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Math Prize For Girls Problems
2022 Math Prize for Girls Problems
2022 Math Prize for Girls Problems
Part of
Math Prize For Girls Problems
Subcontests
(20)
6
1
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Math Prize 2022 Problem 6
An L-shaped region is formed by attaching two
2
2
2
by
5
5
5
rectangles to adjacent sides of a
2
2
2
by
2
2
2
square as shown below. [asy] size(6cm); draw((0,0)--(7,0)--(7,2)--(2,2)--(2,7)--(0,7)--cycle); real eps = 0.45; draw(box( (0,0), (eps,eps) )); draw(box( (7,0), (7-eps,eps) )); draw(box( (7,2), (7-eps,2-eps) )); draw(box( (0,7), (eps,7-eps) )); draw(box( (2,7), (2-eps,7-eps) )); label("
7
7
7
", (0,3.5), dir(180)); label("
7
7
7
", (3.5,0), dir(270)); label("
2
2
2
", (7,1), dir(0)); label("
5
5
5
", (4.5,2), dir(90)); label("
5
5
5
", (2,4.5), dir(0)); label("
2
2
2
", (1,7), dir(90)); [/asy] The resulting shape has an area of
24
24
24
square units. How many ways are there to tile this shape with
2
2
2
by
1
1
1
dominos (each of which may be placed horizontally or vertically)?
18
1
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Math Prize 2022 Problem 18
Let
A
A
A
be the locus of points
(
α
,
β
,
γ
)
(\alpha, \beta, \gamma)
(
α
,
β
,
γ
)
in the
α
β
γ
\alpha\beta\gamma
α
β
γ
-coordinate space that satisfy the following properties:(I) We have
α
\alpha
α
,
β
\beta
β
,
γ
>
0
\gamma > 0
γ
>
0
. (II) We have
α
+
β
+
γ
=
π
\alpha + \beta + \gamma = \pi
α
+
β
+
γ
=
π
. (III) The intersection of the three cylinders in the
x
y
z
xyz
x
yz
-coordinate space given by the equations \begin{eqnarray*} y^2 + z^2 & = & \sin^2 \alpha \\ z^2 + x^2 & = & \sin^2 \beta \\ x^2 + y^2 & = & \sin^2 \gamma \end{eqnarray*} is nonempty. Determine the area of
A
A
A
.
20
1
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Math Prize 2022 Problem 20
Let
a
n
=
n
(
2
n
+
1
)
a_n = n(2n+1)
a
n
=
n
(
2
n
+
1
)
. Evaluate
∣
∑
1
≤
j
<
k
≤
36
sin
(
π
6
(
a
k
−
a
j
)
)
∣
.
\biggl | \sum_{1 \le j < k \le 36} \sin\bigl( \frac{\pi}{6}(a_k-a_j) \bigr) \biggr |.
1
≤
j
<
k
≤
36
∑
sin
(
6
π
(
a
k
−
a
j
)
)
.
19
1
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Math Prize 2022 Problem 19
Let
S
−
S_-
S
−
be the semicircular arc defined by
(
x
+
1
)
2
+
(
y
−
3
2
)
2
=
1
4
and
x
≤
−
1.
(x + 1)^2 + (y - \frac{3}{2})^2 = \frac{1}{4} \text{ and } x \le -1.
(
x
+
1
)
2
+
(
y
−
2
3
)
2
=
4
1
and
x
≤
−
1.
Let
S
+
S_+
S
+
be the semicircular arc defined by
(
x
−
1
)
2
+
(
y
−
3
2
)
2
=
1
4
and
x
≥
1.
(x - 1)^2 + (y - \frac{3}{2})^2 = \frac{1}{4} \text{ and } x \ge 1.
(
x
−
1
)
2
+
(
y
−
2
3
)
2
=
4
1
and
x
≥
1.
Let
R
R
R
be the locus of points
P
P
P
such that
P
P
P
is the intersection of two lines, one of the form
A
x
+
B
y
=
1
Ax + By = 1
A
x
+
B
y
=
1
where
(
A
,
B
)
∈
S
−
(A, B) \in S_-
(
A
,
B
)
∈
S
−
and the other of the form
C
x
+
D
y
=
1
Cx + Dy = 1
C
x
+
Dy
=
1
where
(
C
,
D
)
∈
S
+
(C, D) \in S_+
(
C
,
D
)
∈
S
+
. What is the area of
R
R
R
?
