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Problems
Contests
National and Regional Contests
USA Contests
USA - Middle School Tournaments
Math Invitational for Girls
2023 MIG
2023 MIG
Part of
Math Invitational for Girls
Subcontests
(18)
18
1
Hide problems
2023 Individual #18
The diagram below shows a rectangle and two triangles with areas
20
20
20
and
4
4
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
4
4
",(5.5,0.5)); label("
20
20
20
",(4,4)); [/asy]
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<span class='latex-bold'>(A) </span> 12\qquad<span class='latex-bold'>(B) </span> 14\qquad<span class='latex-bold'>(C) </span> 16\qquad<span class='latex-bold'>(D) </span> 18\qquad<span class='latex-bold'>(E) </span> 20
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20
16
1
Hide problems
2023 Individual #16
Masaru randomly paints
50
%
50\%
50%
of the area of a square. What is the probability that at least
60
%
60\%
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]
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<span class='latex-bold'>(A) </span> 25\%\qquad<span class='latex-bold'>(B) </span> 30\%\qquad<span class='latex-bold'>(C) </span> 35\%\qquad<span class='latex-bold'>(D) </span> 40\%\qquad<span class='latex-bold'>(E) </span> 45\%
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17
1
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2023 Individual #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
1
1
and
2
2
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?
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<span class='latex-bold'>(A) </span> 4~\text{mph}\qquad<span class='latex-bold'>(B) </span> 4.5~\text{mph}\qquad<span class='latex-bold'>(C) </span> 5~\text{mph}\qquad<span class='latex-bold'>(D) </span> 5.5~\text{mph}\qquad<span class='latex-bold'>(E) </span> 6~\text{mph}
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15
1
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2023 Individual #15
Given that
a
>
2
b
a>2b
a
>
2
b
and
b
>
2
c
b>2c
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>
2
c
and
a
a
a
,
b
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b
, and
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c
c
are nonzero, which of the following statements must be true?
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<span class='latex-bold'>(A) </span> a+b>c\qquad<span class='latex-bold'>(B) </span> a-c>0\qquad<span class='latex-bold'>(C) </span> abc>0\qquad<span class='latex-bold'>(D) </span> \frac{a}{b}>2\qquad<span class='latex-bold'>(E) </span> \text{none of these}
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14
1
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2023 Individual #14
Kylie randomly selects two vertices of a rectangle. What is the probability that the two chosen vertices are adjacent?
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<span class='latex-bold'>(A) </span> \dfrac13\qquad<span class='latex-bold'>(B) </span> \dfrac12\qquad<span class='latex-bold'>(C) </span> \dfrac23\qquad<span class='latex-bold'>(D) </span> \dfrac56\qquad<span class='latex-bold'>(E) </span> 1
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13
1
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2023 Individual #13
Five cards numbered
1
,
2
,
3
,
4
,
1,2,3,4,
1
,
2
,
3
,
4
,
and
5
5
5
are given to Paige, Quincy, Ronald, Selena, and Terrence. Paige, Quincy, and Ronald have the following conversation:[*]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.Who received the card numbered
3
3
3
?
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<span class='latex-bold'>(A) </span> \text{Paige}\qquad<span class='latex-bold'>(B) </span> \text{Quincy}\qquad<span class='latex-bold'>(C) </span> \text{Ronald}\qquad<span class='latex-bold'>(D) </span> \text{Selena}\qquad<span class='latex-bold'>(E) </span> \text{Terrence}
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Terrence
12
1
Hide problems
2023 Individual #12
There are ten apples and
p
p
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
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<span class='latex-bold'>(A) </span> 19\qquad<span class='latex-bold'>(B) </span> 28\qquad<span class='latex-bold'>(C) </span> 30\qquad<span class='latex-bold'>(D) </span> 42\qquad<span class='latex-bold'>(E) </span> 45
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11
1
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2023 Individual #11
A semi-palindrome is a four-digit number whose first two digits and last two digits are identical. For instance,
2323
2323
2323
and
5757
5757
5757
are semi-palindromes, but
1001
1001
1001
and
2324
2324
2324
are not. What is the difference between the largest semi-palindrome and smallest semi-palindrome?
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8989
<span class='latex-bold'>(A) </span> 7979\qquad<span class='latex-bold'>(B) </span> 8080\qquad<span class='latex-bold'>(C) </span> 8181\qquad<span class='latex-bold'>(D) </span> 8484\qquad<span class='latex-bold'>(E) </span> 8989
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8989
10
1
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2023 Individual #10
In the equation below,
x
x
x
is a nonzero real number such that
1
729
(
3
t
)
=
3
x
.
\frac1{729}\left(3^t\right)=3^x.
729
1
(
3
t
)
=
3
x
.
Which of the following is equal to
t
t
t
?
