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National and Regional Contests
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USA - Middle School Tournaments
Math Invitational for Girls
2022 MIG
2022 MIG
Part of
Math Invitational for Girls
Subcontests
(25)
1
2
Hide problems
2022 A Individual #1
What is
4
0
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3
1
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2
−
1
3
4^0 - 3^1 - 2^2 - 1^3
4
0
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3
1
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2
2
−
1
3
?
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5
<span class='latex-bold'>(A) </span>{-}8\qquad<span class='latex-bold'>(B) </span>{-}7\qquad<span class='latex-bold'>(C) </span>{-}5\qquad<span class='latex-bold'>(D) </span>4\qquad<span class='latex-bold'>(E) </span>5
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2022 B Individual #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?
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<span class='latex-bold'>(A) </span>\$ 3\qquad<span class='latex-bold'>(B) </span>\$ 3.5\qquad<span class='latex-bold'>(C) </span>\$ 4\qquad<span class='latex-bold'>(D) </span>\$4.5\qquad<span class='latex-bold'>(E) </span>\$ 5
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2
2
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2022 A Individual #2
A machine takes
6
6
6
seconds to make
4
4
4
coins. How long does it take for the machine to make
22
22
22
coins? The machine makes coins at the same constant rate.
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<span class='latex-bold'>(A) </span>30\qquad<span class='latex-bold'>(B) </span>33\qquad<span class='latex-bold'>(C) </span>36\qquad<span class='latex-bold'>(D) </span>39\qquad<span class='latex-bold'>(E) </span>42
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2022 B Individual #2
Let
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x
be a number such that
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10000x+2=4
10000
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=
4
. What is the value of
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1
5000x+1
5000
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1
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<span class='latex-bold'>(A) </span>{-}1\qquad<span class='latex-bold'>(B) </span>0\qquad<span class='latex-bold'>(C) </span>1\qquad<span class='latex-bold'>(D) </span>2\qquad<span class='latex-bold'>(E) </span>3
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2
Hide problems
2022 A Individual #3
Jar
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A
and Jar
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each contain
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beans. The number of beans in jar
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A
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is doubled, and the number of beans in jar
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B
B
is halved. How many beans are now in jars
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A
A
and
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B
B
?
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<span class='latex-bold'>(A) </span>15\qquad<span class='latex-bold'>(B) </span>20\qquad<span class='latex-bold'>(C) </span>25\qquad<span class='latex-bold'>(D) </span>30\qquad<span class='latex-bold'>(E) </span>40
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2022 B Individual #3
Real numbers
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satisfy
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w+x+y+z = 5
w
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+
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. What is the value of
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+
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x+y
x
+
y
?
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<span class='latex-bold'>(A) </span>-\frac{1}{2}\qquad<span class='latex-bold'>(B) </span>1\qquad<span class='latex-bold'>(C) </span>\frac{3}{2}\qquad<span class='latex-bold'>(D) </span>2\qquad<span class='latex-bold'>(E) </span>3
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2
Hide problems
2022 A Individual #4
Which of the following answer choices is equivalent to
a
3
b
2
c
\sqrt{a^3b^2c}
a
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c
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<span class='latex-bold'>(A) </span>ab\sqrt{ac}\qquad<span class='latex-bold'>(B) </span>bc\sqrt{ac}\qquad<span class='latex-bold'>(C) </span>b\sqrt{ac}\qquad<span class='latex-bold'>(D) </span>abc\sqrt{ab}\qquad<span class='latex-bold'>(E) </span>a\sqrt{bc}
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2022 B Individual #4
There exists a real number
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n
n
such that
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=
5
4^n=5
4
n
=
5
. What is the value of
8
n
8^n
8
n
?
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5
<span class='latex-bold'>(A) </span>5\sqrt{5}\qquad<span class='latex-bold'>(B) </span>10\qquad<span class='latex-bold'>(C) </span>25\qquad<span class='latex-bold'>(D) </span>50\qquad<span class='latex-bold'>(E) </span>25\sqrt{5}
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5
5
2
Hide problems
2022 A Individual #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
×
1
−
2
4 + 3 \times 1 - 2
4
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3
×
1
−
2
?
