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Purple Comet Problems
2019 Purple Comet Problems
2019 Purple Comet Problems
Part of
Purple Comet Problems
Subcontests
(30)
26
1
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Purple Comet 2019 HS problem 26
Let
D
D
D
be a regular dodecahedron, which is a polyhedron with
20
20
20
vertices,
30
30
30
edges, and
12
12
12
regular pentagon faces. A tetrahedron is a polyhedron with
4
4
4
vertices,
6
6
6
edges, and
4
4
4
triangular faces. Find the number of tetrahedra with positive volume whose vertices are vertices of
D
D
D
. https://cdn.artofproblemsolving.com/attachments/c/d/44d11fa3326780941d0b6756fb2e5989c2dc5a.png
27
1
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Purple Comet 2019 HS problem 27
Binhao has a fair coin. He writes the number
+
1
+1
+
1
on a blackboard. Then he flips the coin. If it comes up heads (H), he writes
+
1
2
+\frac12
+
2
1
, and otherwise, if he flips tails (T), he writes
−
1
2
-\frac12
−
2
1
. Then he flips the coin again. If it comes up heads, he writes
+
1
4
+\frac14
+
4
1
, and otherwise he writes
−
1
4
-\frac14
−
4
1
. Binhao continues to flip the coin, and on the nth flip, if he flips heads, he writes
+
1
2
n
+ \frac{1}{2n}
+
2
n
1
, and otherwise he writes
−
1
2
n
- \frac{1}{2n}
−
2
n
1
. For example, if Binhao flips HHTHTHT, he writes
1
+
1
2
+
1
4
−
1
8
+
1
16
−
1
32
+
1
64
−
1
128
1 + \frac12 + \frac14 - \frac18 + \frac{1}{16} -\frac{1}{32} + \frac{1}{64} -\frac{1}{128}
1
+
2
1
+
4
1
−
8
1
+
16
1
−
32
1
+
64
1
−
128
1
. The probability that Binhao will generate a series whose sum is greater than
1
7
\frac17
7
1
is
p
q
\frac{p}{q}
q
p
, where
p
p
p
and
q
q
q
are relatively prime positive integers. Find
p
+
10
q
p + 10q
p
+
10
q
.
30
1
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Purple Comet 2019 HS problem 30
A derangement of the letters
A
B
C
D
E
F
ABCDEF
A
BC
D
EF
is a permutation of these letters so that no letter ends up in the position it began such as
B
D
E
C
F
A
BDECFA
B
D
ECF
A
. An inversion in a permutation is a pair of letters
x
y
xy
x
y
where
x
x
x
appears before
y
y
y
in the original order of the letters, but
y
y
y
appears before
x
x
x
in the permutation. For example, the derangement
B
D
E
C
F
A
BDECFA
B
D
ECF
A
has seven inversions:
A
B
,
A
C
,
A
D
,
A
E
,
A
F
,
C
D
AB, AC, AD, AE, AF, CD
A
B
,
A
C
,
A
D
,
A
E
,
A
F
,
C
D
, and
C
E
CE
CE
. Find the total number of inversions that appear in all the derangements of
A
B
C
D
E
F
ABCDEF
A
BC
D
EF
.
29
1
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Purple Comet 2019 HS problem 29
In a right circular cone,
A
A
A
is the vertex,
B
B
B
is the center of the base, and
C
C
C
is a point on the circumference of the base with
B
C
=
1
BC = 1
BC
=
1
and
A
B
=
4
AB = 4
A
B
=
4
. There is a trapezoid
A
B
C
D
ABCD
A
BC
D
with
A
B
‾
∥
C
D
‾
\overline{AB} \parallel \overline{CD}
A
B
∥
C
D
. A right circular cylinder whose surface contains the points
A
,
C
A, C
A
,
C
, and
D
D
D
intersects the cone such that its axis of symmetry is perpendicular to the plane of the trapezoid, and
C
D
‾
\overline{CD}
C
D
is a diameter of the cylinder. A sphere radius
r
r
r
lies inside the cone and inside the cylinder. The greatest possible value of
r
r
r
is
a
b
−
c
d
\frac{a\sqrt{b}-c}{d}
d
a
b
−
c
, where
a
,
b
,
c
a, b, c
a
,
b
,
c
, and
d
d
d
are positive integers,
a
a
a
and
d
d
d
are relatively prime, and
b
b
b
is not divisible by the square of any prime. Find
a
+
b
+
c
+
d
a + b + c + d
a
+
b
+
c
+
d
.
