Found problems: 25
2022 MIG, 5
Jamie accidentally misinterprets the rules of the order of operations, and adds or subtracts before multiplying or dividing. What would be her result for the equation $4 + 3 \times 1 - 2$?
$\textbf{(A) }{-}7\qquad\textbf{(B) }{-}5\qquad\textbf{(C) }5\qquad\textbf{(D) }7\qquad\textbf{(E) }9$
2022 MIG, 10
The diagram below shows a square of area $36$ separated into two rectangles and a smaller square. One of the rectangles has an area of $12$. What is the smallest rectangle's area?
[asy]
size(70);
draw((0,0)--(2,0)--(2,6)--(0,6)--cycle);
draw((2,2)--(6,2)--(6,6)--(2,6)--cycle);
draw((2,2)--(6,2)--(6,0)--(2,0)--cycle);
label("$12$",(1,3));
label("$?$",(4,4));
label("$?$",(4,1));
[/asy]
$\textbf{(A) }8\qquad\textbf{(B) }9\qquad\textbf{(C) }12\qquad\textbf{(D) }16\qquad\textbf{(E) }\text{Not Enough Information}$
2022 MIG, 18
If the six-digit number $\underline{2}\, \underline{0}\, \underline{2} \, \underline{1} \, \underline{a} \, \underline{b}$ is divisible by $9$, what is the greatest possible value of $a \cdot b$?
$\textbf{(A) }18\qquad\textbf{(B) }20\qquad\textbf{(C) }36\qquad\textbf{(D) }40\qquad\textbf{(E) }42$
2022 MIG, 11
The sum of $n$ consecutive integers is divisible by $n$ for some $n > 1$. For which $n$ is this always true?
$\textbf{(A) }\text{even }n\qquad\textbf{(B) }\text{odd }n\text{ divisible by }3\qquad\textbf{(C) }\text{odd }n\qquad\textbf{(D) }\text{prime }n\qquad\textbf{(E) }\text{no such }n\text{ exists}$
2022 MIG, 6
Two different $3 \times 3$ grids are chosen within a $5 \times 5$ grid. What is the least number of unit grids contained in the overlap of the two $3 \times 3$ grids?
$\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }3\qquad\textbf{(D) }4\qquad\textbf{(E) }6$
2022 MIG, 12
For a certain value of $x$, the sum of the digits of $10^x - 100$ is equal to $45$. What is $x$?
$\textbf{(A) }4\qquad\textbf{(B) }5\qquad\textbf{(C) }6\qquad\textbf{(D) }7\qquad\textbf{(E) }8$
2022 MIG, 14
Four coins are placed in a line. A passerby walks by and flips each coin, and stops if she ever obtains two adjacent heads. If the passerby manages to flip all four coins, how many possible head-tail combinations exist for her four flips?
$\textbf{(A) }4\qquad\textbf{(B) }6\qquad\textbf{(C) }8\qquad\textbf{(D) }10\qquad\textbf{(E) }12$
2022 MIG, 2
A machine takes $6$ seconds to make $4$ coins. How long does it take for the machine to make $22$ coins? The machine makes coins at the same constant rate.
$\textbf{(A) }30\qquad\textbf{(B) }33\qquad\textbf{(C) }36\qquad\textbf{(D) }39\qquad\textbf{(E) }42$
2022 MIG, 15
A function $f(a \tfrac bc)$ for a simplified mixed fraction $a \tfrac bc$ returns $\tfrac{a + b}{c}$. For instance, $f(2 \tfrac 57) = 1$ and $f(\tfrac45) = \tfrac45$. What is the sum of the three smallest positive rational $x$ where $f(x) = \tfrac 29$?
$\textbf{(A) }\dfrac52\qquad\textbf{(B) }\dfrac{68}{27}\qquad\textbf{(C) }\dfrac{23}{9}\qquad\textbf{(D) }\dfrac{74}{27}\qquad\textbf{(E) }\dfrac{13}4$
2022 MIG, 4
Which of the following answer choices is equivalent to $\sqrt{a^3b^2c}$?
