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Math Open At Andover problems
2019 MOAA
2019 MOAA
Part of
Math Open At Andover problems
Subcontests
(14)
Speed
1
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2019 MOAA Speed Round - Math Open At Andover
p1. What is
20
×
19
+
20
÷
(
2
−
7
)
20\times 19 + 20 \div (2 - 7)
20
×
19
+
20
÷
(
2
−
7
)
? p2. Will has three spinners. The first has three equally sized sections numbered
1
1
1
,
2
2
2
,
3
3
3
; the second has four equally sized sections numbered
1
1
1
,
2
2
2
,
3
3
3
,
4
4
4
; and the third has five equally sized sections numbered
1
1
1
,
2
2
2
,
3
3
3
,
4
4
4
,
5
5
5
. When Will spins all three spinners, the probability that the same number appears on all three spinners is
p
p
p
. Compute
1
p
\frac{1}{p}
p
1
.p3. Three girls and five boys are seated randomly in a row of eight desks. Let
p
p
p
be the probability that the students at the ends of the row are both boys. If
p
p
p
can be expressed in the form
m
n
\frac{m}{n}
n
m
for relatively prime positive integers
m
m
m
and
n
n
n
, compute
m
+
n
m + n
m
+
n
. p4. Jaron either hits a home run or strikes out every time he bats. Last week, his batting average was
.
300
.300
.300
. (Jaron's batting average is the number of home runs he has hit divided by the number of times he has batted.) After hitting
10
10
10
home runs and striking out zero times in the last week, Jaron has now raised his batting average to
.
310
.310
.310
. How many home runs has Jaron now hit? p5. Suppose that the sum
1
1
⋅
4
+
1
4
⋅
7
+
.
.
.
+
1
97
⋅
100
\frac{1}{1 \cdot 4} +\frac{1}{4 \cdot 7}+ ...+\frac{1}{97 \cdot 100}
1
⋅
4
1
+
4
⋅
7
1
+
...
+
97
⋅
100
1
is expressible as
m
n
\frac{m}{n}
n
m
for relatively prime positive integers
m
m
m
and
n
n
n
. Compute
m
+
n
m + n
m
+
n
. p6. Let
A
B
C
D
ABCD
A
BC
D
be a unit square with center
O
O
O
, and
△
O
E
F
\vartriangle OEF
△
OEF
be an equilateral triangle with center
A
A
A
. Suppose that
M
M
M
is the area of the region inside the square but outside the triangle and
N
N
N
is the area of the region inside the triangle but outside the square, and let
x
=
∣
M
−
N
∣
x = |M -N|
x
=
∣
M
−
N
∣
be the positive difference between
M
M
M
and
N
N
N
. If
x
=
1
8
(
p
−
q
)
x =\frac1 8(p -\sqrt{q})
x
=
8
1
(
p
−
q
)
for positive integers
p
p
p
and
q
q
q
, find
p
+
q
p + q
p
+
q
. p7. Find the number of seven-digit numbers such that the sum of any two consecutive digits is divisible by
3
3
3
. For example, the number
1212121
1212121
1212121
satisfies this property. p8. There is a unique positive integer
x
x
x
such that
x
x
x^x
x
x
has
703
703
703
positive factors. What is
x
x
x
? p9. Let
x
x
x
be the number of digits in
2
2019
2^{2019}
2
2019
and let
y
y
y
be the number of digits in
5
2019
5^{2019}
5
2019
. Compute
x
+
y
x + y
x
+
y
. p10. Let
A
B
C
ABC
A
BC
be an isosceles triangle with
A
B
=
A
C
=
13
AB = AC = 13
A
B
=
A
C
=
13
and
B
C
=
10
BC = 10
BC
=
10
. Consider the set of all points
D
D
D
in three-dimensional space such that
B
C
D
BCD
BC
D
is an equilateral triangle. This set of points forms a circle
ω
\omega
ω
. Let
E
E
E
and
F
F
F
be points on
ω
\omega
ω
such that
A
E
AE
A
E
and
A
F
AF
A
F
are tangent to
ω
\omega
ω
. If
E
F
2
EF^2
E
F
2
can be expressed in the form
m
n
\frac{m}{n}
n
m
, where
m
m
m
and
n
n
n
are relatively prime positive integers, determine
m
+
n
m + n
m
+
n
. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.