28
1
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Purple Comet 2019 HS problem 28
There are positive integers
m
m
m
and
n
n
n
such that
m
2
−
n
=
32
m^2 -n = 32
m
2
−
n
=
32
and
m
+
n
5
+
m
−
n
5
\sqrt[5]{m +\sqrt{n}}+ \sqrt[5]{m -\sqrt{n}}
5
m
+
n
+
5
m
−
n
is a real root of the polynomial
x
5
−
10
x
3
+
20
x
−
40
x^5 - 10x^3 + 20x - 40
x
5
−
10
x
3
+
20
x
−
40
. Find
m
+
n
m + n
m
+
n
.
25
1
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Purple Comet 2019 HS problem 25
The letters
A
A
A
B
B
C
C
AAABBCC
AAA
BBCC
are arranged in random order. The probability no two adjacent letters will be the same is
m
n
\frac{m}{n}
n
m
, where
m
m
m
and
n
n
n
are relatively prime positive integers. Find
m
+
n
m + n
m
+
n
.
24
1
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Purple Comet 2019 HS problem 24
A
12
12
12
-sided polygon is inscribed in a circle with radius
r
r
r
. The polygon has six sides of length
6
3
6\sqrt3
6
3
that alternate with six sides of length
2
2
2
. Find
r
2
r^2
r
2
.
23
1
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Purple Comet 2019 HS problem 23
Find the number of ordered pairs of integers
(
x
,
y
)
(x, y)
(
x
,
y
)
such that
x
2
y
−
y
2
x
=
3
(
2
+
1
x
y
)
\frac{x^2}{y}- \frac{y^2}{x}= 3 \left( 2+ \frac{1}{xy}\right)
y
x
2
−
x
y
2
=
3
(
2
+
x
y
1
)
22
1
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Purple Comet 2019 HS problem 22
Let
a
a
a
and
b
b
b
positive real numbers such that
(
65
a
2
+
2
a
b
+
b
2
)
(
a
2
+
8
a
b
+
65
b
2
)
=
(
8
a
2
+
39
a
b
+
7
b
2
)
2
(65a^2 + 2ab + b^2)(a^2 + 8ab + 65b^2) = (8a^2 + 39ab + 7b^2)^2
(
65
a
2
+
2
ab
+
b
2
)
(
a
2
+
8
ab
+
65
b
2
)
=
(
8
a
2
+
39
ab
+
7
b
2
)
2
. Then one possible value of
a
b
\frac{a}{b}
b
a
satis es
2
(
a
b
)
=
m
+
n
2 \left(\frac{a}{b}\right) = m +\sqrt{n}
2
(
b
a
)
=
m
+
n
, where
m
m
m
and
n
n
n
are positive integers. Find
m
+
n
m + n
m
+
n
.
21
1
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Purple Comet 2019 HS problem 21
Each of the
48
48
48
faces of eight
1
×
1
×
1
1\times 1\times 1
1
×
1
×
1
cubes is randomly painted either blue or green. The probability that these eight cubes can then be assembled into a
2
×
2
×
2
2\times 2\times 2
2
×
2
×
2
cube in a way so that its surface is solid green can be written
p
m
q
n
\frac{p^m}{q^n}
q
n
p
m
, where
p
p
p
and
q
q
q
are prime numbers and
m
m
m
and
n
n
n
are positive integers. Find
p
+
q
+
m
+
n
p + q + m + n
p
+
q
+
m
+
n
.
