Found problems: 25
2022 MIG, 14
Two truth tellers and two liars are positioned in a line, where every person is distinguishable. How many ways are there to position these four people such that everyone claims that all people directly adjacent to them are liars?
$\textbf{(A) }4\qquad\textbf{(B) }6\qquad\textbf{(C) }8\qquad\textbf{(D) }12\qquad\textbf{(E) }16$
2022 MIG, 15
There exists a fraction $x$ that satisfies $ \sqrt{x^2+5} - x = \tfrac{1}{3}$. What is the sum of the numerator and denominator of this fraction?
$\textbf{(A) }8\qquad\textbf{(B) }21\qquad\textbf{(C) }25\qquad\textbf{(D) }32\qquad\textbf{(E) }34$
2022 MIG, 9
A circle with area $\tfrac{36}{\pi}$ has the same perimeter as a square with what side length?
$\textbf{(A) }\frac{9}{\pi}\qquad\textbf{(B) }3\qquad\textbf{(C) }\pi\qquad\textbf{(D) }6\qquad\textbf{(E) }\pi^2$
2022 MIG, 24
Jenn draws a scalene triangle, and measures the heights from each of the vertices to its opposite side. She discovers that the three height lengths are all roots of the polynomial $x^3 - 3.9 x^2 + 4.4 x - 1.2.$ What is the length of the inradius of the triangle?
$\textbf{(A) }\frac{3}{13}\qquad\textbf{(B) }\frac{3}{11}\qquad\textbf{(C) }\frac{2}{7}\qquad\textbf{(D) }\frac{8}{15}\qquad\textbf{(E) }\frac{9}{14}$
2022 MIG, 4
There exists a real number $n$ such that $4^n=5$. What is the value of $8^n$?
$\textbf{(A) }5\sqrt{5}\qquad\textbf{(B) }10\qquad\textbf{(C) }25\qquad\textbf{(D) }50\qquad\textbf{(E) }25\sqrt{5}$
2022 MIG, 21
A herder has forgotten the number of cows she has, and does not want to count them all of them. She remembers these four facts about the number of cows:
[list]
[*]It has $3$ digits.
[*]It is a palindrome.
[*]The middle digit is a multiple of $4$.
[*]It is divisible by $11$.
[/list]
What is the sum of all possible numbers of cows that the herder has?
$\textbf{(A) }343\qquad\textbf{(B) }494\qquad\textbf{(C) }615\qquad\textbf{(D) }635\qquad\textbf{(E) }726$
2022 MIG, 18
Two equilateral triangles are glued, and their opposite vertices are connected. If the larger equilateral triangle has an area of $225$ and the smaller equilateral triangle has an area of $100$, what is the area of the shaded region?
[asy]
size(4cm);
draw((0,0)--(3,0)--(3/2,3sqrt(3)/2)--(0,0));
draw((0,0)--(2,0)--(1,-sqrt(3))--(0,0));
draw((1,-sqrt(3))--(3/2,3sqrt(3)/2));
filldraw((0,0)--(6/5,0)--(3/2,3sqrt(3)/2)--cycle, gray);
[/asy]
$\textbf{(A) }60\qquad\textbf{(B) }90\qquad\textbf{(C) }96\qquad\textbf{(D) }108\qquad\textbf{(E) }120$
2022 MIG, 25
In the below diagram, the three rectangles are similar. Find the area of rectangle $ABCD$.