Accuracy
1
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2019 MOAA Accuracy Round - Math Open At Andover
p1. Farmer John wants to bring some cows to a pasture with grass that grows at a constant rate. Initially, the pasture has some nonzero amount of grass and it will stop growing if there is no grass left. The pasture sustains
100
100
100
cows for ten days. The pasture can also sustain
100
100
100
cows for five days, and then
120
120
120
cows for three more days. If cows eat at a constant rate, fund the maximum number of cows Farmer John can bring to the pasture so that they can be sustained indefinitely. p2. Sam is learning basic arithmetic. He may place either the operation
+
+
+
or
−
-
−
in each of the blank spots between the numbers below:
5
_
8
_
9
_
7
_
2
_
3
5\,\, \_ \,\, 8\,\, \_ \,\,9\,\, \_ \,\,7\,\,\_ \,\,2\,\,\_ \,\,3
5
_
8
_
9
_
7
_
2
_
3
In how many ways can he place the operations so the result is divisible by
3
3
3
? p3. Will loves the color blue, but he despises the color red. In the
5
×
6
5\times 6
5
×
6
rectangular grid below, how many rectangles are there containing at most one red square and with sides contained in the gridlines? https://cdn.artofproblemsolving.com/attachments/1/7/7ce55bdc9e05c7c514dddc7f8194f3031b93c4.pngp4. Let
r
1
,
r
2
,
r
3
r_1, r_2, r_3
r
1
,
r
2
,
r
3
be the three roots of a cubic polynomial
P
(
x
)
P(x)
P
(
x
)
. Suppose that
P
(
2
)
+
P
(
−
2
)
P
(
0
)
=
200.
\frac{P(2) + P(-2)}{P(0)}= 200.
P
(
0
)
P
(
2
)
+
P
(
−
2
)
=
200.
If
1
r
1
r
2
+
1
r
2
r
3
+
1
r
3
r
1
=
m
n
\frac{1}{r_1r_2}+ \frac{1}{r_2r_3}+\frac{1}{r_3r_1}= \frac{m}{n}
r
1
r
2
1
+
r
2
r
3
1
+
r
3
r
1
1
=
n
m
for relatively prime positive integers
m
m
m
and
n
n
n
, compute
m
+
n
m + n
m
+
n
. p5. Consider a rectangle
A
B
C
D
ABCD
A
BC
D
with
A
B
=
3
AB = 3
A
B
=
3
and
B
C
=
1
BC = 1
BC
=
1
. Let
O
O
O
be the intersection of diagonals
A
C
AC
A
C
and
B
D
BD
B
D
. Suppose that the circumcircle of
△
A
D
O
\vartriangle ADO
△
A
D
O
intersects line
A
B
AB
A
B
again at
E
≠
A
E \ne A
E
=
A
. Then, the length
B
E
BE
BE
can be written as
m
n
\frac{m}{n}
n
m
for relatively prime positive integers
m
m
m
and
n
n
n
. Find
m
+
n
m + n
m
+
n
. p6. Let
A
B
C
D
ABCD
A
BC
D
be a square with side length
100
100
100
and
M
M
M
be the midpoint of side
A
B
AB
A
B
. The circle with center
M
M
M
and radius
50
50
50
intersects the circle with center
D
D
D
and radius
100
100
100
at point
E
E
E
.
C
E
CE
CE
intersects
A
B
AB
A
B
at
F
F
F
. If
A
F
=
m
n
AF = \frac{m}{n}
A
F
=
n
m
for relatively prime positive integers
m
m
m
and
n
n
n
, find
m
+
n
m + n
m
+
n
. p7. How many pairs of real numbers
(
x
,
y
)
(x, y)
(
x
,
y
)
, with
0
<
x
,
y
<
1
0 < x, y < 1
0
<
x
,
y
<
1
satisfy the property that both
3
x
+
5
y
3x + 5y
3
x
+
5
y
and
5
x
+
2
y
5x + 2y
5
x
+
2
y
are integers? p8. Sebastian is coloring a circular spinner with
4
4
4
congruent sections. He randomly chooses one of four colors for each of the sections. If two or more adjacent sections have the same color, he fuses them and considers them as one section. (Sections meeting at only one point are not adjacent.) Suppose that the expected number of sections in the final colored spinner is equal to
m
n
\frac{m}{n}
n
m
for relatively prime positive integers
m
m
m
and
n
n
n
. Compute
m
+
n
m + n
m
+
n
. p9. Let
A
B
C
ABC
A
BC
be a triangle and
D
D
D
be a point on the extension of segment
B
C
BC
BC
past
C
C
C
. Let the line through
A
A
A
perpendicular to
B
C
BC
BC
be
ℓ
\ell
ℓ
. The line through
B
B
B
perpendicular to
A
D
AD
A
D
and the line through
C
C
C
perpendicular to
A
D
AD
A
D
intersect
ℓ
\ell
ℓ
at
H
1
H_1
H
1
and
H
2
H_2
H
2
, respectively. If
A
B
=
13
AB = 13
A
B
=
13
,
B
C
=
14
BC = 14
BC
=
14
,
C
A
=
15
CA = 15
C
A
=
15
, and
H
1
H
2
=
1001
H_1H_2 = 1001
H
1
H
2
=
1001
, find
C
D
CD
C
D
. p10. Find the sum of all positive integers
k
k
k
such that \frac21 -\frac{3}{2 \times 1}+\frac{4}{3\times 2\times 1} + ...+ (-1)^{k+1} \frac{k+1}{k\times (k - 1)\times ... \times 2\times 1} \ge 1 + \frac{1}{700^3} PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.