9
2
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Purple Comet 2019 MS problem 9
A semicircle has diameter
A
D
‾
\overline{AD}
A
D
with
A
D
=
30
AD = 30
A
D
=
30
. Points
B
B
B
and
C
C
C
lie on
A
D
‾
\overline{AD}
A
D
, and points
E
E
E
and
F
F
F
lie on the arc of the semicircle. The two right triangles
△
B
C
F
\vartriangle BCF
△
BCF
and
△
C
D
E
\vartriangle CDE
△
C
D
E
are congruent. The area of
△
B
C
F
\vartriangle BCF
△
BCF
is
m
n
m\sqrt{n}
m
n
, where
m
m
m
and
n
n
n
are positive integers, and
n
n
n
is not divisible by the square of any prime. Find
m
+
n
m + n
m
+
n
. https://cdn.artofproblemsolving.com/attachments/b/c/c10258e2e15cab74abafbac5ff50b1d0fd42e6.png
Purple Comet 2019 HS problem 9
Find the positive integer
n
n
n
such that
32
32
32
is the product of the real number solutions of
x
log
2
(
x
3
)
−
n
=
13
x^{\log_2(x^3)-n} = 13
x
l
o
g
2
(
x
3
)
−
n
=
13
7
2
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Purple Comet 2019 MS problem 7
The diagram shows some squares whose sides intersect other squares at the midpoints of their sides. The shaded region has total area
7
7
7
. Find the area of the largest square. https://cdn.artofproblemsolving.com/attachments/3/a/c3317eefe9b0193ca15f36599be3f6c22bb099.png
Purple Comet 2019 HS problem 7
Find the number of real numbers
x
x
x
that satisfy the equation
(
3
x
)
x
+
2
+
(
4
x
)
x
+
2
−
(
6
x
)
x
+
2
=
1
(3^x)^{x+2} + (4^x)^{x+2} - (6^x)^{x+2} = 1
(
3
x
)
x
+
2
+
(
4
x
)
x
+
2
−
(
6
x
)
x
+
2
=
1
5
2
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Purple Comet 2019 MS problem 5
The diagram below shows four congruent squares and some of their diagonals. Let
T
T
T
be the number of triangles and
R
R
R
be the number of rectangles that appear in the diagram. Find
T
+
R
T + R
T
+
R
. https://cdn.artofproblemsolving.com/attachments/1/5/f756bbe67c09c19e811011cb6b18d0ff44be8b.png
Purple Comet 2019 HS problem 5
Evaluate
(
2
+
2
)
2
2
2
⋅
(
3
+
3
+
3
+
3
)
3
(
3
+
3
+
3
)
3
⋅
(
6
+
6
+
6
+
6
+
6
+
6
)
6
(
6
+
6
+
6
+
6
)
6
\frac{(2 + 2)^2}{2^2} \cdot \frac{(3 + 3 + 3 + 3)^3}{(3 + 3 + 3)^3} \cdot \frac{(6 + 6 + 6 + 6 + 6 + 6)^6}{(6 + 6 + 6 + 6)^6}
2
2
(
2
+
2
)
2
⋅
(
3
+
3
+
3
)
3
(
3
+
3
+
3
+
3
)
3
⋅
(
6
+
6
+
6
+
6
)
6
(
6
+
6
+
6
+
6
+
6
+
6
)
6
3
2
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Purple Comet 2019 MS problem 3
The diagram below shows a shaded region bounded by two concentric circles where the outer circle has twice the radius of the inner circle. The total boundary of the shaded region has length
36
π
36\pi
36
π
. Find
n
n
n
such that the area of the shaded region is
n
π
n\pi
nπ
. https://cdn.artofproblemsolving.com/attachments/4/5/c9ffdc41c633cc61127ef585a45ee5e6c0f88d.png
Purple Comet 2019 HS problem 3
The mean of
1
2
,
3
4
\frac12 , \frac34
2
1
,
4
3
, and
5
6
\frac56
6
5
differs from the mean of
7
8
\frac78
8
7
and
9
10
\frac{9}{10}
10
9
by
m
n
\frac{m}{n}
n
m
, where
m
m
m
and
n
n
n
are relatively prime positive integers. Find
m
+
n
m + n
m
+
n
.
1
2
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Purple Comet 2019 MS problem 1
The diagram shows a polygon made by removing six
2
×
2
2\times 2
2
×
2
squares from the sides of an
8
×
12
8\times 12
8
×
12
rectangle. Find the perimeter of this polygon. https://cdn.artofproblemsolving.com/attachments/6/3/c23510c821c159d31aff0e6688edebc81e2737.png
Purple Comet 2019 HS problem 1
Ivan, Stefan, and Katia divided
150
150
150
pieces of candy among themselves so that Stefan and Katia each got twice as many pieces as Ivan received. Find the number of pieces of candy Ivan received.
