Found problems: 18
2023 MIG, 15
Given that $a>2b$ and $b>2c$ and $a$, $b$, and $c$ are nonzero, which of the following statements must be true?
$\textbf{(A) } a+b>c\qquad\textbf{(B) } a-c>0\qquad\textbf{(C) } abc>0\qquad\textbf{(D) } \frac{a}{b}>2\qquad\textbf{(E) } \text{none of these}$
2023 MIG, 11
A [i]semi-palindrome[/i] is a four-digit number whose first two digits and last two digits are identical. For instance, $2323$ and $5757$ are semi-palindromes, but $1001$ and $2324$ are not. What is the difference between the largest semi-palindrome and smallest semi-palindrome?
$\textbf{(A) } 7979\qquad\textbf{(B) } 8080\qquad\textbf{(C) } 8181\qquad\textbf{(D) } 8484\qquad\textbf{(E) } 8989$
2023 MIG, 3
A square with sides of length $6$ has the same area as a rectangle with a length of $9$. What is the width of the rectangle?
$\textbf{(A) } 2\qquad\textbf{(B) } \frac73\qquad\textbf{(C) } 3\qquad\textbf{(D) } \frac{10}{3}\qquad\textbf{(E) } 4$
2023 MIG, 6
If $a+3b = 9$ and $a+11b =21$, what is the missing coefficient in the expression $2a+\underline{?}b = 27$?
$\textbf{(A) } 5\qquad\textbf{(B) } 6\qquad\textbf{(C) } 9\qquad\textbf{(D) } 12\qquad\textbf{(E) } 14$
2023 MIG, 1
What is $1-2+3-4$?
$\textbf{(A) } {-}2\qquad\textbf{(B) } {-}1\qquad\textbf{(C) } 1\qquad\textbf{(D) } 4\qquad\textbf{(E) } 9$
2023 MIG, 9
Which answer choice correctly fills the blank in the statement below?
"The probability of flipping heads on a fair coin is the equal to the probability of rolling a $\underline{~~~~~~~~~~~}$ on a fair dice."
$\textbf{(A) }\text{prime number}\qquad\textbf{(B) }\text{number divisible by 3}\qquad\textbf{(C) }\text{number with four factors}\qquad\textbf{(D) }2~\text{or}~3\qquad\textbf{(E) }4$
2023 MIG, 7
At a length of $104$ miles, the Danyang-Kushan Bridge holds the title for being the longest bridge in the world. A car travels at a constant speed of $39$ miles per hour across the Danyang-Kushan Bridge. How long does it take the car to travel across the entire bridge?
$\textbf{(A) }\text{2 hours, 12 minutes} \qquad \textbf{(B) }\text{2 hours, 20 minutes} \qquad \textbf{(C) }\text{2 hours, 25 minutes}$\\
$\textbf{(D) }\text{2 hours, 30 minutes} \qquad \textbf{(E) }\text{2 hours, 40 minutes}$
2023 MIG, 8
Anna is buying fruits at a grocery store. If she loses a nickel, she still has enough money to buy exactly $16$ lemons. Similarly, if she loses a quarter, she has enough money to buy exactly $14$ lemons. What is the cost of each lemon?
$\textbf{(A) } \$0.05\qquad\textbf{(B) } \$0.10\qquad\textbf{(C) } \$0.15\qquad\textbf{(D) } \$0.20\qquad\textbf{(E) } \$0.25$
2023 MIG, 10
In the equation below, $x$ is a nonzero real number such that
\[\frac1{729}\left(3^t\right)=3^x.\]
Which of the following is equal to $t$?
$\textbf{(A) } \dfrac16x\qquad\textbf{(B) } \dfrac13x\qquad\textbf{(C) } 6x\qquad\textbf{(D) } x-6\qquad\textbf{(E) } x+6$
2023 MIG, 16
Masaru randomly paints $50\%$ of the area of a square. What is the probability that at least $60\%$ of the left side of the square is painted?
[asy]
size(2cm);
defaultpen(fontsize(7));
draw((0,0)--(4,0)--(4,4)--(0,4)--cycle,linewidth(1.5));
fill(circle((0.3,0.3),0.3),paleblue);
fill(circle((1,1),0.5),palered);
fill(circle((0.8,2),0.8),purple);
fill(circle((3,3),0.6),orange);
fill(circle((3.4,1.5),0.6),mediumgreen);
fill(circle((1.4,1.6),0.4),yellow);
fill(circle((2.5,2.8),0.4),cyan);
fill(circle((1,3.5),0.5),red);
fill((3,0)--(4,0)--(4,0.4)--(3.5,0.8)--cycle,magenta);
fill((2,4)--(3,4)--(3.1,3.8)--(2.7,3.5)--(2.4,3.1)--cycle,olive);
draw((2,0)--(2,4),dashed);
[/asy]
$\textbf{(A) } 25\%\qquad\textbf{(B) } 30\%\qquad\textbf{(C) } 35\%\qquad\textbf{(D) } 40\%\qquad\textbf{(E) } 45\%$
2023 MIG, 5
In the regular hexagon shown below, how many diagonals are longer than the red diagonal?
