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Math Open At Andover problems
2018 MOAA
2018 MOAA
Part of
Math Open At Andover problems
Subcontests
(12)
Sets 7-12
1
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2018 MOAA Gunga Bowl - Math Open At Andover - last 6 sets - 18 problems
Set 7 p19. Let circles
ω
1
\omega_1
ω
1
and
ω
2
\omega_2
ω
2
, with centers
O
1
O_1
O
1
and
O
2
O_2
O
2
, respectively, intersect at
X
X
X
and
Y
Y
Y
. A lies on
ω
1
\omega_1
ω
1
and
B
B
B
lies on
ω
2
\omega_2
ω
2
such that
A
O
1
AO_1
A
O
1
and
B
O
2
BO_2
B
O
2
are both parallel to
X
Y
XY
X
Y
, and
A
A
A
and
B
B
B
lie on the same side of
O
1
O
2
O_1O_2
O
1
O
2
. If
X
Y
=
60
XY = 60
X
Y
=
60
,
∠
X
A
Y
=
4
5
o
\angle XAY = 45^o
∠
X
A
Y
=
4
5
o
, and
∠
X
B
Y
=
3
0
o
\angle XBY = 30^o
∠
XB
Y
=
3
0
o
, then the length of
A
B
AB
A
B
can be expressed in the form
a
−
b
2
+
c
3
\sqrt{a - b\sqrt2 + c\sqrt3}
a
−
b
2
+
c
3
, where
a
,
b
,
c
a, b, c
a
,
b
,
c
are positive integers. Determine
a
+
b
+
c
a + b + c
a
+
b
+
c
. p20. If
x
x
x
is a positive real number such that
x
x
2
=
2
80
x^{x^2}= 2^{80}
x
x
2
=
2
80
, find the largest integer not greater than
x
3
x^3
x
3
. p21. Justin has a bag containing
750
750
750
balls, each colored red or blue. Sneaky Sam takes out a random number of balls and replaces them all with green balls. Sam notices that of the balls left in the bag, there are
15
15
15
more red balls than blue balls. Justin then takes out
500
500
500
of the balls chosen randomly. If
E
E
E
is the expected number of green balls that Justin takes out, determine the greatest integer less than or equal to
E
E
E
. Set 8These three problems are interdependent; each problem statement in this set will use the answers to the other two problems in this set. As such, let the positive integers
A
,
B
,
C
A, B, C
A
,
B
,
C
be the answers to problems
22
22
22
,
23
23
23
, and
24
24
24
, respectively, for this set.p22. Let
W
X
Y
Z
WXYZ
W
X
Y
Z
be a rectangle with
W
X
=
5
B
WX =\sqrt{5B}
W
X
=
5
B
and
X
Y
=
5
C
XY =\sqrt{5C}
X
Y
=
5
C
. Let the midpoint of
X
Y
XY
X
Y
be
M
M
M
and the midpoint of
Y
Z
YZ
Y
Z
be
N
N
N
. If
X
N
XN
XN
and
W
Y
W Y
WY
intersect at
P
P
P
, determine the area of
M
P
N
Y
MPNY
MPN
Y
. p23. Positive integers
x
,
y
,
z
x, y, z
x
,
y
,
z
satisfy
x
y
≡
A
(
m
o
d
5
)
xy \equiv A \,\, (mod 5)
x
y
≡
A
(
m
o
d
5
)
y
z
≡
2
A
+
C
(
m
o
d
7
)
yz \equiv 2A + C\,\, (mod 7)
yz
≡
2
A
+
C
(
m
o
d
7
)
z
x
≡
C
+
3
(
m
o
d
9
)
.
zx \equiv C + 3 \,\, (mod 9).
z
x
≡
C
+
3
(
m
o
d
9
)
.
(Here, writing
a
≡
b
(
m
o
d
m
)
a \equiv b \,\, (mod m)
a
≡
b
(
m
o
d
m
)
is equivalent to writing
m
∣
a
−
b
m | a - b
m
∣
a
−
b
.) Given that
3
∤
x
3 \nmid x
3
∤
x
,
3
∤
z
3 \nmid z
3
∤
z
, and
9
∣
y
9 | y
9∣
y
, find the minimum possible value of the product
x
y
z
xyz
x
yz
. p24. Suppose
x
x
x
and
y
y
y
are real numbers such that
x
+
y
=
A
x + y = A
x
+
y
=
A
x
y
=
1
36
B
2
.
xy =\frac{1}{36}B^2.
x
y
=
36
1
B
2
.