Sets 6-9
1
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2019 MOAA Gunga Bowl - Math Open At Andover - last 4 sets - 12 problems
Set 6 p16. Let
n
!
=
n
×
(
n
−
1
)
×
.
.
.
×
2
×
1
n! = n \times (n - 1) \times ... \times 2 \times 1
n
!
=
n
×
(
n
−
1
)
×
...
×
2
×
1
. Find the maximum positive integer value of
x
x
x
such that the quotient
160
!
16
0
x
\frac{160!}{160^x}
16
0
x
160
!
is an integer. p17. Let
△
O
A
B
\vartriangle OAB
△
O
A
B
be a triangle with
∠
O
A
B
=
9
0
o
\angle OAB = 90^o
∠
O
A
B
=
9
0
o
. Draw points
C
,
D
,
E
,
F
,
G
C, D, E, F, G
C
,
D
,
E
,
F
,
G
in its plane so that
△
O
A
B
∼
△
O
B
C
∼
△
O
C
D
∼
△
O
D
E
∼
△
O
E
F
∼
△
O
F
G
,
\vartriangle OAB \sim \vartriangle OBC \sim \vartriangle OCD \sim \vartriangle ODE \sim \vartriangle OEF \sim \vartriangle OFG,
△
O
A
B
∼
△
OBC
∼
△
OC
D
∼
△
O
D
E
∼
△
OEF
∼
△
OFG
,
and none of these triangles overlap. If points
O
,
A
,
G
O, A, G
O
,
A
,
G
lie on the same line, then let
x
x
x
be the sum of all possible values of
O
G
O
A
\frac{OG}{OA }
O
A
OG
. Then,
x
x
x
can be expressed in the form
m
/
n
m/n
m
/
n
for relatively prime positive integers
m
,
n
m, n
m
,
n
. Compute
m
+
n
m + n
m
+
n
. p18. Let
f
(
x
)
f(x)
f
(
x
)
denote the least integer greater than or equal to
x
x
x^{\sqrt{x}}
x
x
. Compute
f
(
1
)
+
f
(
2
)
+
f
(
3
)
+
f
(
4
)
f(1)+f(2)+f(3)+f(4)
f
(
1
)
+
f
(
2
)
+
f
(
3
)
+
f
(
4
)
. Set 7 The Fibonacci sequence
{
F
n
}
\{F_n\}
{
F
n
}
is defined as
F
0
=
0
F_0 = 0
F
0
=
0
,
F
1
=
1
F_1 = 1
F
1
=
1
and
F
n
+
2
=
F
n
+
1
+
F
n
F_{n+2} = F_{n+1} + F_n
F
n
+
2
=
F
n
+
1
+
F
n
for all integers
n
≥
0
n \ge 0
n
≥
0
.p19. Find the least odd prime factor of
(
F
3
)
20
+
(
F
4
)
20
+
(
F
5
)
20
(F_3)^{20} + (F_4)^{20} + (F_5)^{20}
(
F
3
)
20
+
(
F
4
)
20
+
(
F
5
)
20
. p20. Let
S
=
1
F
3
F
5
+
1
F
4
F
6
+
1
F
5
F
7
+
1
F
6
F
8
+
.
.
.