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<span class='latex-bold'>(A) </span> \dfrac16x\qquad<span class='latex-bold'>(B) </span> \dfrac13x\qquad<span class='latex-bold'>(C) </span> 6x\qquad<span class='latex-bold'>(D) </span> x-6\qquad<span class='latex-bold'>(E) </span> x+6
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9
1
Hide problems
2023 Individual #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."
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prime number
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number divisible by 3
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number with four factors
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<span class='latex-bold'>(A) </span>\text{prime number}\qquad<span class='latex-bold'>(B) </span>\text{number divisible by 3}\qquad<span class='latex-bold'>(C) </span>\text{number with four factors}\qquad<span class='latex-bold'>(D) </span>2~\text{or}~3\qquad<span class='latex-bold'>(E) </span>4
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prime number
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number divisible by 3
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number with four factors
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8
1
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2023 Individual #8
Anna is buying fruits at a grocery store. If she loses a nickel, she still has enough money to buy exactly
16
16
16
lemons. Similarly, if she loses a quarter, she has enough money to buy exactly
14
14
14
lemons. What is the cost of each lemon?
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<span class='latex-bold'>(A) </span> \$0.05\qquad<span class='latex-bold'>(B) </span> \$0.10\qquad<span class='latex-bold'>(C) </span> \$0.15\qquad<span class='latex-bold'>(D) </span> \$0.20\qquad<span class='latex-bold'>(E) </span> \$0.25
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$0.25
7
1
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2023 Individual #7
At a length of
104
104
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
39
39
miles per hour across the Danyang-Kushan Bridge. How long does it take the car to travel across the entire bridge?
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2 hours, 12 minutes
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<span class='latex-bold'>(A) </span>\text{2 hours, 12 minutes} \qquad <span class='latex-bold'>(B) </span>\text{2 hours, 20 minutes} \qquad <span class='latex-bold'>(C) </span>\text{2 hours, 25 minutes}
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\\
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<span class='latex-bold'>(D) </span>\text{2 hours, 30 minutes} \qquad <span class='latex-bold'>(E) </span>\text{2 hours, 40 minutes}
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2 hours, 40 minutes
6
1
Hide problems
2023 Individual #6
If
a
+
3
b
=
9
a+3b = 9
a
+
3
b
=
9
and
a
+
11
b
=
21
a+11b =21
a
+
11
b
=
21
, what is the missing coefficient in the expression
2
a
+
?
‾
b
=
27
2a+\underline{?}b = 27
2
a
+
?
b
=
27
?
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5
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<span class='latex-bold'>(A) </span> 5\qquad<span class='latex-bold'>(B) </span> 6\qquad<span class='latex-bold'>(C) </span> 9\qquad<span class='latex-bold'>(D) </span> 12\qquad<span class='latex-bold'>(E) </span> 14
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14
5
1
Hide problems
2023 Individual #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]
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4
<span class='latex-bold'>(A) </span> 0\qquad<span class='latex-bold'>(B) </span> 1\qquad<span class='latex-bold'>(C) </span> 2\qquad<span class='latex-bold'>(D) </span> 3\qquad<span class='latex-bold'>(E) </span> 4
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4
1
Hide problems
2023 Individual #4
Which operation makes the following expression true:
(
4
‾
1
)
×
(
3
‾
2
−
1
)
=
2
(4 \underline{~~~~} 1) \times (3 \underline{~~~~} 2 - 1) = 2
(
4
1
)
×
(
3
2
−
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)
=
2
?
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There is no such operation
<span class='latex-bold'>(A) </span> +\qquad<span class='latex-bold'>(B) </span> -\qquad<span class='latex-bold'>(C) </span> \times\qquad<span class='latex-bold'>(D) </span> \div\qquad<span class='latex-bold'>(E) </span> \text{There is no such operation}
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There is no such operation
3
1
Hide problems
2023 Individual #3
A square with sides of length
6
6
6
has the same area as a rectangle with a length of
9
9
9
. What is the width of the rectangle?
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<span class='latex-bold'>(A) </span> 2\qquad<span class='latex-bold'>(B) </span> \frac73\qquad<span class='latex-bold'>(C) </span> 3\qquad<span class='latex-bold'>(D) </span> \frac{10}{3}\qquad<span class='latex-bold'>(E) </span> 4
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2
1
Hide problems
2023 Individual #2
What is the sum of all
x
x
x
that satisfy
∣
2
x
−
4
∣
=
2
|2x-4| = 2
∣2
x
−
4∣
=
2
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<span class='latex-bold'>(A) </span> 1\qquad<span class='latex-bold'>(B) </span> 2\qquad<span class='latex-bold'>(C) </span> 3\qquad<span class='latex-bold'>(D) </span> 4\qquad<span class='latex-bold'>(E) </span> 5
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5
1
1
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2023 Individual #1
What is
1
−
2
+
3
−
4
1-2+3-4
1
−
2
+
3
−
4
?
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<span class='latex-bold'>(A) </span> {-}2\qquad<span class='latex-bold'>(B) </span> {-}1\qquad<span class='latex-bold'>(C) </span> 1\qquad<span class='latex-bold'>(D) </span> 4\qquad<span class='latex-bold'>(E) </span> 9
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