[asy]
size(6cm);
draw((0,0)--(10,0)--(10,8.16496580928)--(0,8.16496580928)--(0,0));
draw((2,0)--(6,3.26598632371)--(2,8.16496580928)--(-2,4.89897949)--(2,0));
draw((10,3.26598632371)--(6,3.26598632371)--(6,0));
label("$A$",(0,8.16496580928), NW);
label("$D$",(0,0), SW);
label("$C$",(10,0), SE);
label("$B$",(10,8.16496580928), NE);
label("$E$",(2,8.16496580928), N);
label("$F$",(2,0), S);
label("$3$",(1,0),S);
label("$3$",(1,8.16496580928),N);
label("$12$",(6,8.16496580928),N);
[/asy]
$\textbf{(A) }75\sqrt{3}\qquad\textbf{(B) }120\sqrt{2}\qquad\textbf{(C) }100\sqrt{3}\qquad\textbf{(D) }180\qquad\textbf{(E) }75\sqrt{6}$
2022 MIG, 7
Alice, Bob, and Charlie are each thinking of a number. Alice's number differs from Bob's number by $2$. Bob's number differs from Charlie's number by $6$. Charlie's number differs from Alice's number by $N$. What is the sum of all possible values for $N$?
$\textbf{(A) }4\qquad\textbf{(B) }6\qquad\textbf{(C) }7\qquad\textbf{(D) }12\qquad\textbf{(E) }14$
2022 MIG, 6
A coin is flipped three times. What is the probability that there are no instances of two consecutive heads or two consecutive tails?
$\textbf{(A) }\frac{1}{8}\qquad\textbf{(B) }\frac{1}{4}\qquad\textbf{(C) }\frac{3}{8}\qquad\textbf{(D) }\frac{5}{8}\qquad\textbf{(E) }\frac{3}{4}$
2022 MIG, 5
What is the perimeter of the smallest rectangle with integer side lengths that fits three non-overlapping squares with areas $4,9,$ and $16$?
$\textbf{(A) }20\qquad\textbf{(B) }24\qquad\textbf{(C) }26\qquad\textbf{(D) }28\qquad\textbf{(E) }32$
2022 MIG, 2
Let $x$ be a number such that $10000x+2=4$. What is the value of $5000x+1$?
$\textbf{(A) }{-}1\qquad\textbf{(B) }0\qquad\textbf{(C) }1\qquad\textbf{(D) }2\qquad\textbf{(E) }3$
2022 MIG, 11
If $N=1000^2-950^2$, what is the largest prime factor of $N$?
$\textbf{(A) }5\qquad\textbf{(B) }13\qquad\textbf{(C) }17\qquad\textbf{(D) }19\qquad\textbf{(E) }29$
2022 MIG, 23
Friends Alice, Betty, and Cathy are playing a game. Betty and Cathy are each given a square number, such that Betty knows Cathy's number and Cathy knows Betty's, but neither of them know their own.
Alice then says: "The sum of the numbers is less than 100."
Betty says: "If Cathy knew the number of possibilities for my number, she would know her own."
Cathy then says: "Now I know my number."
What is Cathy's number?
$\textbf{(A) }16\qquad\textbf{(B) }25\qquad\textbf{(C) }36\qquad\textbf{(D) }49\qquad\textbf{(E) }64$
2022 MIG, 22
How many ways are there to color each of the $8$ cells below red or blue such that no two blue cells are adjacent?
[asy]
size(3cm);
draw((0,0)--(4,0)--(4,1)--(0,1)--(0,0));
draw((1,-1)--(1,2)--(3,2)--(3,-1)--(1,-1));
draw((2,-1)--(2,2));
[/asy]
$\textbf{(A) }48\qquad\textbf{(B) }50\qquad\textbf{(C) }52\qquad\textbf{(D) }54\qquad\textbf{(E) }56$
2022 MIG, 16
An ant climbs either two inches or three inches each day. In how many ways can the ant climb twelve inches, if the order of its climbing sequence matters?
$\textbf{(A) }8\qquad\textbf{(B) }9\qquad\textbf{(C) }10\qquad\textbf{(D) }12\qquad\textbf{(E) }14$
2022 MIG, 13
Sarah is leading a class of $35$ students. Initially, all students are standing. Each time Sarah waves her hands, a prime number of standing students sit down. If no one is left standing after Sarah waves her hands $3$ times, what is the greatest possible number of students that could have been standing before her third wave?