S = \frac{1}{F_3F_5}+\frac{1}{F_4F_6}+\frac{1}{F_5F_7}+\frac{1}{F_6F_8}+...
S
=
F
3
F
5
1
+
F
4
F
6
1
+
F
5
F
7
1
+
F
6
F
8
1
+
...
Compute
420
S
420S
420
S
. p21. Consider the number
Q
=
0.000101020305080130210340550890144...
,
Q = 0.000101020305080130210340550890144... ,
Q
=
0.000101020305080130210340550890144...
,
the decimal created by concatenating every Fibonacci number and placing a 0 right after the decimal point and between each Fibonacci number. Find the greatest integer less than or equal to
1
Q
\frac{1}{Q}
Q
1
. Set 8 p22. In five dimensional hyperspace, consider a hypercube
C
0
C_0
C
0
of side length
2
2
2
. Around it, circumscribe a hypersphere
S
0
S_0
S
0
, so all
32
32
32
vertices of
C
0
C_0
C
0
are on the surface of
S
0
S_0
S
0
. Around
S
0
S_0
S
0
, circumscribe a hypercube
C
1
C_1
C
1
, so that
S
0
S_0
S
0
is tangent to all hyperfaces of
C
1
C_1
C
1
. Continue in this same fashion for
S
1
S_1
S
1
,
C
2
C_2
C
2
,
S
2
S_2
S
2
, and so on. Find the side length of
C
4
C_4
C
4
. p23. Suppose
△
A
B
C
\vartriangle ABC
△
A
BC
satisfies
A
C
=
10
2
AC = 10\sqrt2
A
C
=
10
2
,
B
C
=
15
BC = 15
BC
=
15
,
∠
C
=
4
5
o
\angle C = 45^o
∠
C
=
4
5
o
. Let
D
,
E
,
F
D, E, F
D
,
E
,
F
be the feet of the altitudes in
△
A
B
C
\vartriangle ABC
△
A
BC
, and let
U
,
V
,
W
U, V , W
U
,
V
,
W
be the points where the incircle of
△
D
E
F
\vartriangle DEF
△
D
EF
is tangent to the sides of
△
D
E
F
\vartriangle DEF
△
D
EF
. Find the area of
△
U
V
W
\vartriangle UVW
△
U
VW
. p24. A polynomial
P
(
x
)
P(x)
P
(
x
)
is called spicy if all of its coefficients are nonnegative integers less than
9
9
9
. How many spicy polynomials satisfy
P
(
3
)
=
2019
P(3) = 2019
P
(
3
)
=
2019
? The next set will consist of three estimation problems. Set 9 Points will be awarded based on the formulae below. Answers are nonnegative integers that may exceed
1
,
000
,
000
1,000,000
1
,
000
,
000
.p25. Suppose a circle of radius
20192019
20192019
20192019
has area
A
A
A
. Let s be the side length of a square with area
A
A
A
. Compute the greatest integer less than or equal to
s
s
s
.If
n
n
n
is the correct answer, an estimate of
e
e
e
gives
max
{
0
,
⌊
1030
(
m
i
n
{
n
e
,
e
n
}
18
⌋
−
1000
}
\max \{ 0, \left\lfloor 1030 ( min \{ \frac{n}{e},\frac{e}{n}\}^{18}\right\rfloor -1000 \}
max
{
0
,
⌊
1030
(
min
{
e
n
,
n
e
}
18
⌋
−
1000
}
points. p26. Given a
50
×
50
50 \times 50
50
×
50
grid of squares, initially all white, define an operation as picking a square and coloring it and the four squares horizontally or vertically adjacent to it blue, if they exist. If a square is already colored blue, it will remain blue if colored again. What is the minimum number of operations necessary to color the entire grid blue?If
n
n
n
is the correct answer, an estimate of
e
e
e
gives
⌊
180
5
∣
n
−
e
∣
+
6
⌋
\left\lfloor \frac{180}{5|n-e|+6}\right\rfloor
⌊
5∣
n
−
e
∣
+
6
180
⌋
points. p27. The sphere packing problem asks what percent of space can be filled with equally sized spheres without overlap. In three dimensions, the answer is
π
3
2
≈
74.05
%
\frac{\pi}{3\sqrt2} \approx 74.05\%
3
2
π
≈
74.05%
of space (confirmed as recently as
2017
!
2017!
2017
!