17
1
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Math Prize 2022 Problem 17
Let
O
O
O
be the set of odd numbers between 0 and 100. Let
T
T
T
be the set of subsets of
O
O
O
of size
25
25
25
. For any finite subset of integers
S
S
S
, let
P
(
S
)
P(S)
P
(
S
)
be the product of the elements of
S
S
S
. Define
n
=
∑
S
∈
T
P
(
S
)
n=\textstyle{\sum_{S \in T}} P(S)
n
=
∑
S
∈
T
P
(
S
)
. If you divide
n
n
n
by 17, what is the remainder?
16
1
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Math Prize 2022 Problem 16
A snail begins a journey starting at the origin of a coordinate plane. The snail moves along line segments of length
10
\sqrt{10}
10
and in any direction such that the horizontal and vertical displacements are both integers. As the snail moves, it leaves a trail tracing out its entire journey. After a while, this trail can form various polygons. What is the smallest possible area of a polygon that could be created by the snail's trail?
15
1
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Math Prize 2022 Problem 15
What is the smallest positive integer
m
m
m
such that
15
!
m
15! \, m
15
!
m
can be expressed in more than one way as a product of
16
16
16
distinct positive integers, up to order?
14
1
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Math Prize 2022 Problem 14
Across the face of a rectangular post-it note, you idly draw lines that are parallel to its edges. Each time you draw a line, there is a
50
%
50\%
50%
chance it'll be in each direction and you never draw over an existing line or the edge of the post-it note. After a few minutes, you notice that you've drawn 20 lines. What is the expected number of rectangles that the post-it note will be partitioned into?
13
1
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Math Prize 2022 Problem 13
The roots of the polynomial
x
4
−
4
i
x
3
+
3
x
2
−
14
i
x
−
44
x^4 - 4ix^3 +3x^2 -14ix - 44
x
4
−
4
i
x
3
+
3
x
2
−
14
i
x
−
44
form the vertices of a parallelogram in the complex plane. What is the area of the parallelogram?
12
1
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Math Prize 2022 Problem 12
Solve the equation
sin
9
∘
sin
2
1
∘
sin
(
10
2
∘
+
x
∘
)
=
sin
3
0
∘
sin
4
2
∘
sin
x
∘
\sin 9^\circ \sin 21^\circ \sin(102^\circ + x^\circ) = \sin 30^\circ \sin 42^\circ \sin x^\circ
sin
9
∘
sin
2
1
∘
sin
(
10
2
∘
+
x
∘
)
=
sin
3
0
∘
sin
4
2
∘
sin
x
∘
for
x
x
x
where
0
<
x
<
90
0 < x < 90
0
<
x
<
90
.
11
1
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Math Prize 2022 Problem 11
Let
A
A
A
,
B
B
B
,
C
C
C
,
D
D
D
,
E
E
E
, and
F
F
F
be
6
6
6
points around a circle, listed in clockwise order. We have
A
B
=
3
2
AB = 3\sqrt{2}
A
B
=
3
2
,
B
C
=
3
3
BC = 3\sqrt{3}
BC
=
3
3
,
C
D
=
6
6
CD = 6\sqrt{6}
C
D
=
6
6
,
D
E
=
4
2
DE = 4\sqrt{2}
D
E
=
4
2
, and
E
F
=
5
2
EF = 5\sqrt{2}
EF
=
5
2
. Given that
A
D
‾
\overline{AD}
A
D
,
B
E
‾
\overline{BE}
BE
, and
C
F
‾
\overline{CF}
CF
are concurrent, determine the square of
A
F
AF
A
F
.
10
1
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Math Prize 2022 Problem 10
An algal cell population is found to have
a
k
a_k
a
k
cells on day
k
k
k
. Each day, the number of cells at least doubles. If
a
0
≥
1
a_0 \ge 1
a
0
≥
1
and
a
3
≤
60
a_3 \le 60
a
3
≤
60
, how many quadruples of integers
(
a
0
,
a
1
,
a
2
,
a
3
)
(a_0, a_1, a_2, a_3)
(
a
0
,
a
1
,
a
2
,
a
3
)
could represent the algal cell population size on the first
4
4
4
days?