Determine
∣
x
−
y
∣
|x - y|
∣
x
−
y
∣
. Set 9 p25. The integer
2017
2017
2017
is a prime which can be uniquely represented as the sum of the squares of two positive integers:
9
2
+
4
4
2
=
2017.
9^2 + 44^2 = 2017.
9
2
+
4
4
2
=
2017.
If
N
=
2017
⋅
128
N = 2017 \cdot 128
N
=
2017
⋅
128
can be uniquely represented as the sum of the squares of two positive integers
a
2
+
b
2
a^2 +b^2
a
2
+
b
2
, determine
a
+
b
a + b
a
+
b
. p26. Chef Celia is planning to unveil her newest creation: a whole-wheat square pyramid filled with maple syrup. She will use a square flatbread with a one meter diagonal and cut out each of the five polygonal faces of the pyramid individually. If each of the triangular faces of the pyramid are to be equilateral triangles, the largest volume of syrup, in cubic meters, that Celia can enclose in her pyramid can be expressed as
a
−
b
c
\frac{a-\sqrt{b}}{c}
c
a
−
b
where
a
,
b
a, b
a
,
b
and
c
c
c
are the smallest possible possible positive integers. What is
a
+
b
+
c
a + b + c
a
+
b
+
c
? p27. In the Cartesian plane, let
ω
\omega
ω
be the circle centered at
(
24
,
7
)
(24, 7)
(
24
,
7
)
with radius
6
6
6
. Points
P
,
Q
P, Q
P
,
Q
, and
R
R
R
are chosen in the plane such that
P
P
P
lies on
ω
\omega
ω
,
Q
Q
Q
lies on the line
y
=
x
y = x
y
=
x
, and
R
R
R
lies on the
x
x
x
-axis. The minimum possible value of
P
Q
+
Q
R
+
R
P
PQ+QR+RP
PQ
+
QR
+
RP
can be expressed in the form
m
\sqrt{m}
m
for some integer
m
m
m
. Find m. Set 10Deja vu? p28. Let
A
B
C
ABC
A
BC
be a triangle with incircle
ω
\omega
ω
. Let
ω
\omega
ω
intersect sides
B
C
BC
BC
,
C
A
CA
C
A
,
A
B
AB
A
B
at
D
,
E
,
F
D, E, F
D
,
E
,
F
, respectively. Suppose
A
B
=
7
AB = 7
A
B
=
7
,
B
C
=
12
BC = 12
BC
=
12
, and
C
A
=
13
CA = 13
C
A
=
13
. If the area of
A
B
C
ABC
A
BC
is
K
K
K
and the area of
D
E
F
DEF
D
EF
is
m
n
⋅
K
\frac{m}{n}\cdot K
n
m
⋅
K
, where
m
m
m
and
n
n
n
are relatively prime positive integers, then compute
m
+
n
m + n
m
+
n
. p29. Sebastian is playing the game Split! again, but this time in a three dimensional coordinate system. He begins the game with one token at
(
0
,
0
,
0
)
(0, 0, 0)
(
0
,
0
,
0
)
. For each move, he is allowed to select a token on any point
(
x
,
y
,
z
)
(x, y, z)
(
x
,
y
,
z
)
and take it off, replacing it with three tokens, one at
(
x
+
1
,
y
,
z
)
(x + 1, y, z)
(
x
+
1
,
y
,
z
)
, one at
(
x
,
y
+
1
,
z
)
(x, y + 1, z)
(
x
,
y
+
1
,
z
)
, and one at
(
x
,
y
,
z
+
1
)
(x, y, z + 1)
(
x
,
y
,
z
+
1
)
At the end of the game, for a token on
(
a
,
b
,
c
)
(a, b, c)
(
a
,
b
,
c
)
, it is assigned a score
1
2
a
+
b
+
c
\frac{1}{2^{a+b+c}}
2
a
+
b
+
c
1
. These scores are summed for his total score. If the highest total score Sebastian can get in
100
100
100
moves is
m
/
n
m/n
m
/
n
, then determine
m
+
n
m + n
m
+
n
. p30. Determine the number of positive
6
6
6
digit integers that satisfy the following properties:
∙
\bullet
∙
All six of their digits are
1
,
5
,
7
1, 5, 7
1
,
5
,
7
, or
8
8
8
,
∙
\bullet
∙
The sum of all the digits is a multiple of
5
5
5
. Set 11 p31. The triangular numbers are defined as
T
n
=
n
(
n
+
1
)
2
T_n =\frac{n(n+1)}{2}
T
n
=
2
n
(
n
+
1
)
. We also define
S
n
=
n
(
n
+
2
)
3
S_n =\frac{n(n+2)}{3}
S
n
=
3
n
(
n
+
2
)
. If the sum
∑
i
=
16
32
(
1
T
i
+
1
S
i
)
=
(
1
T
16
+
1
S
16
)
+
(
1
T
17
+
1
S
17
)
+
.