), so we say that the packing density of spheres in three dimensions is about
0.74
0.74
0.74
. In fact, mathematicians have found optimal packing densities for certain other dimensions as well, one being eight-dimensional space. Let d be the packing density of eight-dimensional hyperspheres in eightdimensional hyperspace. Compute the greatest integer less than
1
0
8
×
d
10^8 \times d
1
0
8
×
d
.If
n
n
n
is the correct answer, an estimate of e gives
max
{
⌊
30
−
1
0
−
5
∣
n
−
e
∣
⌋
,
0
}
\max \left\{ \lfloor 30-10^{-5}|n - e|\rfloor, 0 \right\}
max
{
⌊
30
−
1
0
−
5
∣
n
−
e
∣
⌋
,
0
}
points.PS. You had better use hide for answers. First sets have be posted [url=https://artofproblemsolving.com/community/c4h2777330p24370124]here. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.
Sets 1-5
1
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2019 MOAA Gunga Bowl - Math Open At Andover - first 5 sets - 15 problems
Set 1 p1. Farmer John has
4000
4000
4000
gallons of milk in a bucket. On the first day, he withdraws
10
%
10\%
10%
of the milk in the bucket for his cows. On each following day, he withdraws a percentage of the remaining milk that is
10
%
10\%
10%
more than the percentage he withdrew on the previous day. For example, he withdraws
20
%
20\%
20%
of the remaining milk on the second day. How much milk, in gallons, is left after the tenth day? p2. Will multiplies the first four positive composite numbers to get an answer of
w
w
w
. Jeremy multiplies the first four positive prime numbers to get an answer of
j
j
j
. What is the positive difference between
w
w
w
and
j
j
j
? p3. In Nathan’s math class of
60
60
60
students,
75
%
75\%
75%
of the students like dogs and
60
%
60\%
60%
of the students like cats. What is the positive difference between the maximum possible and minimum possible number of students who like both dogs and cats? Set 2 p4. For how many integers
x
x
x
is
x
4
−
1
x^4 - 1
x
4
−
1
prime? p5. Right triangle
△
A
B
C
\vartriangle ABC
△
A
BC
satisfies
∠
B
A
C
=
9
0
o
\angle BAC = 90^o
∠
B
A
C
=
9
0
o
. Let
D
D
D
be the foot of the altitude from
A
A
A
to
B
C
BC
BC
. If
A
D
=
60
AD = 60
A
D
=
60
and
A
B
=
65
AB = 65
A
B
=
65
, find the area of
△
A
B
C
\vartriangle ABC
△
A
BC
. p6. Define
n
!
=
n
×
(
n
−
1
)
×
.
.
.
×
1
n! = n \times (n - 1) \times ... \times 1
n
!
=
n
×
(
n
−
1
)
×
...
×
1
. Given that
3
!
+
4
!
+
5
!
=
a
2
+
b
2
+
c
2
3! + 4! + 5! = a^2 + b^2 + c^2
3
!
+
4
!
+
5
!
=
a
2
+
b
2
+
c
2
for distinct positive integers
a
,
b
,
c
a, b, c
a
,
b
,
c
, find
a
+
b
+
c
a + b + c
a
+
b
+
c
. Set 3 p7. Max nails a unit square to the plane. Let M be the number of ways to place a regular hexagon (of any size) in the same plane such that the square and hexagon share at least
2
2
2
vertices. Vincent, on the other hand, nails a regular unit hexagon to the plane. Let
V
V
V
be the number of ways to place a square (of any size) in the same plane such that the square and hexagon share at least
2
2
2
vertices. Find the nonnegative difference between
M
M
M
and
V
V
V
. p8. Let a be the answer to this question, and suppose
a
>
0
a > 0
a
>
0
. Find
a
+
a
+
a
+
.
.
.
\sqrt{a +\sqrt{a +\sqrt{a +...}}}
a
+
a
+
a
+
...
. p9. How many ordered pairs of integers
(
x
,
y
)
(x, y)
(
x
,
y
)
are there such that
x
2
−
y
2
=
2019
x^2 - y^2 = 2019
x
2
−
y
2
=
2019
? Set 4 p10. Compute
p
3
+
q
3
+
r
3
−
3
p
q
r
p
+
q
+
r
\frac{p^3 + q^3 + r^3 - 3pqr}{p + q + r}
p
+
q
+
r
p
3
+
q
3
+
r
3
−
3
pq
r
where
p
=
17
p = 17
p
=
17
,
q
=
7
q = 7
q
=
7
, and
r
=
8
r = 8
r
=
8
. p11. The unit squares of a
3
×
3
3 \times 3
3
×
3
grid are colored black and white. Call a coloring good if in each of the four
2
×
2
2 \times 2
2
×
2
squares in the
3
×
3
3 \times 3
3
×
3
grid, there is either exactly one black square or exactly one white square. How many good colorings are there? Consider rotations and reflections of the same pattern distinct colorings. p12. Define a
k
k
k
-respecting string as a sequence of
k
k
k
consecutive positive integers
a
1
a_1
a
1
,
a
2
a_2
a
2
,
.