$\textbf{(A) }ab\sqrt{ac}\qquad\textbf{(B) }bc\sqrt{ac}\qquad\textbf{(C) }b\sqrt{ac}\qquad\textbf{(D) }abc\sqrt{ab}\qquad\textbf{(E) }a\sqrt{bc}$
2022 MIG, 7
Consider the rectangular strip of length $12$ below, divided into three rectangles. The distance between the centers of two of the rectangles is $4$. What is the length of the other rectangle?
[asy]
size(120);
draw((0,0)--(12,0)--(12,1)--(0,1)--cycle);
draw((8,1)--(8,0));
draw((3,1)--(3,0));
dot((1.5,0.5));
dot((5.5,0.5));
draw((1.5,0.5)--(5.5,0.5));
[/asy]
$\textbf{(A) }2.5\qquad\textbf{(B) }3\qquad\textbf{(C) }3.5\qquad\textbf{(D) }4\qquad\textbf{(E) }4.5$
2022 MIG, 22
Jerry and Aaron both pick two integers from $1$ to $6$, inclusive, and independently and secretly tell their numbers to Dennis.
Dennis then announces, "Aaron's number is at least three times Jerry's number."
Aaron says, "I still don't know Jerry's number."
Jerry then replies, "Oh, now I know Aaron's number."
What is the sum of their numbers?
$\textbf{(A) }4\qquad\textbf{(B) }5\qquad\textbf{(C) }6\qquad\textbf{(D) }7\qquad\textbf{(E) }8$
2022 MIG, 25
For all positive integers $a > 1$, there are divisors of $2021a$ that are not divisors of $2021$. If there are twelve unshared divisors, including $2021a$, which of the following answer choices could be a possible value of $a$?
$\textbf{(A) }9\qquad\textbf{(B) }10\qquad\textbf{(C) }16\qquad\textbf{(D) }18\qquad\textbf{(E) }19$
2022 MIG, 24
Cows Alpha and Beta are tied by eight-meter ropes, on the midpoints of adjacent sides of a rectangular fence. Both cows are outside the fence; Alpha can wander in a region with an area of $34\pi$ square meters and Beta can wander in a region with an area of $40\pi$ square meters. What is the area enclosed by the rectangular fence?
$\textbf{(A) }45\qquad\textbf{(B) }48\qquad\textbf{(C) }96\qquad\textbf{(D) }120\qquad\textbf{(E) }144$
2022 MIG, 23
Elax creates a partially filled $4 \times 4$ grid, and is trying to write in positive integers such that any four cells that share no rows and columns always sum to a number $S$. Given that the sum of the numbers in the top row is also $S$, what is the missing cell number?
[asy]
size(100);
add(grid(4,4));
label("$11$", (0.5,1.5));
label("$10$", (0.5,2.5));
label("?", (0.5,3.5));
label("$8$", (1.5,3.5));
label("$7$", (2.5,2.5));
label("$4$", (3.5,0.5));
label("$9$", (3.5,1.5));
[/asy]
$\textbf{(A) }4\qquad\textbf{(B) }6\qquad\textbf{(C) }9\qquad\textbf{(D) }10\qquad\textbf{(E) }12$
2022 MIG, 9
How many integer values of $x$ satisfy \[\dfrac32 < \dfrac9x < \dfrac 73?\]
$\textbf{(A) }1\qquad\textbf{(B) }2\qquad\textbf{(C) }3\qquad\textbf{(D) }4\qquad\textbf{(E) }5$
2022 MIG, 20
The area of the dark gray triangle depicted below is $35$, and a segment is divided into lengths $14$ and $10$ as shown below. What is the area of the light gray triangle?