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<span class='latex-bold'>(A) </span>{-}7\qquad<span class='latex-bold'>(B) </span>{-}5\qquad<span class='latex-bold'>(C) </span>5\qquad<span class='latex-bold'>(D) </span>7\qquad<span class='latex-bold'>(E) </span>9
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2022 B Individual #5
What is the perimeter of the smallest rectangle with integer side lengths that fits three non-overlapping squares with areas
4
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9
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4,9,
4
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9
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and
16
16
16
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<span class='latex-bold'>(A) </span>20\qquad<span class='latex-bold'>(B) </span>24\qquad<span class='latex-bold'>(C) </span>26\qquad<span class='latex-bold'>(D) </span>28\qquad<span class='latex-bold'>(E) </span>32
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6
2
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2022 A Individual #6
Two different
3
×
3
3 \times 3
3
×
3
grids are chosen within a
5
×
5
5 \times 5
5
×
5
grid. What is the least number of unit grids contained in the overlap of the two
3
×
3
3 \times 3
3
×
3
grids?
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<span class='latex-bold'>(A) </span>0\qquad<span class='latex-bold'>(B) </span>1\qquad<span class='latex-bold'>(C) </span>3\qquad<span class='latex-bold'>(D) </span>4\qquad<span class='latex-bold'>(E) </span>6
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2022 B Individual #6
A coin is flipped three times. What is the probability that there are no instances of two consecutive heads or two consecutive tails?
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<span class='latex-bold'>(A) </span>\frac{1}{8}\qquad<span class='latex-bold'>(B) </span>\frac{1}{4}\qquad<span class='latex-bold'>(C) </span>\frac{3}{8}\qquad<span class='latex-bold'>(D) </span>\frac{5}{8}\qquad<span class='latex-bold'>(E) </span>\frac{3}{4}
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3
7
2
Hide problems
2022 A Individual #7
Consider the rectangular strip of length
12
12
12
below, divided into three rectangles. The distance between the centers of two of the rectangles is
4
4
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]
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<span class='latex-bold'>(A) </span>2.5\qquad<span class='latex-bold'>(B) </span>3\qquad<span class='latex-bold'>(C) </span>3.5\qquad<span class='latex-bold'>(D) </span>4\qquad<span class='latex-bold'>(E) </span>4.5
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4.5
2022 B Individual #7
Alice, Bob, and Charlie are each thinking of a number. Alice's number differs from Bob's number by
2
2
2
. Bob's number differs from Charlie's number by
6
6
6
. Charlie's number differs from Alice's number by
N
N
N
. What is the sum of all possible values for
N
N
N
?
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<span class='latex-bold'>(A) </span>4\qquad<span class='latex-bold'>(B) </span>6\qquad<span class='latex-bold'>(C) </span>7\qquad<span class='latex-bold'>(D) </span>12\qquad<span class='latex-bold'>(E) </span>14
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20
2
Hide problems
2022 A Individual #20
The area of the dark gray triangle depicted below is
35
35
35
, and a segment is divided into lengths
14
14
14
and
10
10
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
14
14
", (7,0), S); label("
10
10
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]
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<span class='latex-bold'>(A) </span>84\qquad<span class='latex-bold'>(B) </span>120\qquad<span class='latex-bold'>(C) </span>132\qquad<span class='latex-bold'>(D) </span>144\qquad<span class='latex-bold'>(E) </span>168
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2022 B Individual #20
In the diagram below,
A
X
AX
A
X
is parallel to
B
Y
BY
B
Y
,
A
B
AB
A
B
is perpendicular to
B
Y
BY
B
Y
, and
A
Z
B
AZB
A
ZB
is an isosceles right triangle. If
A
B
=
7
AB = 7
A
B
=
7
and
X
Y
=
25
XY =25
X
Y
=
25
, what is the length of
A
X
AX
A
X
? [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
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B
",(0,0),S); label("
Y
Y
Y
",(-31/2,0),S); label("
A
A
A
",(0,7),N); label("
X
X
X
",(17/2,7),N); label("
Z
Z
Z
",(-7/2,7/2),N); [/asy]
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<span class='latex-bold'>(A) </span>\frac{17}{3}\qquad<span class='latex-bold'>(B) </span>\frac{17}{2}\qquad<span class='latex-bold'>(C) </span>9\qquad<span class='latex-bold'>(D) </span>\frac{51}{4}\qquad<span class='latex-bold'>(E) </span>12
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8
2
Hide problems
2022 A Individual #8
Write a list of the first
10
10
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?