.
.
+
(
1
T
32
+
1
S
32
)
\sum_{i=16}^{32} \left(\frac{1}{T_i}+\frac{1}{S_i}\right)= \left(\frac{1}{T_{16}}+\frac{1}{S_{16}}\right)+\left(\frac{1}{T_{17}}+\frac{1}{S_{17}}\right)+...+\left(\frac{1}{T_{32}}+\frac{1}{S_{32}}\right)
i
=
16
∑
32
(
T
i
1
+
S
i
1
)
=
(
T
16
1
+
S
16
1
)
+
(
T
17
1
+
S
17
1
)
+
...
+
(
T
32
1
+
S
32
1
)
can be written in the form
a
/
b
a/b
a
/
b
, where
a
a
a
and
b
b
b
are positive integers with
g
c
d
(
a
,
b
)
=
1
gcd(a, b) = 1
g
c
d
(
a
,
b
)
=
1
, then find
a
+
b
a + b
a
+
b
. p32. Farmer Will is considering where to build his house in the Cartesian coordinate plane. He wants to build his house on the line
y
=
x
y = x
y
=
x
, but he also has to minimize his travel time for his daily trip to his barnhouse at
(
24
,
15
)
(24, 15)
(
24
,
15
)
and back. From his house, he must first travel to the river at
y
=
2
y = 2
y
=
2
to fetch water for his animals. Then, he heads for his barnhouse, and promptly leaves for the long strip mall at the line
y
=
3
x
y =\sqrt3 x
y
=
3
x
for groceries, before heading home. If he decides to build his house at
(
x
0
,
y
0
)
(x_0, y_0)
(
x
0
,
y
0
)
such that the distance he must travel is minimized,
x
0
x_0
x
0
can be written in the form
a
b
−
c
d
\frac{a\sqrt{b}-c}{d}
d
a
b
−
c
, where
a
,
b
,
c
,
d
a, b, c, d
a
,
b
,
c
,
d
are positive integers,
b
b
b
is not divisible by the square of a prime, and
g
c
d
(
a
,
c
,
d
)
=
1
gcd(a, c, d) = 1
g
c
d
(
a
,
c
,
d
)
=
1
. Compute
a
+
b
+
c
+
d
a+b+c+d
a
+
b
+
c
+
d
. p33. Determine the greatest positive integer
n
n
n
such that the following two conditions hold:
∙
\bullet
∙
n
2
n^2
n
2
is the difference of consecutive perfect cubes;
∙
\bullet
∙
2
n
+
287
2n + 287
2
n
+
287
is the square of an integer. Set 12The answers to these problems are nonnegative integers that may exceed
1000000
1000000
1000000
. You will be awarded points as described in the problems. p34. The “Collatz sequence” of a positive integer n is the longest sequence of distinct integers
(
x
i
)
i
≥
0
(x_i)_{i\ge 0}
(
x
i
)
i
≥
0
with
x
0
=
n
x_0 = n
x
0
=
n
and
x
n
+
1
=
{
x
n
2
i
f
x
n
i
s
e
v
e
n
3
x
n
+
1
i
f
x
n
i
s
o
d
d
.
x_{n+1} =\begin{cases} \frac{x_n}{2} & if \,\, x_n \,\, is \,\, even \\ 3x_n + 1 & if \,\, x_n \,\, is \,\, odd \end{cases}.
x
n
+
1
=
{
2
x
n
3
x
n
+
1
i
f
x
n
i
s
e
v
e
n
i
f
x
n
i
s
o
dd
.
