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Math Prize For Girls Problems
2021 Math Prize for Girls Problems
2021 Math Prize for Girls Problems
Part of
Math Prize For Girls Problems
Subcontests
(20)
20
1
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Math Prize 2021 Problem 20
Let
G
G
G
be the set of points
(
x
,
y
)
(x, y)
(
x
,
y
)
such that
x
x
x
and
y
y
y
are positive integers less than or equal to 6. A magic grid is an assignment of an integer to each point in
G
G
G
such that, for every square with horizontal and vertical sides and all four vertices in
G
G
G
, the sum of the integers assigned to the four vertices is the same as the corresponding sum for any other such square. A magic grid is formed so that the product of all 36 integers is the smallest possible value greater than 1. What is this product?
19
1
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Math Prize 2021 Problem 19
Let
T
T
T
be a regular tetrahedron. Let
t
t
t
be the regular tetrahedron whose vertices are the centers of the faces of
T
T
T
. Let
O
O
O
be the circumcenter of either tetrahedron. Given a point
P
P
P
different from
O
O
O
, let
m
(
P
)
m(P)
m
(
P
)
be the midpoint of the points of intersection of the ray
O
P
→
\overrightarrow{OP}
OP
with
t
t
t
and
T
T
T
. Let
S
S
S
be the set of eight points
m
(
P
)
m(P)
m
(
P
)
where
P
P
P
is a vertex of either
t
t
t
or
T
T
T
. What is the volume of the convex hull of
S
S
S
divided by the volume of
t
t
t
?
18
1
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Math Prize 2021 Problem 18
Let
N
N
N
be the set of square-free positive integers less than or equal to 50. (A square-free number is an integer that is not divisible by a perfect square bigger than 1.) How many 3-element subsets
S
S
S
of
N
N
N
are there such that the greatest common divisor of all 3 numbers in
S
S
S
is 1, but no pair of numbers in
S
S
S
is relatively prime?
17
1
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Math Prize 2021 Problem 17
In the coordinate plane, let
A
=
(
−
8
,
0
)
A = (-8, 0)
A
=
(
−
8
,
0
)
,
B
=
(
8
,
0
)
B = (8, 0)
B
=
(
8
,
0
)
, and
C
=
(
t
,
6
)
C = (t, 6)
C
=
(
t
,
6
)
. What is the maximum value of
sin
m
∠
C
A
B
⋅
sin
m
∠
C
B
A
\sin m\angle CAB \cdot \sin m\angle CBA
sin
m
∠
C
A
B
⋅
sin
m
∠
CB
A
, over all real numbers
t
t
t
?
16
1
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Math Prize 2021 Problem 16
Let
G
G
G
be the set of points
(
x
,
y
)
(x, y)
(
x
,
y
)
such that
x
x
x
and
y
y
y
are positive integers less than or equal to 20. Say that a ray in the coordinate plane is ocular if it starts at
(
0
,
0
)
(0, 0)
(
0
,
0
)
and passes through at least one point in
G
G
G
. Let
A
A
A
be the set of angle measures of acute angles formed by two distinct ocular rays. Determine
min
a
∈
A
tan
a
.
\min_{a \in A} \tan a.
a
∈
A
min
tan
a
.
15
1
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Math Prize 2021 Problem 15
There are 300 points in space. Four planes
A
A
A
,
B
B
B
,
C
C
C
, and
D
D
D
each have the property that they split the 300 points into two equal sets. (No plane contains one of the 300 points.) What is the maximum number of points that can be found inside the tetrahedron whose faces are on
A
A
A
,
B
B
B
,
C
C
C
, and
D
D
D
?
14
1
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Math Prize 2021 Problem 14
Let
S
S
S
be the set of monic polynomials in
x
x
x
of degree 6 all of whose roots are members of the set
{
−
1
,
0
,
1
}
\{ -1, 0, 1\}
{
−
1
,
0
,
1
}
. Let
P
P
P
be the sum of the polynomials in
S
S
S
. What is the coefficient of
x
4
x^4
x
4
in
P
(
x
)
P(x)
P
(
x
)
?