13
2
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Purple Comet 2019 MS problem 13
Squares
A
B
C
D
ABCD
A
BC
D
and
A
E
F
G
AEFG
A
EFG
each with side length
12
12
12
overlap so that
△
A
E
D
\vartriangle AED
△
A
E
D
is an equilateral triangle as shown. The area of the region that is in the interior of both squares which is shaded in the diagram is
m
n
m\sqrt{n}
m
n
, where
m
m
m
and
n
n
n
are positive integers, and
n
n
n
is not divisible by the square of any prime. Find
m
+
n
m + n
m
+
n
. https://cdn.artofproblemsolving.com/attachments/c/2/a2f8d2a090a6342610c43b3fed8a87fa5d7f03.png
Purple Comet 2019 HS problem 13
There are relatively prime positive integers
m
m
m
and
n
n
n
so that the parabola with equation
y
=
4
x
2
y = 4x^2
y
=
4
x
2
is tangent to the parabola with equation
x
=
y
2
+
m
n
x = y^2 + \frac{m}{n}
x
=
y
2
+
n
m
. Find
m
+
n
m + n
m
+
n
.
16
2
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Purple Comet 2019 MS problem 16
Four congruent semicircular half-disks are arranged inside a circle with radius
4
4
4
so that each semicircle is internally tangent to the circle, and the diameters of the semicircles form a
2
×
2
2\times 2
2
×
2
square centered at the center of the circle as shown. The radius of each semicircular half-disk is
m
n
\frac{m}{n}
n
m
, where
m
m
m
and
n
n
n
are relatively prime positive integers. Find
m
+
n
m + n
m
+
n
. https://cdn.artofproblemsolving.com/attachments/f/e/8c0b9fdd69f6b54d39708da94ef2b2d039cb1e.png
Purple Comet 2019 HS problem 16
Find the number of ordered triples of sets
(
T
1
,
T
2
,
T
3
)
(T_1, T_2, T_3)
(
T
1
,
T
2
,
T
3
)
such that 1. each of
T
1
,
T
2
T_1, T_2
T
1
,
T
2
, and
T
3
T_3
T
3
is a subset of
{
1
,
2
,
3
,
4
}
\{1, 2, 3, 4\}
{
1
,
2
,
3
,
4
}
, 2.
T
1
⊆
T
2
∪
T
3
T_1 \subseteq T_2 \cup T_3
T
1
⊆
T
2
∪
T
3
, 3.
T
2
⊆
T
1
∪
T
3
T_2 \subseteq T_1 \cup T_3
T
2
⊆
T
1
∪
T
3
, and 4.
T
3
⊆
T
1
∪
T
2
T_3\subseteq T_1 \cup T_2
T
3
⊆
T
1
∪
T
2
.
19
2
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Purple Comet 2019 MS problem 19
Rectangle
A
B
C
D
ABCD
A
BC
D
has sides
A
B
=
10
AB = 10
A
B
=
10
and
A
D
=
7
AD = 7
A
D
=
7
. Point
G
G
G
lies in the interior of
A
B
C
D
ABCD
A
BC
D
a distance
2
2
2
from side
C
D
‾
\overline{CD}
C
D
and a distance
2
2
2
from side
B
C
‾
\overline{BC}
BC
. Points
H
,
I
,
J
H, I, J
H
,
I
,
J
, and
K
K
K
are located on sides
B
C
‾
,
A
B
‾
,
A
D
‾
\overline{BC}, \overline{AB}, \overline{AD}
BC
,
A
B
,
A
D
, and
C
D
‾
\overline{CD}
C
D
, respectively, so that the path
G
H
I
J
K
G
GHIJKG
G
H
I
J
K
G
is as short as possible. Then
A
J
=
m
n
AJ = \frac{m}{n}
A
J
=
n
m
, where
m
m
m
and
n
n
n
are relatively prime positive integers. Find
m
+
n
m + n
m
+
n
.
Purple Comet 2019 HS problem 19
Find the remainder when
∏
n
=
3
33
2
n
4
−
25
n
3
+
33
n
2
\prod_{n=3}^{33}2n^4 - 25n^3 + 33n^2
∏
n
=
3
33
2
n
4
−
25
n
3
+
33
n
2
is divided by
2019
2019
2019
.