It is conjectured that all Collatz sequences have a finite number of elements, terminating at
1
1
1
. This has been confirmed via computer program for all numbers up to
2
64
2^{64}
2
64
. There is a unique positive integer
n
<
1
0
9
n < 10^9
n
<
1
0
9
such that its Collatz sequence is longer than the Collatz sequence of any other positive integer less than
1
0
9
10^9
1
0
9
. What is this integer
n
n
n
?An estimate of
e
e
e
gives
max
{
⌊
32
−
11
3
log
10
(
∣
n
−
e
∣
+
1
)
⌋
,
0
}
\max\{\lfloor 32 - \frac{11}{3}\log_{10}(|n - e| + 1)\rfloor, 0\}
max
{⌊
32
−
3
11
lo
g
10
(
∣
n
−
e
∣
+
1
)⌋
,
0
}
points. p35. We define a graph
G
G
G
as a set
V
(
G
)
V (G)
V
(
G
)
of vertices and a set
E
(
G
)
E(G)
E
(
G
)
of distinct edges connecting those vertices. A graph
H
H
H
is a subgraph of
G
G
G
if the vertex set
V
(
H
)
V (H)
V
(
H
)
is a subset of
V
(
G
)
V (G)
V
(
G
)
and the edge set
E
(
H
)
E(H)
E
(
H
)
is a subset of
E
(
G
)
E(G)
E
(
G
)
. Let
e
x
(
k
,
H
)
ex(k, H)
e
x
(
k
,
H
)
denote the maximum number of edges in a graph with
k
k
k
vertices without a subgraph of
H
H
H
. If
K
i
K_i
K
i
denotes a complete graph on
i
i
i
vertices, that is, a graph with
i
i
i
vertices and all
(
i
2
)
{i \choose 2}
(
2
i
)
edges between them present, determine
n
=
∑
i
=
2
2018
e
x
(
2018
,
K
i
)
.
n =\sum_{i=2}^{2018} ex(2018, K_i).
n
=
i
=
2
∑
2018
e
x
(
2018
,
K
i
)
.
An estimate of
e
e
e
gives
max
{
⌊
32
−
3
log
10
(
∣
n
−
e
∣
+
1
)
⌋
,
0
}
\max\{\lfloor 32 - 3\log_{10}(|n - e| + 1)\rfloor, 0\}
max
{⌊
32
−
3
lo
g
10
(
∣
n
−
e
∣
+
1
)⌋
,
0
}
points. p36. Write down an integer between
1
1
1
and
100
100
100
, inclusive. This number will be denoted as
n
i
n_i
n
i
, where your Team ID is
i
i
i
. Let
S
S
S
be the set of Team ID’s for all teams that submitted an answer to this problem. For every ordered triple of distinct Team ID’s
(
a
,
b
,
c
)
(a, b, c)
(
a
,
b
,
c
)
such that a, b, c ∈ S, if all roots of the polynomial
x
3
+
n
a
x
2
+
n
b
x
+
n
c
x^3 + n_ax^2 + n_bx + n_c
x
3
+
n
a
x
2
+
n
b
x
+
n
c
are real, then the teams with ID’s
a
,
b
,
c
a, b, c
a
,
b
,
c
will each receive one virtual banana. If you receive
v
b
v_b
v
b
virtual bananas in total and
∣
S
∣
≥
3
|S| \ge 3
∣
S
∣
≥
3
teams submit an answer to this problem, you will be awarded
⌊
32
v
b
3
(
∣
S
∣
−
1
)
(
∣
S
∣
−
2
)
⌋
\left\lfloor \frac{32v_b}{3(|S| - 1)(|S| - 2)}\right\rfloor
⌊
3
(
∣
S
∣
−
1
)
(
∣
S
∣
−
2
)
32
v
b
⌋
points for this problem. If
∣
S
∣
≤
2
|S| \le 2
∣
S
∣
≤
2
, the team(s) that submitted an answer to this problem will receive
32
32
32
points for this problem. PS. You had better use hide for answers. First sets have been posted [url=https://artofproblemsolving.com/community/c4h2777264p24369138]here.Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.