13
1
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Math Prize 2021 Problem 13
There are 2021 light bulbs in a row, labeled 1 through 2021, each with an on/off switch. They all start in the off position when 1011 people walk by. The first person flips the switch on every bulb; the second person flips the switch on every 3rd bulb (bulbs 3, 6, etc.); the third person flips the switch on every 5th bulb; and so on. In general, the
k
k
k
th person flips the switch on every
(
2
k
−
1
)
(2k - 1)
(
2
k
−
1
)
th light bulb, starting with bulb
2
k
−
1
2k - 1
2
k
−
1
. After all 1011 people have gone by, how many light bulbs are on?
12
1
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Math Prize 2021 Problem 12
Let
P
1
P_1
P
1
,
P
2
P_2
P
2
,
P
3
P_3
P
3
,
P
4
P_4
P
4
,
P
5
P_5
P
5
, and
P
6
P_6
P
6
be six parabolas in the plane, each congruent to the parabola
y
=
x
2
/
16
y = x^2/16
y
=
x
2
/16
. The vertices of the six parabolas are evenly spaced around a circle. The parabolas open outward with their axes being extensions of six of the circle's radii. Parabola
P
1
P_1
P
1
is tangent to
P
2
P_2
P
2
, which is tangent to
P
3
P_3
P
3
, which is tangent to
P
4
P_4
P
4
, which is tangent to
P
5
P_5
P
5
, which is tangent to
P
6
P_6
P
6
, which is tangent to
P
1
P_1
P
1
. What is the diameter of the circle?
11
1
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Math Prize 2021 Problem 11
Say that a sequence
a
1
a_1
a
1
,
a
2
a_2
a
2
,
a
3
a_3
a
3
,
a
4
a_4
a
4
,
a
5
a_5
a
5
,
a
6
a_6
a
6
,
a
7
a_7
a
7
,
a
8
a_8
a
8
is cool if * the sequence contains each of the integers 1 through 8 exactly once, and * every pair of consecutive terms in the sequence are relatively prime. In other words,
a
1
a_1
a
1
and
a
2
a_2
a
2
are relatively prime,
a
2
a_2
a
2
and
a
3
a_3
a
3
are relatively prime,
…
\ldots
…
, and
a
7
a_7
a
7
and
a
8
a_8
a
8
are relatively prime.How many cool sequences are there?
10
1
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Math Prize 2021 Problem 10
Let
P
P
P
be the product of all the entries in row 2021 of Pascal's triangle (the row that begins 1, 2021,
…
\ldots
…
). What is the largest integer
j
j
j
such that
P
P
P
is divisible by
10
1
j
101^j
10
1
j
?
9
1
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Math Prize 2021 Problem 9
Let
H
H
H
be a regular hexagon with area 360. Three distinct vertices
X
X
X
,
Y
Y
Y
, and
Z
Z
Z
are picked randomly, with all possible triples of distinct vertices equally likely. Let
A
A
A
,
B
B
B
, and
C
C
C
be the unpicked vertices. What is the expected value (average value) of the area of the intersection of
△
A
B
C
\triangle ABC
△
A
BC
and
△
X
Y
Z
\triangle XYZ
△
X
Y
Z
?
8
1
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Math Prize 2021 Problem 8
In
△
A
B
C
\triangle ABC
△
A
BC
, let point
D
D
D
be on
B
C
‾
\overline{BC}
BC
such that the perimeters of
△
A
D
B
\triangle ADB
△
A
D
B
and
△
A
D
C
\triangle ADC
△
A
D
C
are equal. Let point
E
E
E
be on
A
C
‾
\overline{AC}
A
C
such that the perimeters of
△
B
E
A
\triangle BEA
△
BE
A
and
△
B
E
C
\triangle BEC
△
BEC
are equal. Let point
F
F
F
be the intersection of
A
B
‾
\overline{AB}
A
B
with the line that passes through
C
C
C
and the intersection of
A
D
‾
\overline{AD}
A
D
and
B
E
‾
\overline{BE}
BE
. Given that
B
D
=
10
BD = 10
B
D
=
10
,
C
D
=
2
CD = 2
C
D
=
2
, and
B
F
/
F
A
=
3
BF/FA = 3
BF
/
F
A
=
3
, what is the perimeter of
△
A
B
C
\triangle ABC
△
A
BC
?
7
1
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Math Prize 2021 Problem 7
Compute the value of the infinite series
∑
k
=
0
∞
cos
(
k
π
/
4
)
2
k
.