.
.
...
...
,
a
k
a_k
a
k
such that
a
i
a_i
a
i
is divisible by
i
i
i
for each
1
≤
i
≤
k
1 \le i \le k
1
≤
i
≤
k
. For example,
7
7
7
,
8
8
8
,
9
9
9
is a
3
3
3
-respecting string because
7
7
7
is divisible by
1
1
1
,
8
8
8
is divisible by
2
2
2
, and
9
9
9
is divisible by
3
3
3
. Let
S
7
S_7
S
7
be the set of the first terms of all
7
7
7
-respecting strings. Find the sum of the three smallest elements in
S
7
S_7
S
7
. Set 5 p13. A triangle and a quadrilateral are situated in the plane such that they have a finite number of intersection points
I
I
I
. Find the sum of all possible values of
I
I
I
. p14. Mr. DoBa continuously chooses a positive integer at random such that he picks the positive integer
N
N
N
with probability
2
−
N
2^{-N}
2
−
N
, and he wins when he picks a multiple of 10. What is the expected number of times Mr. DoBa will pick a number in this game until he wins? p15. If
a
,
b
,
c
,
d
a, b, c, d
a
,
b
,
c
,
d
are all positive integers less than
5
5
5
, not necessarily distinct, find the number of ordered quadruples
(
a
,
b
,
c
,
d
)
(a, b, c, d)
(
a
,
b
,
c
,
d
)
such that
a
b
−
c
d
a^b - c^d
a
b
−
c
d
is divisible by
5
5
5
. PS. You had better use hide for answers. Last 4 sets have been posted [url=https://artofproblemsolving.com/community/c4h2777362p24370554]here. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.
8
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2019 MOAA Team P8
Suppose that
(
2
)
5
+
1
2
+
1
×
2
5
+
1
2
+
1
×
4
5
+
1
4
+
1
×
1
6
5
+
1
16
+
1
=
m
7
+
3
2
\frac{(\sqrt2)^5 + 1}{\sqrt2 + 1} \times \frac{2^5 + 1}{2 + 1} \times \frac{4^5 + 1}{4 + 1} \times \frac{16^5 + 1}{16 + 1} =\frac{m}{7 + 3\sqrt2}
2
+
1
(
2
)
5
+
1
×
2
+
1
2
5
+
1
×
4
+
1
4
5
+
1
×
16
+
1
1
6
5
+
1
=
7
+
3
2
m
for some integer
m
m
m
. How many
0
0
0
’s are in the binary representation of
m
m
m
? (For example, the number
20
=
1010
0
2
20 = 10100_2
20
=
1010
0
2
has three
0
0
0
’s in its binary representation.)
6
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2019 MOAA Team P6
Let
f
(
x
,
y
)
=
⌊
5
x
2
y
⌋
+
⌈
5
y
2
x
⌉
f(x, y) = \left\lfloor \frac{5x}{2y} \right\rfloor + \left\lceil \frac{5y}{2x} \right\rceil
f
(
x
,
y
)
=
⌊
2
y
5
x
⌋
+
⌈
2
x
5
y
⌉
. Suppose
x
,
y
x, y
x
,
y
are chosen independently uniformly at random from the interval
(
0
,
1
]
(0, 1]
(
0
,
1
]
. Let
p
p
p
be the probability that
f
(
x
,
y
)
<
6
f(x, y) < 6
f
(
x
,
y
)
<
6
. If
p
p
p
can be expressed in the form
m
/
n
m/n
m
/
n
for relatively prime positive integers
m
m
m
and
n
n
n
, compute
m
+
n
m + n
m
+
n
.(Note:
⌊
x
⌋
\lfloor x\rfloor
⌊
x
⌋
is defined as the greatest integer less than or equal to
x
x
x
and
⌈
x
⌉
\lceil x \rceil
⌈
x
⌉
is defined as the least integer greater than or equal to
x
x
x
.)