Sets 1-6
1
Hide problems
2018 MOAA Gunga Bowl - Math Open At Andover - first 6 sets - 18 problems
Set 1 p1. Find
1
+
2
+
3
+
4
+
5
+
6
+
7
+
8
+
9
+
10
+
11
1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11
1
+
2
+
3
+
4
+
5
+
6
+
7
+
8
+
9
+
10
+
11
. p2. Find
1
⋅
11
+
2
⋅
10
+
3
⋅
9
+
4
⋅
8
+
5
⋅
7
+
6
⋅
6
1 \cdot 11 + 2 \cdot 10 + 3 \cdot 9 + 4 \cdot 8 + 5 \cdot 7 + 6 \cdot 6
1
⋅
11
+
2
⋅
10
+
3
⋅
9
+
4
⋅
8
+
5
⋅
7
+
6
⋅
6
. p3. Let
1
1
⋅
2
+
1
2
⋅
3
+
1
3
⋅
4
+
1
4
⋅
5
+
1
5
⋅
6
+
1
6
⋅
7
+
1
7
⋅
8
+
1
8
⋅
9
+
1
9
⋅
10
+
1
10
⋅
11
=
m
n
\frac{1}{1\cdot 2} +\frac{1}{2\cdot 3} +\frac{1}{3\cdot 4} +\frac{1}{4\cdot 5} +\frac{1}{5\cdot 6} +\frac{1}{6\cdot 7} +\frac{1}{7\cdot 8} +\frac{1}{8\cdot 9} +\frac{1}{9\cdot 10} +\frac{1}{10\cdot 11} =\frac{m}{n}
1
⋅
2
1
+
2
⋅
3
1
+
3
⋅
4
1
+
4
⋅
5
1
+
5
⋅
6
1
+
6
⋅
7
1
+
7
⋅
8
1
+
8
⋅
9
1
+
9
⋅
10
1
+
10
⋅
11
1
=
n
m
, where
m
m
m
and
n
n
n
are positive integers that share no prime divisors. Find
m
+
n
m + n
m
+
n
. Set 2 p4. Define
0
!
=
1
0! = 1
0
!
=
1
and let
n
!
=
n
⋅
(
n
−
1
)
!
n! = n \cdot (n - 1)!
n
!
=
n
⋅
(
n
−
1
)!
for all positive integers
n
n
n
. Find the value of
(
2
!
+
0
!
)
(
1
!
+
8
!
)
(2! + 0!)(1! + 8!)
(
2
!
+
0
!)
(
1
!
+
8
!)
. p5. Rachel’s favorite number is a positive integer
n
n
n
. She gives Justin three clues about it:
∙
\bullet
∙
n
n
n
is prime.
∙
\bullet
∙
n
2
−
5
n
+
6
≠
0
n^2 - 5n + 6 \ne 0
n
2
−
5
n
+
6
=
0
.