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<span class='latex-bold'>(A) </span>2\qquad<span class='latex-bold'>(B) </span>3\qquad<span class='latex-bold'>(C) </span>4\qquad<span class='latex-bold'>(D) </span>5\qquad<span class='latex-bold'>(E) </span>6
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2022 B Individual #8
Let
A
B
C
ABC
A
BC
be a triangle and
D
D
D
be a point on segment
B
C
BC
BC
. If
△
A
B
D
\triangle ABD
△
A
B
D
is equilateral and
∠
A
C
B
=
1
4
∘
\angle ACB = 14^{\circ}
∠
A
CB
=
1
4
∘
, what is
∠
D
A
C
\angle{DAC}
∠
D
A
C
?
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6
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<span class='latex-bold'>(A) </span>26^{\circ}\qquad<span class='latex-bold'>(B) </span>34^{\circ}\qquad<span class='latex-bold'>(C) </span>46^{\circ}\qquad<span class='latex-bold'>(D) </span>50^{\circ}\qquad<span class='latex-bold'>(E) </span>54^{\circ}
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∘
9
2
Hide problems
2022 A Individual #9
How many integer values of
x
x
x
satisfy
3
2
<
9
x
<
7
3
?
\dfrac32 < \dfrac9x < \dfrac 73?
2
3
<
x
9
<
3
7
?
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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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2022 B Individual #9
A circle with area
36
π
\tfrac{36}{\pi}
π
36
has the same perimeter as a square with what side length?
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<span class='latex-bold'>(A) </span>\frac{9}{\pi}\qquad<span class='latex-bold'>(B) </span>3\qquad<span class='latex-bold'>(C) </span>\pi\qquad<span class='latex-bold'>(D) </span>6\qquad<span class='latex-bold'>(E) </span>\pi^2
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10
2
Hide problems
2022 A Individual #10
The diagram below shows a square of area
36
36
36
separated into two rectangles and a smaller square. One of the rectangles has an area of
12
12
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
12
12
",(1,3)); label("
?
?
?
",(4,4)); label("
?
?
?
",(4,1)); [/asy]
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Not Enough Information
<span class='latex-bold'>(A) </span>8\qquad<span class='latex-bold'>(B) </span>9\qquad<span class='latex-bold'>(C) </span>12\qquad<span class='latex-bold'>(D) </span>16\qquad<span class='latex-bold'>(E) </span>\text{Not Enough Information}
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Not Enough Information
2022 B Individual #10
What is the maximum possible value of
5
−
∣
6
x
−
80
∣
5-|6x-80|
5
−
∣6
x
−
80∣
over all integers
x
x
x
?
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<span class='latex-bold'>(A) </span>{-}1\qquad<span class='latex-bold'>(B) </span>0\qquad<span class='latex-bold'>(C) </span>1\qquad<span class='latex-bold'>(D) </span>3\qquad<span class='latex-bold'>(E) </span>5
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11
2
Hide problems
2022 A Individual #11
The sum of
n
n
n
consecutive integers is divisible by
n
n
n
for some
n
>
1
n > 1
n
>
1
. For which
n
n
n
is this always true?
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<span class='latex-bold'>(A) </span>\text{even }n\qquad<span class='latex-bold'>(B) </span>\text{odd }n\text{ divisible by }3\qquad<span class='latex-bold'>(C) </span>\text{odd }n\qquad<span class='latex-bold'>(D) </span>\text{prime }n\qquad<span class='latex-bold'>(E) </span>\text{no such }n\text{ exists}
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no such
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2022 B Individual #11
If
N
=
100
0
2
−
95
0
2
N=1000^2-950^2
N
=
100
0
2
−
95
0
2
, what is the largest prime factor of
N
N
N
?
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<span class='latex-bold'>(A) </span>5\qquad<span class='latex-bold'>(B) </span>13\qquad<span class='latex-bold'>(C) </span>17\qquad<span class='latex-bold'>(D) </span>19\qquad<span class='latex-bold'>(E) </span>29
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12
2
Hide problems
2022 A Individual #12
For a certain value of
x
x
x
, the sum of the digits of
1
0
x
−
100
10^x - 100
1
0
x
−
100
is equal to
45
45
45
. What is
x
x
x
?
