-0.38,0)--(1,0), N);
[/asy]
$\textbf{(A) }8\qquad\textbf{(B) }8+4\pi\qquad\textbf{(C) }16+4\pi\qquad\textbf{(D) }16+8\pi\qquad\textbf{(E) }16\pi$