∙
\bullet
∙
n
n
n
is a divisor of
252
252
252
. What is Rachel’s favorite number? p6. Shen eats eleven blueberries on Monday. Each day after that, he eats five more blueberries than the day before. For example, Shen eats sixteen blueberries on Tuesday. How many blueberries has Shen eaten in total before he eats on the subsequent Monday? Set 3 p7. Triangle
A
B
C
ABC
A
BC
satisfies
A
B
=
7
AB = 7
A
B
=
7
,
B
C
=
12
BC = 12
BC
=
12
, and
C
A
=
13
CA = 13
C
A
=
13
. If the area of
A
B
C
ABC
A
BC
can be expressed in the form
m
n
m\sqrt{n}
m
n
, where
n
n
n
is not divisible by the square of a prime, then determine
m
+
n
m + n
m
+
n
. p8. Sebastian is playing the game Split! on a coordinate plane. He begins the game with one token at
(
0
,
0
)
(0, 0)
(
0
,
0
)
. For each move, he is allowed to select a token on any point
(
x
,
y
)
(x, y)
(
x
,
y
)
and take it off the plane, replacing it with two tokens, one at
(
x
+
1
,
y
)
(x + 1, y)
(
x
+
1
,
y
)
, and one at
(
x
,
y
+
1
)
(x, y + 1)
(
x
,
y
+
1
)
. At the end of the game, for a token on
(
a
,
b
)
(a, b)
(
a
,
b
)
, it is assigned a score
1
2
a
+
b
\frac{1}{2^{a+b}}
2
a
+
b
1
. These scores are summed for his total score. Determine the highest total score Sebastian can get in
100
100
100
moves. p9. Find the number of positive integers
n
n
n
satisfying the following two properties:
∙
\bullet
∙
n
n
n
has either four or five digits, where leading zeros are not permitted,
∙
\bullet
∙
The sum of the digits of
n
n
n
is a multiple of
3
3
3
. Set 4 p10. A unit square rotated
4
5
o
45^o
4
5
o
about a vertex, Sweeps the area for Farmer Khiem’s pen. If
n
n
n
is the space the pigs can roam, Determine the floor of
100
n
100n
100
n
.If
n
n
n
is the area a unit square sweeps out when rotated 4
5
5
5
degrees about a vertex, determine
⌊
100
n
⌋
\lfloor 100n \rfloor
⌊
100
n
⌋
. Here
⌊
x
⌋
\lfloor x \rfloor
⌊
x
⌋
denotes the greatest integer less than or equal to
x
x
x
.https://cdn.artofproblemsolving.com/attachments/b/1/129efd0dbd56dc0b4fb742ac80eaf2447e106d.pngp11. Michael is planting four trees, In a grid, three rows of three, If two trees are close, Then both are bulldozed, So how many ways can it be?In a three by three grid of squares, determine the number of ways to select four squares such that no two share a side. p12. Three sixty-seven Are the last three digits of
n
n
n
cubed. What is
n
n
n
?If the last three digits of
n
3
n^3
n
3
are
367
367
367
for a positive integer
n
n
n
less than
1000
1000
1000
, determine
n
n
n
. Set 5 p13. Determine
97
+
56
3
4
+
97
−
56
3
4
\sqrt[4]{97 + 56\sqrt{3}} + \sqrt[4]{97 - 56\sqrt{3}}
4
97
+
56
3
+
4
97
−
56
3
. p14. Triangle
△
A
B
C
\vartriangle ABC
△
A
BC
is inscribed in a circle
ω
\omega
ω
of radius
12
12
12
so that
∠
B
=
6
8
o
\angle B = 68^o
∠
B
=
6
8
o
and
∠
C
=
6
4
o
\angle C = 64^o
∠
C
=
6
4
o
. The perpendicular from
A
A
A
to
B
C
BC
BC
intersects
ω
\omega
ω
at
D
D
D
, and the angle bisector of
∠
B
\angle B
∠
B
intersects
ω
\omega
ω
at
E
E
E
. What is the value of
D
E
2
DE^2
D
E
2
? p15. Determine the sum of all positive integers
n
n
n
such that
4
n
4
+
1
4n^4 + 1
4
n
4
+
1
is prime. Set 6 p16. Suppose that
p
,
q
,
r
p, q, r
p
,
q
,
r
are primes such that
p
q
r
=
11
(
p
+
q
+
r
)
pqr = 11(p + q + r)
pq
r
=
11
(
p
+
q
+
r
)
such that
p
≥
q
≥
r
p\ge q \ge r
p
≥
q
≥
r
. Determine the sum of all possible values of
p
p
p
. p17. Let the operation
⊕
\oplus
⊕
satisfy
a
⊕
b
=
1
1
/
a
+
1
/
b
a \oplus b =\frac{1}{1/a+1/b}
a
⊕
b
=
1/
a
+
1/
b
1
. Suppose
N
=
(
.
.
.
(
(
2
⊕
2
)
⊕
2
)
⊕
.