20
2
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Purple Comet 2019 MS problem 20
Harold has
3
3
3
red checkers and
3
3
3
black checkers. Find the number of distinct ways that Harold can place these checkers in stacks. Two ways of stacking checkers are the same if each stack of the rst way matches a corresponding stack in the second way in both size and color arrangement. So, for example, the
3
3
3
stack arrangement
R
B
R
,
B
R
,
B
RBR, BR, B
RBR
,
BR
,
B
is distinct from
R
B
R
,
R
B
,
B
RBR, RB, B
RBR
,
RB
,
B
, but the
4
4
4
stack arrangement
R
B
,
B
R
,
B
,
R
RB, BR, B, R
RB
,
BR
,
B
,
R
is the same as
B
,
B
R
,
R
,
R
B
B, BR, R, RB
B
,
BR
,
R
,
RB
.
Purple Comet 2019 HS problem 20
In the diagram below, points
D
,
E
D, E
D
,
E
, and
F
F
F
are on the inside of equilateral
△
A
B
C
\vartriangle ABC
△
A
BC
such that
D
D
D
is on
A
E
‾
,
E
\overline{AE}, E
A
E
,
E
is on
C
F
‾
,
F
\overline{CF}, F
CF
,
F
is on
B
D
‾
\overline{BD}
B
D
, and the triangles
△
A
E
C
,
△
B
D
A
\vartriangle AEC, \vartriangle BDA
△
A
EC
,
△
B
D
A
, and
△
C
F
B
\vartriangle CFB
△
CFB
are congruent. Given that
A
B
=
10
AB = 10
A
B
=
10
and
D
E
=
6
DE = 6
D
E
=
6
, the perimeter of
△
B
D
A
\vartriangle BDA
△
B
D
A
is
a
+
b
c
d
\frac{a+b\sqrt{c}}{d}
d
a
+
b
c
, where
a
,
b
,
c
a, b, c
a
,
b
,
c
, and
d
d
d
are positive integers,
b
b
b
and
d
d
d
are relatively prime, and
c
c
c
is not divisible by the square of any prime. Find
a
+
b
+
c
+
d
a + b + c + d
a
+
b
+
c
+
d
. https://cdn.artofproblemsolving.com/attachments/8/6/98da82fc1c26fa13883a47ba6d45a015622b20.png
18
2
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Purple Comet 2019 MS problem 18
Suppose that
a
,
b
,
c
a, b, c
a
,
b
,
c
, and
d
d
d
are real numbers simultaneously satisfying
a
+
b
−
c
−
d
=
3
a + b - c - d = 3
a
+
b
−
c
−
d
=
3
a
b
−
3
b
c
+
c
d
−
3
d
a
=
4
ab - 3bc + cd - 3da = 4
ab
−
3
b
c
+
c
d
−
3
d
a
=
4
3
a
b
−
b
c
+
3
c
d
−
d
a
=
5
3ab - bc + 3cd - da = 5
3
ab
−
b
c
+
3
c
d
−
d
a
=
5
Find
11
(
a
−
c
)
2
+
17
(
b
−
d
)
2
11(a - c)^2 + 17(b -d)^2
11
(
a
−
c
)
2
+
17
(
b
−
d
)
2
.
Purple Comet 2019 HS problem 18
A container contains five red balls. On each turn, one of the balls is selected at random, painted blue, and returned to the container. The expected number of turns it will take before all fi ve balls are colored blue is
m
n
\frac{m}{n}
n
m
, where
m
m
m
and
n
n
n
are relatively prime positive integers. Find
m
+
n
m + n
m
+
n
.
17
2
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Purple Comet 2019 MS problem 17
Find the greatest integer
n
n
n
such that
5
n
5^n
5
n
divides
2019
!
−
2018
!
+
2017
!
2019! - 2018! + 2017!
2019
!
−
2018
!
+
2017
!
.