10
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2019 MOAA Team P10
Let
S
S
S
be the set of all four digit palindromes (a palindrome is a number that reads the same forwards and backwards). The average value of
∣
m
−
n
∣
|m - n|
∣
m
−
n
∣
over all ordered pairs
(
m
,
n
)
(m, n)
(
m
,
n
)
, where
m
m
m
and
n
n
n
are (not necessarily distinct) elements of
S
S
S
, is equal to
p
/
q
p/q
p
/
q
, for relatively prime positive integers
p
p
p
and
q
q
q
. Find
p
+
q
p + q
p
+
q
.
9
1
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2019 MOAA Team P9
Jonathan finds all ordered triples
(
a
,
b
,
c
)
(a, b, c)
(
a
,
b
,
c
)
of positive integers such that
a
b
c
=
720
abc = 720
ab
c
=
720
. For each ordered triple, he writes their sum
a
+
b
+
c
a + b + c
a
+
b
+
c
on the board. (Numbers may appear more than once.) What is the sum of all the numbers written on the board?
7
1
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2019 MOAA Team P7
Suppose
A
B
C
ABC
A
BC
is a triangle inscribed in circle
ω
\omega
ω
. Let
A
′
A'
A
′
be the point on
ω
\omega
ω
so that
A
A
′
AA'
A
A
′
is a diameter, and let
G
G
G
be the centroid of
A
B
C
ABC
A
BC
. Given that
A
B
=
13
AB = 13
A
B
=
13
,
B
C
=
14
BC = 14
BC
=
14
, and
C
A
=
15
CA = 15
C
A
=
15
, let
x
x
x
be the area of triangle
A
G
A
′
AGA'
A
G
A
′
. If
x
x
x
can be expressed in the form
m
/
n
m/n
m
/
n
, where m and n are relatively prime positive integers, compute
100
n
+
m
100n + m
100
n
+
m
.
5
1
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2019 MOAA Team P5
Let
A
B
C
ABC
A
BC
be a triangle with
A
B
=
A
C
=
10
AB = AC = 10
A
B
=
A
C
=
10
and
B
C
=
12
BC = 12
BC
=
12
. Define
ℓ
A
\ell_A
ℓ
A
as the line through
A
A
A
perpendicular to
A
B
‾
\overline{AB}
A
B
. Similarly,
ℓ
B
\ell_B
ℓ
B
is the line through
B
B
B
perpendicular to
B
C
‾
\overline{BC}
BC
and
ℓ
C
\ell_C
ℓ
C
is the line through
C
C
C
perpendicular to
C
A
‾
\overline{CA}
C
A
. These three lines
ℓ
A
,
ℓ
B
,
ℓ
C
\ell_A, \ell_B, \ell_C
ℓ
A
,
ℓ
B
,
ℓ
C
form a triangle with perimeter
m
/
n
m/n
m
/
n
for relatively prime positive integers
m
m
m
and
n
n
n
. Find
m
+
n
m + n
m
+
n
.
4
1
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2019 MOAA Team P4
Brandon wants to split his orchestra of
20
20
20
violins,
15
15
15
violas,
10
10
10
cellos, and
5
5
5
basses into three distinguishable groups, where all of the players of each instrument are indistinguishable. He wants each group to have at least one of each instrument and for each group to have more violins than violas, more violas than cellos, and more cellos than basses. How many ways are there for Brandon to split his orchestra following these conditions?
3
1
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2019 MOAA Team P3
For how many ordered pairs of positive integers
(
a
,
b
)
(a, b)
(
a
,
b
)
such that
a
≤
50
a \le 50
a
≤
50
is it true that
x
2
−
a
x
+
b
x^2 - ax + b
x
2
−
a
x
+
b
has integer roots?
2
1
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2019 MOAA Team P2
The lengths of the two legs of a right triangle are the two distinct roots of the quadratic
x
2
−
36
x
+
70
x^2 - 36x + 70
x
2
−
36
x
+
70
. What is the length of the triangle’s hypotenuse?
1
1
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2019 MOAA Team P1
Jeffrey stands on a straight horizontal bridge that measures
20000
20000
20000
meters across. He wishes to place a pole vertically at the center of the bridge so that the sum of the distances from the top of the pole to the two ends of the bridge is
20001
20001
20001
meters. To the nearest meter, how long of a pole does Jeffrey need?