.
.
2
)
,
N = (...((2 \oplus 2) \oplus 2) \oplus ... 2),
N
=
(
...
((
2
⊕
2
)
⊕
2
)
⊕
...2
)
,
where there are
2018
2018
2018
instances of
⊕
\oplus
⊕
. If
N
N
N
can be expressed in the form
m
/
n
m/n
m
/
n
, where
m
m
m
and
n
n
n
are relatively prime positive integers, then determine
m
+
n
m + n
m
+
n
. p18. What is the remainder when
201
8
1001
−
1
2017
\frac{2018^{1001} - 1}{2017}
2017
201
8
1001
−
1
is divided by
2017
2017
2017
? PS. You had better use hide for answers. Last sets have been posted [url=https://artofproblemsolving.com/community/c4h2777307p24369763]here. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.
10
1
Hide problems
2018 MOAA Team P10
Vincent is playing a game with Evil Bill. The game uses an infinite number of red balls, an infinite number of green balls, and a very large bag. Vincent first picks two nonnegative integers
g
g
g
and
k
k
k
such that
g
<
k
≤
2016
g < k \le 2016
g
<
k
≤
2016
, and Evil Bill places
g
g
g
green balls and
2016
−
g
2016 - g
2016
−
g
red balls in the bag, so that there is a total of
2016
2016
2016
balls in the bag. Vincent then picks a ball of either color and places it in the bag. Evil Bill then inspects the bag. If the ratio of green balls to total balls in the bag is ever exactly
k
2016
\frac{k}{2016}
2016
k
, then Evil Bill wins. If the ratio of green balls to total balls is greater than
k
2016
\frac{k}{2016}
2016
k
, then Vincent wins. Otherwise, Vincent and Evil Bill repeat the previous two actions (Vincent picks a ball and Evil Bill inspects the bag). If
S
S
S
is the sum of all possible values of
k
k
k
that Vincent could choose and be able to win, determine the largest prime factor of
S
S
S
.
9
1
Hide problems
2018 MOAA Team P9
Quadrilateral
A
B
C
D
ABCD
A
BC
D
with
A
C
=
800
AC = 800
A
C
=
800
is inscribed in a circle, and
E
,
W
,
X
,
Y
,
Z
E, W, X, Y, Z
E
,
W
,
X
,
Y
,
Z
are the midpoints of segments
B
D
BD
B
D
,
A
B
AB
A
B
,
B
C
BC
BC
,
C
D
CD
C
D
,
D
A
DA
D
A
, respectively. If the circumcenters of
E
W
Z
EW Z
E
W
Z
and
E
X
Y
EXY
EX
Y
are
O
1
O_1
O
1
and
O
2
O_2
O
2
, respectively, determine
O
1
O
2
O_1O_2
O
1
O
2
.
8
1
Hide problems
2018 MOAA Team P8
Suppose that k and x are positive integers such that
k
2
=
(
1
+
3
2
)
x
+
(
1
−
3
2
)
x
.
\frac{k}{2}=\left( \sqrt{1 +\frac{\sqrt3}{2}}\right)^x+\left( \sqrt{1 -\frac{\sqrt3}{2}}\right)^x.
2
k
=
1
+
2
3
x
+
1
−
2
3
x
.
Find the sum of all possible values of
k
k
k
7
1
Hide problems
2018 MOAA Team P7
For a positive integer
k
k
k
, define the
k
k
k
-pop of a positive integer
n
n
n
as the infinite sequence of integers
a
1
,
a
2
,
.
.
.
a_1, a_2, ...
a
1
,
a
2
,
...
such that
a
1
=
n
a_1 = n
a
1
=
n
and
a
i
+
1
=
⌊
a
i
k
⌋
,
i
=
1
,
2
,
.