Purple Comet 2019 HS problem 17
The following diagram shows equilateral triangle
△
A
B
C
\vartriangle ABC
△
A
BC
and three other triangles congruent to it. The other three triangles are obtained by sliding copies of
△
A
B
C
\vartriangle ABC
△
A
BC
a distance
1
8
A
B
\frac18 AB
8
1
A
B
along a side of
△
A
B
C
\vartriangle ABC
△
A
BC
in the directions from
A
A
A
to
B
B
B
, from
B
B
B
to
C
C
C
, and from
C
C
C
to
A
A
A
. The shaded region inside all four of the triangles has area
300
300
300
. Find the area of
△
A
B
C
\vartriangle ABC
△
A
BC
. https://cdn.artofproblemsolving.com/attachments/3/a/8d724563c7411547d3161076015b247e882122.png
15
2
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Purple Comet 2019 MS problem 15
Let
a
,
b
,
c
a, b, c
a
,
b
,
c
, and
d
d
d
be prime numbers with
a
≤
b
≤
c
≤
d
>
0
a \le b \le c \le d > 0
a
≤
b
≤
c
≤
d
>
0
. Suppose
a
2
+
2
b
2
+
c
2
+
2
d
2
=
2
(
a
b
+
b
c
−
c
d
+
d
a
)
a^2 + 2b^2 + c^2 + 2d^2 = 2(ab + bc - cd + da)
a
2
+
2
b
2
+
c
2
+
2
d
2
=
2
(
ab
+
b
c
−
c
d
+
d
a
)
. Find
4
a
+
3
b
+
2
c
+
d
4a + 3b + 2c + d
4
a
+
3
b
+
2
c
+
d
.
Purple Comet 2019 HS problem 15
Suppose
a
a
a
is a real number such that
sin
(
π
⋅
cos
a
)
=
cos
(
π
⋅
sin
a
)
\sin(\pi \cdot \cos a) = \cos(\pi \cdot \sin a)
sin
(
π
⋅
cos
a
)
=
cos
(
π
⋅
sin
a
)
. Evaluate
35
sin
2
(
2
a
)
+
84
cos
2
(
4
a
)
35 \sin^2(2a) + 84 \cos^2(4a)
35
sin
2
(
2
a
)
+
84
cos
2
(
4
a
)
.
14
2
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Purple Comet 2019 MS problem 14
For real numbers
a
a
a
and
b
b
b
, let
f
(
x
)
=
a
x
+
b
f(x) = ax + b
f
(
x
)
=
a
x
+
b
and
g
(
x
)
=
x
2
−
x
g(x) = x^2 - x
g
(
x
)
=
x
2
−
x
. Suppose that
g
(
f
(
2
)
)
=
2
,
g
(
f
(
3
)
)
=
0
g(f(2)) = 2, g(f(3)) = 0
g
(
f
(
2
))
=
2
,
g
(
f
(
3
))
=
0
, and
g
(
f
(
4
)
)
=
6
g(f(4)) = 6
g
(
f
(
4
))
=
6
. Find
g
(
f
(
5
)
)
g(f(5))
g
(
f
(
5
))
.
Purple Comet 2019 HS problem 14
The circle centered at point
A
A
A
with radius
19
19
19
and the circle centered at point
B
B
B
with radius
32
32
32
are both internally tangent to a circle centered at point
C
C
C
with radius
100
100
100
such that point
C
C
C
lies on segment
A
B
‾
\overline{AB}
A
B
. Point
M
M
M
is on the circle centered at
A
A
A
and point
N
N
N
is on the circle centered at
B
B
B
such that line
M
N
MN
MN
is a common internal tangent of those two circles. Find the distance
M
N
MN
MN
. https://cdn.artofproblemsolving.com/attachments/3/d/1933ce259c229d49e21b9a2dcadddea2a6b404.png
12
2
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Purple Comet 2019 MS problem 12
Find the number of ordered triples of positive integers
(
a
,
b
,
c
)
(a, b, c)
(
a
,
b
,
c
)
, where
a
,
b
,
c
a, b,c
a
,
b
,
c
is a strictly increasing arithmetic progression,
a
+
b
+
c
=
2019
a + b + c = 2019
a
+
b
+
c
=
2019
, and there is a triangle with side lengths
a
,
b
a, b
a
,
b
, and
c
c
c
.