[asy]
size(150);
filldraw((0,0)--(0,12)--(24,-60/7)--cycle, lightgray);
filldraw((14,0)--(14,5)--(0,12)--cycle, gray);
draw((0,0)--(24,0)--(0,12)--cycle);
draw((0,0)--(24,0)--(24,-60/7)--cycle);
draw((0,12)--(24,-60/7));
draw((14,5)--(14,0));
dot((0,0));
dot((0,12));
dot((14,5));
dot((24,0));
dot((14,0));
dot((24,-60/7));
label("$14$", (7,0), S);
label("$10$", (19,0), S);
draw((0,2/3)--(2/3,2/3)--(2/3,0));
draw((14,2/3)--(14+2/3,2/3)--(14+2/3,0));
draw((24-2/3,0)--(24-2/3,-2/3)--(24,-2/3));
[/asy]
$\textbf{(A) }84\qquad\textbf{(B) }120\qquad\textbf{(C) }132\qquad\textbf{(D) }144\qquad\textbf{(E) }168$
2022 MIG, 16
Let $P$ be a point on side $\overline{AB}$ of equilateral triangle $ABC$. If $BP = 6$ and $CP = 9$, what is the length of $AB$?
$\textbf{(A) }2\sqrt5\qquad\textbf{(B) }3+\sqrt6\qquad\textbf{(C) }3\sqrt5\qquad\textbf{(D) }3\sqrt6 + 3\qquad\textbf{(E) }6\sqrt2$
2022 MIG, 17
Jane and Jena sit at non-adjacent chairs of a four-chair circular table. In a turn, one person can move to an adjacent chair without a person. Jane moves in the first turn, and alternates with Jena afterwards. In how many ways can Jena be adjacent to Jane after nine moves?
$\textbf{(A) }16\qquad\textbf{(B) }18\qquad\textbf{(C) }32\qquad\textbf{(D) }162\qquad\textbf{(E) }512$
2022 MIG, 3
Jar $A$ and Jar $B$ each contain $10$ beans. The number of beans in jar $A$ is doubled, and the number of beans in jar $B$ is halved. How many beans are now in jars $A$ and $B$?
$\textbf{(A) }15\qquad\textbf{(B) }20\qquad\textbf{(C) }25\qquad\textbf{(D) }30\qquad\textbf{(E) }40$
2022 MIG, 21
Let $T(p)$ denote the number of right triangles with integer side lengths and one of its side lengths being $p$. Which of the following values of $p$ produces the greatest possible value of $T(p)$ among all five answer choices?
$\textbf{(A) }24\qquad\textbf{(B) }27\qquad\textbf{(C) }28\qquad\textbf{(D) }36\qquad\textbf{(E) }54$
2022 MIG, 1
What is $4^0 - 3^1 - 2^2 - 1^3$?
$\textbf{(A) }{-}8\qquad\textbf{(B) }{-}7\qquad\textbf{(C) }{-}5\qquad\textbf{(D) }4\qquad\textbf{(E) }5$
2022 MIG, 13
Consider the numbers $1$ through $6$ numbered on the coins below. Ella takes a coin from each of the three columns. Bella takes a coin from each of the remaining two columns. Cassandra takes the remaining coin. In how many ways could they have taken out the six coins?
[asy]
size(100);
draw(Circle((0,0),0.45));
label("$1$",(0,0));
draw(Circle((0,1),0.45));
label("$2$",(0,1));
draw(Circle((0,2),0.45));
label("$3$",(0,2));
draw(Circle((1,0),0.45));
label("$5$",(1,0));
draw(Circle((1,1),0.45));
label("$4$",(1,1));
draw(Circle((2,0),0.45));
label("$6$",(2,0));
[/asy]
$\textbf{(A) }9\qquad\textbf{(B) }12\qquad\textbf{(C) }15\qquad\textbf{(D) }18\qquad\textbf{(E) }20$
2022 MIG, 19
A three-digit number $N$ is equal to $36$ times the sum of its digits. Find the sum of all possible values of $N$.
$\textbf{(A) }576\qquad\textbf{(B) }648\qquad\textbf{(C) }972\qquad\textbf{(D) }1152\qquad\textbf{(E) }1620$
2022 MIG, 8
Write a list of the first $10$ positive integers in increasing order. Erase any number adjacent to a prime; if two primes are adjacent, do not erase either prime. Apply this process twice. How many positive integers remain in the list?
$\textbf{(A) }2\qquad\textbf{(B) }3\qquad\textbf{(C) }4\qquad\textbf{(D) }5\qquad\textbf{(E) }6$