.
a_{i+1}= \left\lfloor \frac{a_i}{k} \right\rfloor , i = 1, 2, ..
a
i
+
1
=
⌊
k
a
i
⌋
,
i
=
1
,
2
,
..
where
⌊
x
⌋
\lfloor x\rfloor
⌊
x
⌋
denotes the greatest integer less than or equal to
x
x
x
. Furthermore, define a positive integer
m
m
m
to be
k
k
k
-pop avoiding if
k
k
k
does not divide any nonzero term in the
k
k
k
-pop of
m
m
m
. For example,
14
14
14
is 3-pop avoiding because
3
3
3
does not divide any nonzero term in the
3
3
3
-pop of
14
14
14
, which is
14
,
4
,
1
,
0
,
0
,
.
.
.
.
14, 4, 1, 0, 0, ....
14
,
4
,
1
,
0
,
0
,
....
Suppose that the number of positive integers less than
1
3
2018
13^{2018}
1
3
2018
which are
13
13
13
-pop avoiding is equal to N. What is the remainder when
N
N
N
is divided by
1000
1000
1000
?
6
1
Hide problems
2018 MOAA Team P6
Consider an
m
×
n
m \times n
m
×
n
grid of unit squares. Let
R
R
R
be the total number of rectangles of any size, and let
S
S
S
be the total number of squares of any size. Assume that the sides of the rectangles and squares are parallel to the sides of the
m
×
n
m \times n
m
×
n
grid. If
R
S
=
759
50
\frac{R}{S} =\frac{759}{50}
S
R
=
50
759
, then determine
m
n
mn
mn
.
5
1
Hide problems
2018 MOAA Team P5
Mr. DoBa likes to listen to music occasionally while he does his math homework. When he listens to classical music, he solves one problem every
3
3
3
minutes. When he listens to rap music, however, he only solves one problem every
5
5
5
minutes. Mr. DoBa listens to a playlist comprised of
60
%
60\%
60%
classical music and
40
%
40\%
40%
rap music. Each song is exactly
4
4
4
minutes long. Suppose that the expected number of problems he solves in an hour does not depend on whether or not Mr. DoBa is listening to music at any given moment, and let
m
m
m
the average number of problems Mr. DoBa solves per minute when he is not listening to music. Determine the value of
1000
m
1000m
1000
m
.
4
1
Hide problems
2018 MOAA Team P4
Michael and Andrew are playing the game Bust, which is played as follows: Michael chooses a positive integer less than or equal to
99
99
99
, and writes it on the board. Andrew then makes a move, which consists of him choosing a positive integer less than or equal to
8
8
8
and increasing the integer on the board by the integer he chose. Play then alternates in this manner, with each person making exactly one move, until the integer on the board becomes greater than or equal to
100
100
100
. The person who made the last move loses. Let S be the sum of all numbers for which Michael could choose initially and win with both people playing optimally. Find S.
3
1
Hide problems
2018 MOAA Team P3
Let
B
E
BE
BE
and
C
F
CF
CF
be altitudes in triangle
A
B
C
ABC
A
BC
. If
A
E
=
24
AE = 24
A
E
=
24
,
E
C
=
60
EC = 60
EC
=
60
, and
B
F
=
31
BF = 31
BF
=
31
, determine
A
F
AF
A
F
.
2
1
Hide problems
2018 MOAA Team P2
If
x
>
0
x > 0
x
>
0
and
x
2
+
1
x
2
=
14
x^2 +\frac{1}{x^2}= 14
x
2
+
x
2
1
=
14
, find
x
5
+
1
x
5
x^5 +\frac{1}{x^5}
x
5
+
x
5
1
.
1
1
Hide problems
2018 MOAA Team P1
In
△
A
B
C
\vartriangle ABC
△
A
BC
,
A
B
=
3
AB = 3
A
B
=
3
,
B
C
=
5
BC = 5
BC
=
5
, and
C
A
=
6
CA = 6
C
A
=
6
. Points
D
D
D
and
E
E
E
are chosen such that
A
C
D
E
ACDE
A
C
D
E
is a square which does not overlap with
△
A
B
C
\vartriangle ABC
△
A
BC
. The length of
B
D
BD
B
D
can be expressed in the form
m
+
n
p
\sqrt{m + n\sqrt{p}}
m
+
n
p
, where
m
m
m
,
n
n
n
, and
p
p
p
are positive integers and
p
p
p
is not divisible by the square of a prime. Determine the value of
m
+
n
+
p
m + n + p
m
+
n
+
p
.