9
1
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Math Prize 2022 Problem 9
Let
△
P
Q
O
\triangle PQO
△
PQO
be the unique right isosceles triangle inscribed in the parabola
y
=
12
x
2
y = 12x^2
y
=
12
x
2
with
P
P
P
in the first quadrant, right angle at
Q
Q
Q
in the second quadrant, and
O
O
O
at the vertex
(
0
,
0
)
(0, 0)
(
0
,
0
)
. Let
△
A
B
V
\triangle ABV
△
A
B
V
be the unique right isosceles triangle inscribed in the parabola
y
=
x
2
/
5
+
1
y = x^2/5 + 1
y
=
x
2
/5
+
1
with
A
A
A
in the first quadrant, right angle at
B
B
B
in the second quadrant, and
V
V
V
at the vertex
(
0
,
1
)
(0, 1)
(
0
,
1
)
. The
y
y
y
-coordinate of
A
A
A
can be uniquely written as
u
q
2
+
v
q
+
w
uq^2 + vq + w
u
q
2
+
v
q
+
w
, where
q
q
q
is the
x
x
x
-coordinate of
Q
Q
Q
and
u
u
u
,
v
v
v
, and
w
w
w
are integers. Determine
u
+
v
+
w
u + v + w
u
+
v
+
w
.
8
1
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Math Prize 2022 Problem 8
Let
S
S
S
be the set of numbers of the form
n
5
−
5
n
3
+
4
n
n^5 - 5n^3 + 4n
n
5
−
5
n
3
+
4
n
, where
n
n
n
is an integer that is not a multiple of
3
3
3
. What is the largest integer that is a divisor of every number in
S
S
S
?
7
1
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Math Prize 2022 Problem 7
The quadrilateral
A
B
C
D
ABCD
A
BC
D
is an isosceles trapezoid with
A
B
=
C
D
=
1
AB = CD = 1
A
B
=
C
D
=
1
,
B
C
=
2
BC = 2
BC
=
2
, and
D
A
=
1
+
3
DA = 1+ \sqrt{3}
D
A
=
1
+
3
. What is the measure of
∠
A
C
D
\angle ACD
∠
A
C
D
in degrees?
5
1
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Math Prize 2022 Problem 5
Given a real number
a
a
a
, the floor of
a
a
a
, written
⌊
a
⌋
\lfloor a \rfloor
⌊
a
⌋
, is the greatest integer less than or equal to
a
a
a
. For how many real numbers
x
x
x
such that
1
≤
x
≤
20
1 \le x \le 20
1
≤
x
≤
20
is
x
2
+
⌊
2
x
⌋
=
⌊
x
2
⌋
+
2
x
?
x^2 + \lfloor 2x \rfloor = \lfloor x^2 \rfloor + 2x \, ?
x
2
+
⌊
2
x
⌋
=
⌊
x
2
⌋
+
2
x
?
4
1
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Math Prize 2022 Problem 4
Determine the largest integer
n
n
n
such that
n
<
103
n < 103
n
<
103
and
n
3
−
1
n^3 - 1
n
3
−
1
is divisible by
103
103
103
.
3
1
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Math Prize 2022 Problem 3
Let
A
B
C
D
ABCD
A
BC
D
be a square face of a cube with edge length
2
2
2
. A plane
P
P
P
that contains
A
A
A
and the midpoint of
B
C
‾
\overline{BC}
BC
splits the cube into two pieces of the same volume. What is the square of the area of the intersection of
P
P
P
and the cube?
2
1
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Math Prize 2022 Problem 2
Let
b
b
b
and
c
c
c
be random integers from the set
{
1
,
2
,
…
,
100
}
\{1, 2, \ldots, 100\}
{
1
,
2
,
…
,
100
}
, chosen uniformly and independently. What is the probability that the roots of the quadratic
x
2
+
b
x
+
c
x^2 + bx + c
x
2
+
b
x
+
c
are real?
1
1
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Math Prize 2022 Problem 1
Determine the real value of
t
t
t
that minimizes the expression
t
2
+
(
t
2
−
1
)
2
+
(
t
−
14
)
2
+
(
t
2
−
46
)
2
.
\sqrt{t^2 + (t^2 - 1)^2} + \sqrt{(t-14)^2 + (t^2 - 46)^2}.
t
2
+
(
t
2
−
1
)
2
+
(
t
−
14
)
2
+
(
t
2
−
46
)
2
.