\sum_{k=0}^{\infty} \frac{\cos(k \pi / 4)}{2^k} \, .
k
=
0
∑
∞
2
k
cos
(
kπ
/4
)
.
6
1
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Math Prize 2021 Problem 6
The number
734
,
851
,
474
,
594
,
578
,
436
,
096
734{,}851{,}474{,}594{,}578{,}436{,}096
734
,
851
,
474
,
594
,
578
,
436
,
096
is equal to
n
6
n^6
n
6
for some positive integer
n
n
n
. What is the value of
n
n
n
?
5
1
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Math Prize 2021 Problem 5
Among all fractions (whose numerator and denominator are positive integers) strictly between
6
17
\tfrac{6}{17}
17
6
and
9
25
\tfrac{9}{25}
25
9
, which one has the smallest denominator?
4
1
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Math Prize 2021 Problem 4
For a positive integer
n
n
n
, let
v
(
n
)
v(n)
v
(
n
)
denote the largest integer
j
j
j
such that
n
n
n
is divisible by
2
j
2^j
2
j
. Let
a
a
a
and
b
b
b
be chosen uniformly and independently at random from among the integers between 1 and 32, inclusive. What is the probability that
v
(
a
)
>
v
(
b
)
v(a) > v(b)
v
(
a
)
>
v
(
b
)
?
3
1
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Math Prize 2021 Problem 3
Let
O
O
O
be the center of an equilateral triangle
A
B
C
ABC
A
BC
of area
1
/
π
1/\pi
1/
π
. As shown in the diagram below, a circle centered at
O
O
O
meets the triangle at points
D
D
D
,
E
E
E
,
F
F
F
,
G
G
G
,
H
H
H
, and
I
I
I
, which trisect each of the triangle's sides. Compute the total area of all six shaded regions. [asy] unitsize(90); pair A = dir(0); pair B = dir(120); pair C = dir(240); draw(A -- B -- C -- cycle); pair D = (2*A + B)/3; pair E = (A + 2*B)/3; pair F = (2*B + C)/3; pair G = (B + 2*C)/3; pair H = (2*C + A)/3; pair I = (C + 2*A)/3; draw(E -- F); draw(G -- H); draw(I -- D); draw(D -- G); draw(E -- H); draw(F -- I); pair O = (0, 0); real r = 1/sqrt(3); draw(circle(O, r)); fill(O -- D -- E -- cycle, gray); fill(O -- F -- G -- cycle, gray); fill(O -- H -- I -- cycle, gray); fill(arc(O, r, -30, 30) -- cycle, gray); fill(arc(0, r, 90, 150) -- cycle, gray); fill(arc(0, r, 210, 270) -- cycle, gray); label("
A
A
A
", A, A); label("
B
B
B
", B, B); label("
C
C
C
", C, C); label("
D
D
D
", D, unit(D)); label("
E
E
E
", E, unit(E)); label("
F
F
F
", F, unit(F)); label("
G
G
G
", G, unit(G)); label("
H
H
H
", H, unit(H)); label("
I
I
I
", I, unit(I)); label("
O
O
O
", O, C); [/asy]
2
1
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Math Prize 2021 Problem 2
Let
m
m
m
and
n
n
n
be positive integers such that
m
4
−
n
4
=
3439
m^4 - n^4 = 3439
m
4
−
n
4
=
3439
. What is the value of
m
n
mn
mn
?
1
1
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Math Prize 2021 Problem 1
A soccer coach named
C
C
C
does a header drill with two players
A
A
A
and
B
B
B
, but they all forgot to put sunscreen on their foreheads. They solve this issue by dunking the ball into a vat of sunscreen before starting the drill. Coach
C
C
C
heads the ball to
A
A
A
, who heads the ball back to
C
C
C
, who then heads the ball to
B
B
B
, who heads the ball back to
C
C
C
; this pattern
C
A
C
B
C
A
C
B
…
CACBCACB\ldots\,
C
A
CBC
A
CB
…
repeats ad infinitum. Each time a person heads the ball,
1
/
10
1/10
1/10
of the sunscreen left on the ball ends up on the person's forehead. In the limit, what fraction of the sunscreen originally on the ball will end up on the coach's forehead?