$\textbf{(A) }23\qquad\textbf{(B) }27\qquad\textbf{(C) }29\qquad\textbf{(D) }31\qquad\textbf{(E) }33$
2022 MIG, 12
Out of a sample of $100$ people, $24$ do not like red or blue, $40$ like both red and blue, and $50$ people like red. How many people like blue but not red?
$\textbf{(A) }24\qquad\textbf{(B) }26\qquad\textbf{(C) }48\qquad\textbf{(D) }64\qquad\textbf{(E) }76$
2022 MIG, 8
Let $ABC$ be a triangle and $D$ be a point on segment $BC$. If $\triangle ABD$ is equilateral and $\angle ACB = 14^{\circ}$, what is $\angle{DAC}$?
$\textbf{(A) }26^{\circ}\qquad\textbf{(B) }34^{\circ}\qquad\textbf{(C) }46^{\circ}\qquad\textbf{(D) }50^{\circ}\qquad\textbf{(E) }54^{\circ}$
2022 MIG, 19
Cozi makes a two-way table on chalkboard describing the right or left hand usage of students and teachers in her school. However, when she returns to the chalkboard from lunch, she is dismayed to find that most of the numbers on her table have been erased, leaving behind:
\begin{tabular}{c c c c}
5 & ? & ? & Total \\
? & ? & 6 & Total \\
? & 11 & ? & \\
Total & Total & & \\
\end{tabular}
Fortunately, Cozi remembers that the difference between two of the missing numbers is equal to $12.$ Which of the following could be the total number of students and teachers on the table?
$\textbf{(A) }14\qquad\textbf{(B) }15\qquad\textbf{(C) }16\qquad\textbf{(D) }17\qquad\textbf{(E) }18$
2022 MIG, 1
In a certain store, all pencils cost the same amount of money. If three pencils can be bought for six dollars, what is the price of two pencils?
$\textbf{(A) }\$ 3\qquad\textbf{(B) }\$ 3.5\qquad\textbf{(C) }\$ 4\qquad\textbf{(D) }\$4.5\qquad\textbf{(E) }\$ 5$
2022 MIG, 20
In the diagram below, $AX$ is parallel to $BY$, $AB$ is perpendicular to $BY$, and $AZB$ is an isosceles right triangle. If $AB = 7$ and $XY =25$, what is the length of $AX$?
[asy]
size(7cm);
draw((0,0)--(0,7)--(17/2,7)--(-31/2,0)--(0,0));
draw((0,0)--(-7/2,7/2)--(0,7));
label("$B$",(0,0),S);
label("$Y$",(-31/2,0),S);
label("$A$",(0,7),N);
label("$X$",(17/2,7),N);
label("$Z$",(-7/2,7/2),N);
[/asy]
$\textbf{(A) }\frac{17}{3}\qquad\textbf{(B) }\frac{17}{2}\qquad\textbf{(C) }9\qquad\textbf{(D) }\frac{51}{4}\qquad\textbf{(E) }12$
2022 MIG, 3
Real numbers $w$, $x$, $y$, and $z$ satisfy $w+x+y = 3$, $x+y+z = 4,$ and $w+x+y+z = 5$. What is the value of $x+y$?
$\textbf{(A) }-\frac{1}{2}\qquad\textbf{(B) }1\qquad\textbf{(C) }\frac{3}{2}\qquad\textbf{(D) }2\qquad\textbf{(E) }3$
2022 MIG, 17
What is the value of
$$(\sqrt{2}-1)^4+\frac{1}{(\sqrt{2}-1)^4}?$$
$\textbf{(A) }32-16\sqrt{2}\qquad\textbf{(B) }30\qquad\textbf{(C) }34\qquad\textbf{(D) }15+15\sqrt{2}\qquad\textbf{(E) }16+16\sqrt{2}$
2022 MIG, 10
What is the maximum possible value of $5-|6x-80|$ over all integers $x$?
$\textbf{(A) }{-}1\qquad\textbf{(B) }0\qquad\textbf{(C) }1\qquad\textbf{(D) }3\qquad\textbf{(E) }5$