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<span class='latex-bold'>(A) </span>4\qquad<span class='latex-bold'>(B) </span>5\qquad<span class='latex-bold'>(C) </span>6\qquad<span class='latex-bold'>(D) </span>7\qquad<span class='latex-bold'>(E) </span>8
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8
2022 B Individual #12
Out of a sample of
100
100
100
people,
24
24
24
do not like red or blue,
40
40
40
like both red and blue, and
50
50
50
people like red. How many people like blue but not red?
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<span class='latex-bold'>(A) </span>24\qquad<span class='latex-bold'>(B) </span>26\qquad<span class='latex-bold'>(C) </span>48\qquad<span class='latex-bold'>(D) </span>64\qquad<span class='latex-bold'>(E) </span>76
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13
2
Hide problems
2022 A Individual #13
Consider the numbers
1
1
1
through
6
6
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
1
1
",(0,0)); draw(Circle((0,1),0.45)); label("
2
2
2
",(0,1)); draw(Circle((0,2),0.45)); label("
3
3
3
",(0,2)); draw(Circle((1,0),0.45)); label("
5
5
5
",(1,0)); draw(Circle((1,1),0.45)); label("
4
4
4
",(1,1)); draw(Circle((2,0),0.45)); label("
6
6
6
",(2,0)); [/asy]
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<span class='latex-bold'>(A) </span>9\qquad<span class='latex-bold'>(B) </span>12\qquad<span class='latex-bold'>(C) </span>15\qquad<span class='latex-bold'>(D) </span>18\qquad<span class='latex-bold'>(E) </span>20
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2022 B Individual #13
Sarah is leading a class of
35
35
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
3
3
times, what is the greatest possible number of students that could have been standing before her third wave?
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<span class='latex-bold'>(A) </span>23\qquad<span class='latex-bold'>(B) </span>27\qquad<span class='latex-bold'>(C) </span>29\qquad<span class='latex-bold'>(D) </span>31\qquad<span class='latex-bold'>(E) </span>33
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14
2
Hide problems
2022 A Individual #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?
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<span class='latex-bold'>(A) </span>4\qquad<span class='latex-bold'>(B) </span>6\qquad<span class='latex-bold'>(C) </span>8\qquad<span class='latex-bold'>(D) </span>10\qquad<span class='latex-bold'>(E) </span>12
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2022 B Individual #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?
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<span class='latex-bold'>(A) </span>4\qquad<span class='latex-bold'>(B) </span>6\qquad<span class='latex-bold'>(C) </span>8\qquad<span class='latex-bold'>(D) </span>12\qquad<span class='latex-bold'>(E) </span>16
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15
2
Hide problems
2022 A Individual #15
A function
f
(
a
b
c
)
f(a \tfrac bc)
f
(
a
c
b
)
for a simplified mixed fraction
a
b
c
a \tfrac bc
a
c
b
returns
a
+
b
c
\tfrac{a + b}{c}
c
a
+
b
. For instance,
f
(
2
5
7
)
=
1
f(2 \tfrac 57) = 1
f
(
2
7
5
)
=
1
and
f
(
4
5
)
=
4
5
f(\tfrac45) = \tfrac45
f
(
5
4
)
=
5
4
. What is the sum of the three smallest positive rational
x
x
x
where
f
(
x
)
=
2
9
f(x) = \tfrac 29
f
(
x
)
=
9
2
?
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9
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<span class='latex-bold'>(A) </span>\dfrac52\qquad<span class='latex-bold'>(B) </span>\dfrac{68}{27}\qquad<span class='latex-bold'>(C) </span>\dfrac{23}{9}\qquad<span class='latex-bold'>(D) </span>\dfrac{74}{27}\qquad<span class='latex-bold'>(E) </span>\dfrac{13}4
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13
2022 B Individual #15
There exists a fraction
x
x
x
that satisfies
x
2
+
5
−
x
=
1
3
\sqrt{x^2+5} - x = \tfrac{1}{3}
x
2
+
5
−
x
=
3
1
. What is the sum of the numerator and denominator of this fraction?
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<span class='latex-bold'>(A) </span>8\qquad<span class='latex-bold'>(B) </span>21\qquad<span class='latex-bold'>(C) </span>25\qquad<span class='latex-bold'>(D) </span>32\qquad<span class='latex-bold'>(E) </span>34
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16
2
Hide problems
2022 A Individual #16
Let
P
P
P
be a point on side
A
B
‾
\overline{AB}
A
B
of equilateral triangle
A
B
C
ABC
A
BC
. If
B
P
=
6
BP = 6
BP
=
6
and
C
P
=
9
CP = 9
CP
=
9
, what is the length of
A
B
AB
A
B
?