[asy]
size(2cm);
draw((0,0)--(2,0)--(3,1.732)--(2,3.464)--(0,3.464)--(-1,1.732)--cycle);
draw((-1,1.732)--(2,0),red);
[/asy]
$\textbf{(A) } 0\qquad\textbf{(B) } 1\qquad\textbf{(C) } 2\qquad\textbf{(D) } 3\qquad\textbf{(E) } 4$
2023 MIG, 4
Which operation makes the following expression true: $(4 \underline{~~~~} 1) \times (3 \underline{~~~~} 2 - 1) = 2$?
$\textbf{(A) } +\qquad\textbf{(B) } -\qquad\textbf{(C) } \times\qquad\textbf{(D) } \div\qquad\textbf{(E) } \text{There is no such operation}$
2023 MIG, 14
Kylie randomly selects two vertices of a rectangle. What is the probability that the two chosen vertices are adjacent?
$\textbf{(A) } \dfrac13\qquad\textbf{(B) } \dfrac12\qquad\textbf{(C) } \dfrac23\qquad\textbf{(D) } \dfrac56\qquad\textbf{(E) } 1$
2023 MIG, 13
Five cards numbered $1,2,3,4,$ and $5$ are given to Paige, Quincy, Ronald, Selena, and Terrence. Paige, Quincy, and Ronald have the following conversation:
[list=disc]
[*]Paige: My number is between is between Selena's number and Quincy's number.
[*]Quincy: My number is between Ronald's number and Terrence's number.
[*]Ronald: My number is between Paige's number and Quincy's number.
[/list]
Who received the card numbered $3$?
$\textbf{(A) } \text{Paige}\qquad\textbf{(B) } \text{Quincy}\qquad\textbf{(C) } \text{Ronald}\qquad\textbf{(D) } \text{Selena}\qquad\textbf{(E) } \text{Terrence}$
2023 MIG, 2
What is the sum of all $x$ that satisfy $|2x-4| = 2$?
$\textbf{(A) } 1\qquad\textbf{(B) } 2\qquad\textbf{(C) } 3\qquad\textbf{(D) } 4\qquad\textbf{(E) } 5$
2023 MIG, 12
There are ten apples and $p$ pears in a basket. Anna eats two apples, and she finds that there are now more pears than apples. She then eats four pears. After eating the pears, she notices that there are more apples than pears. What is the sum of all possible values of $p$?
$\textbf{(A) } 19\qquad\textbf{(B) } 28\qquad\textbf{(C) } 30\qquad\textbf{(D) } 42\qquad\textbf{(E) } 45$
2023 MIG, 18
The diagram below shows a rectangle and two triangles with areas $20$ and $4$. What is the area of the shaded triangle?
[asy]
size(3cm);
draw((0,0)--(7,0)--(7,5)--(0,5)--(0,0));
draw((4,0)--(0,5));
draw((7,0)--(0,5));
draw((4,0)--(7,5));
filldraw((0,0)--(4,0)--(0,5)--cycle, lightgray);
label("$4$",(5.5,0.5));
label("$20$",(4,4));
[/asy]
$\textbf{(A) } 12\qquad\textbf{(B) } 14\qquad\textbf{(C) } 16\qquad\textbf{(D) } 18\qquad\textbf{(E) } 20$
2023 MIG, 17
Adeline, Bonnie, and Cathy are walking along a long flat path, with their initial distances shown below.
[asy]
size(10cm);
draw((0,0)--(12,0)--(28,0));
label("Adeline",(0,1));
label("Bonnie",(12,1));
label("Cathy",(28,1));
label("12 miles",(6,-1));
label("16 miles",(20,-1));
dot((0,0));
dot((12,0));
dot((28,0));
[/asy]
Adeline and Bonnie walk towards each other at constant speeds of $1$ and $2$ miles per hour, respectively. Cathy walks in the same direction as Bonnie. If all three girls meet each other at the same time, what is Cathy's walking speed, in miles per hour?
$\textbf{(A) } 4~\text{mph}\qquad\textbf{(B) } 4.5~\text{mph}\qquad\textbf{(C) } 5~\text{mph}\qquad\textbf{(D) } 5.5~\text{mph}\qquad\textbf{(E) } 6~\text{mph}$