Purple Comet 2019 HS problem 12
The following diagram shows four adjacent
2
×
2
2\times 2
2
×
2
squares labeled
1
,
2
,
3
1, 2, 3
1
,
2
,
3
, and
4
4
4
. A line passing through the lower left vertex of square
1
1
1
divides the combined areas of squares
1
,
3
1, 3
1
,
3
, and
4
4
4
in half so that the shaded region has area
6
6
6
. The difference between the areas of the shaded region within square
4
4
4
and the shaded region within square
1
1
1
is
p
q
\frac{p}{q}
q
p
, where
p
p
p
and
q
q
q
are relatively prime positive integers. Find
p
+
q
p + q
p
+
q
. https://cdn.artofproblemsolving.com/attachments/7/4/b9554ccd782af15c680824a1fbef278a4f736b.png
11
2
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Purple Comet 2019 MS problem 11
Find the number of positive integers less than or equal to
2019
2019
2019
that are no more than
10
10
10
away from a perfect square.
Purple Comet 2019 HS problem 11
Let
m
>
n
m > n
m
>
n
be positive integers such that
3
(
3
m
n
−
2
)
2
−
2
(
3
m
−
3
n
)
2
=
2019
3(3mn - 2)^2 - 2(3m -3n)^2 = 2019
3
(
3
mn
−
2
)
2
−
2
(
3
m
−
3
n
)
2
=
2019
. Find
3
m
+
n
3m + n
3
m
+
n
.
10
2
Hide problems
Purple Comet 2019 MS problem 10
Let N be the greatest positive integer that can be expressed using all seven Roman numerals
I
,
V
,
X
,
L
,
C
,
D
I, V, X, L, C,D
I
,
V
,
X
,
L
,
C
,
D
, and
M
M
M
exactly once each, and let n be the least positive integer that can be expressed using these numerals exactly once each. Find
N
−
n
N - n
N
−
n
. Note that the arrangement
C
M
CM
CM
is never used in a number along with the numeral
D
D
D
.
Purple Comet 2019 HS problem 10
Find the number of positive integers less than
2019
2019
2019
that are neither multiples of
3
3
3
nor have any digits that are multiples of
3
3
3
.
8
2
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Purple Comet 2019 MS problem 8
In the subtraction PURPLE
−
-
−
COMET
=
=
=
MEET each distinct letter represents a distinct decimal digit, and no leading digit is
0
0
0
. Find the greatest possible number represented by PURPLE.
Purple Comet 2019 HS problem 8
The diagram below shows a
12
12
12
by
20
20
20
rectangle split into four strips of equal widths all surrounding an isosceles triangle. Find the area of the shaded region. https://cdn.artofproblemsolving.com/attachments/9/e/ed6be5110d923965c64887a2ca8e858c977700.png
6
2
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Purple Comet 2019 MS problem 6
Find the value of
n
n
n
such that
2019
+
n
2019
−
n
=
5
\frac{2019 + n}{2019 - n}= 5
2019
−
n
2019
+
n
=
5
Purple Comet 2019 HS problem 6
A pentagon has four interior angles each equal to
11
0
o
110^o
11
0
o
. Find the degree measure of the fifth interior angle.
4
2
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Purple Comet 2019 MS problem 4
Of the students attending a school athletic event,
80
%
80\%
80%
of the boys were dressed in the school colors,
60
%
60\%
60%
of the girls were dressed in the school colors, and
45
%
45\%
45%
of the students were girls. Find the percentage of students attending the event who were wearing the school colors.
Purple Comet 2019 HS problem 4
The diagram below shows a sequence of equally spaced parallel lines with a triangle whose vertices lie on these lines. The segment
C
D
‾
\overline{CD}
C
D
is
6
6
6
units longer than the segment
A
B
‾
\overline{AB}
A
B
. Find the length of segment
E
F
‾
\overline{EF}
EF
. https://cdn.artofproblemsolving.com/attachments/8/0/abac87d63d366bf4c4e913fdb1022798379a73.png
2
2
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Purple Comet 2019 MS problem 2
Evaluate
1
+
2
−
3
−
4
+
5
+
6
−
7
−
8
+
.
.
.
+
2018
−
2019
1 + 2 - 3 - 4 + 5 + 6 - 7 - 8 + ... + 2018 - 2019
1
+
2
−
3
−
4
+
5
+
6
−
7
−
8
+
...
+
2018
−
2019
.
Purple Comet 2019 HS problem 2
The large square in the diagram below with sides of length
8
8
8
is divided into
16
16
16
congruent squares. Find the area of the shaded region. https://cdn.artofproblemsolving.com/attachments/6/e/cf828197aa2585f5eab2320a43b80616072135.png