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<span class='latex-bold'>(A) </span>2\sqrt5\qquad<span class='latex-bold'>(B) </span>3+\sqrt6\qquad<span class='latex-bold'>(C) </span>3\sqrt5\qquad<span class='latex-bold'>(D) </span>3\sqrt6 + 3\qquad<span class='latex-bold'>(E) </span>6\sqrt2
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2
2022 B Individual #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?
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<span class='latex-bold'>(A) </span>8\qquad<span class='latex-bold'>(B) </span>9\qquad<span class='latex-bold'>(C) </span>10\qquad<span class='latex-bold'>(D) </span>12\qquad<span class='latex-bold'>(E) </span>14
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2022 A Individual #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?
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<span class='latex-bold'>(A) </span>16\qquad<span class='latex-bold'>(B) </span>18\qquad<span class='latex-bold'>(C) </span>32\qquad<span class='latex-bold'>(D) </span>162\qquad<span class='latex-bold'>(E) </span>512
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2022 B Individual #17
What is the value of
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(\sqrt{2}-1)^4+\frac{1}{(\sqrt{2}-1)^4}?
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<span class='latex-bold'>(A) </span>32-16\sqrt{2}\qquad<span class='latex-bold'>(B) </span>30\qquad<span class='latex-bold'>(C) </span>34\qquad<span class='latex-bold'>(D) </span>15+15\sqrt{2}\qquad<span class='latex-bold'>(E) </span>16+16\sqrt{2}
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2022 A Individual #18
If the six-digit number
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\underline{2}\, \underline{0}\, \underline{2} \, \underline{1} \, \underline{a} \, \underline{b}
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<span class='latex-bold'>(A) </span>18\qquad<span class='latex-bold'>(B) </span>20\qquad<span class='latex-bold'>(C) </span>36\qquad<span class='latex-bold'>(D) </span>40\qquad<span class='latex-bold'>(E) </span>42
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2022 B Individual #18
Two equilateral triangles are glued, and their opposite vertices are connected. If the larger equilateral triangle has an area of
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and the smaller equilateral triangle has an area of
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, 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]
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<span class='latex-bold'>(A) </span>60\qquad<span class='latex-bold'>(B) </span>90\qquad<span class='latex-bold'>(C) </span>96\qquad<span class='latex-bold'>(D) </span>108\qquad<span class='latex-bold'>(E) </span>120
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2022 A Individual #19
A three-digit number
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times the sum of its digits. Find the sum of all possible values of
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N
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<span class='latex-bold'>(A) </span>576\qquad<span class='latex-bold'>(B) </span>648\qquad<span class='latex-bold'>(C) </span>972\qquad<span class='latex-bold'>(D) </span>1152\qquad<span class='latex-bold'>(E) </span>1620
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2022 B Individual #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
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12.
12.
Which of the following could be the total number of students and teachers on the table?
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<span class='latex-bold'>(A) </span>14\qquad<span class='latex-bold'>(B) </span>15\qquad<span class='latex-bold'>(C) </span>16\qquad<span class='latex-bold'>(D) </span>17\qquad<span class='latex-bold'>(E) </span>18
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21
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2022 A Individual #21
Let
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T(p)
T
(
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denote the number of right triangles with integer side lengths and one of its side lengths being
p
p
p
. Which of the following values of
p
p
p
produces the greatest possible value of
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(
p
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T(p)
T
(
p
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among all five answer choices?
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<span class='latex-bold'>(A) </span>24\qquad<span class='latex-bold'>(B) </span>27\qquad<span class='latex-bold'>(C) </span>28\qquad<span class='latex-bold'>(D) </span>36\qquad<span class='latex-bold'>(E) </span>54
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2022 B Individual #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: [*]It has
3
3
3
digits. [*]It is a palindrome. [*]The middle digit is a multiple of
4
4
4
. [*]It is divisible by
11
11
11
.What is the sum of all possible numbers of cows that the herder has?
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<span class='latex-bold'>(A) </span>343\qquad<span class='latex-bold'>(B) </span>494\qquad<span class='latex-bold'>(C) </span>615\qquad<span class='latex-bold'>(D) </span>635\qquad<span class='latex-bold'>(E) </span>726
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22
2
Hide problems
2022 A Individual #22
Jerry and Aaron both pick two integers from
1
1
1
to
6
6
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?
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<span class='latex-bold'>(A) </span>4\qquad<span class='latex-bold'>(B) </span>5\qquad<span class='latex-bold'>(C) </span>6\qquad<span class='latex-bold'>(D) </span>7\qquad<span class='latex-bold'>(E) </span>8
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2022 B Individual #22
How many ways are there to color each of the
8
8
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]
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<span class='latex-bold'>(A) </span>48\qquad<span class='latex-bold'>(B) </span>50\qquad<span class='latex-bold'>(C) </span>52\qquad<span class='latex-bold'>(D) </span>54\qquad<span class='latex-bold'>(E) </span>56
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23
2
Hide problems
2022 A Individual #23
Elax creates a partially filled
4
×
4
4 \times 4
4
×
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
S
S
. Given that the sum of the numbers in the top row is also
S
S
S
, what is the missing cell number?[asy] size(100); add(grid(4,4)); label("
11
11
11
", (0.5,1.5)); label("
10
10
10
", (0.5,2.5)); label("?", (0.5,3.5)); label("
8
8
8
", (1.5,3.5)); label("
7
7
7
", (2.5,2.5)); label("
4
4
4
", (3.5,0.5)); label("
9
9
9
", (3.5,1.5)); [/asy]
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<span class='latex-bold'>(A) </span>4\qquad<span class='latex-bold'>(B) </span>6\qquad<span class='latex-bold'>(C) </span>9\qquad<span class='latex-bold'>(D) </span>10\qquad<span class='latex-bold'>(E) </span>12
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2022 B Individual #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?
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24
2
Hide problems
2022 A Individual #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
π
34\pi
34
π
square meters and Beta can wander in a region with an area of
40
π
40\pi
40
π
square meters. What is the area enclosed by the rectangular fence?
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<span class='latex-bold'>(A) </span>45\qquad<span class='latex-bold'>(B) </span>48\qquad<span class='latex-bold'>(C) </span>96\qquad<span class='latex-bold'>(D) </span>120\qquad<span class='latex-bold'>(E) </span>144
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2022 B Individual #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.
x^3 - 3.9 x^2 + 4.4 x - 1.2.
x
3
−
3.9
x
2
+
4.4
x
−
1.2.
What is the length of the inradius of the triangle?
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<span class='latex-bold'>(A) </span>\frac{3}{13}\qquad<span class='latex-bold'>(B) </span>\frac{3}{11}\qquad<span class='latex-bold'>(C) </span>\frac{2}{7}\qquad<span class='latex-bold'>(D) </span>\frac{8}{15}\qquad<span class='latex-bold'>(E) </span>\frac{9}{14}
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9
25
2
Hide problems
2022 A Individual #25
For all positive integers
a
>
1
a > 1
a
>
1
, there are divisors of
2021
a
2021a
2021
a
that are not divisors of
2021
2021
2021
. If there are twelve unshared divisors, including
2021
a
2021a
2021
a
, which of the following answer choices could be a possible value of
a
a
a
?
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<span class='latex-bold'>(A) </span>9\qquad<span class='latex-bold'>(B) </span>10\qquad<span class='latex-bold'>(C) </span>16\qquad<span class='latex-bold'>(D) </span>18\qquad<span class='latex-bold'>(E) </span>19
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2022 B Individual #25
In the below diagram, the three rectangles are similar. Find the area of rectangle
A
B
C
D
ABCD
A
BC
D
.[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
A
A
",(0,8.16496580928), NW); label("
D
D
D
",(0,0), SW); label("
C
C
C
",(10,0), SE); label("
B
B
B
",(10,8.16496580928), NE); label("
E
E
E
",(2,8.16496580928), N); label("
F
F
F
",(2,0), S); label("
3
3
3
",(1,0),S); label("
3
3
3
",(1,8.16496580928),N); label("
12
12
12
",(6,8.16496580928),N); [/asy]
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<span class='latex-bold'>(A) </span>75\sqrt{3}\qquad<span class='latex-bold'>(B) </span>120\sqrt{2}\qquad<span class='latex-bold'>(C) </span>100\sqrt{3}\qquad<span class='latex-bold'>(D) </span>180\qquad<span class='latex-bold'>(E) </span>75\sqrt{6}
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6