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Math Open At Andover problems
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MOAA Individual Speed General Rounds
Part of
Math Open At Andover problems
Subcontests
(2)
2018I Sample
1
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2018 MOAA Individual Round Sample - Math Open At Andover
p1. Will is distributing his money to three friends. Since he likes some friends more than others, the amount of money he gives each is in the ratio of
5
:
3
:
2
5 : 3 : 2
5
:
3
:
2
. If the person who received neither the least nor greatest amount of money was given
42
42
42
dollars, how many dollars did Will distribute in all? p2. Fan, Zhu, and Ming are driving around a circular track. Fan drives
24
24
24
times as fast as Ming and Zhu drives
9
9
9
times as fast as Ming. All three drivers start at the same point on the track and keep driving until Fan and Zhu pass Ming at the same time. During this interval, how many laps have Fan and Zhu driven together? p3. Mr. DoBa is playing a game with Gunga the Gorilla. They both agree to think of a positive integer from
1
1
1
to
120
120
120
, inclusive. Let the sum of their numbers be
n
n
n
. Let the remainder of the operation
n
2
4
\frac{n^2}{4}
4
n
2
be
r
r
r
. If
r
r
r
is
0
0
0
or
1
1
1
, Mr. DoBa wins. Otherwise, Gunga wins. Let the probability that Mr. DoBa wins a given round of this game be
p
p
p
. What is
120
p
120p
120
p
? p4. Let S be the set
{
1
,
2
,
3
,
4
,
5
,
6
,
7
,
8
,
9
,
10
}
\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}
{
1
,
2
,
3
,
4
,
5
,
6
,
7
,
8
,
9
,
10
}
. How many subsets of
S
S
S
are there such that if
a
a
a
is the number of even numbers in the subset and
b
b
b
is the number of odd numbers in the subset, then
a
a
a
and
b
b
b
are either both odd or both even? By definition, subsets of
S
S
S
are unordered and only contain distinct elements that belong to
S
S
S
. p5. Phillips Academy has five clusters,
W
Q
N
WQN
W
QN
,
W
Q
S
WQS
W
QS
,
P
K
N
PKN
P
K
N
,
F
L
G
FLG
F
L
G
and
A
B
B
ABB
A
BB
. The Blue Key heads are going to visit all five clusters in some order, except
W
Q
S
WQS
W
QS
must be visited before
W
Q
N
WQN
W
QN
. How many total ways can they visit the five clusters? p6. An astronaut is in a spaceship which is a cube of side length
6
6
6
. He can go outside but has to be within a distance of
3
3
3
from the spaceship, as that is the length of the rope that tethers him to the ship. Out of all the possible points he can reach, the surface area of the outer surface can be expressed as
m
+
n
π
m+n\pi
m
+
nπ
, where
m
m
m
and
n
n
n
are relatively prime positive integers. What is
m
+
n
m + n
m
+
n
? p7. Let
A
B
C
D
ABCD
A
BC
D
be a square and
E
E
E
be a point in its interior such that
C
D
E
CDE
C
D
E
is an equilateral triangle. The circumcircle of
C
D
E
CDE
C
D
E
intersects sides
A
D
AD
A
D
and
B
C
BC
BC
at
D
D
D
,
F
F
F
and
C
C
C
,
G
G
G
, respectively. If
A
B
=
30
AB = 30
A
B
=
30
, the area of
A
F
G
B
AFGB
A
FGB
can be expressed as
a
−
b
c
a-b\sqrt{c}
a
−
b
c
, where
a
a
a
,
b
b
b
, and
c
c
c
are positive integers and c is not divisible by the square of any prime. Find
a
+
b
+
c
a + b + c
a
+
b
+
c
. p8. Suppose that
x
,
y
,
z
x, y, z
x
,
y
,
z
satisfy the equations
x
+
y
+
z
=
3
x + y + z = 3
x
+
y
+
z
=
3
x
2
+
y
2
+
z
2
=
3
x^2 + y^2 + z^2 = 3
x
2
+
y
2
+
z
2
=
3
x
3
+
y
3
+
z
3
=
3
x^3 + y^3 + z^3 = 3
x
3
+
y
3
+
z
3
=
3
Let the sum of all possible values of
x
x
x
be
N
N
N
. What is
12000
N
12000N
12000
N
? p9. In circle
O
O
O
inscribe triangle
△
A
B
C
\vartriangle ABC
△
A
BC
so 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
D
D
D
be the midpoint of arc
B
C
BC
BC
, and let
A
D
AD
A
D
intersect
B
C
BC
BC
at
E
E
E
. Determine the value of
D
E
⋅
D
A
DE \cdot DA
D
E
⋅
D
A
. p10. How many ways are there to color the vertices of a regular octagon in
3
3
3
colors such that no two adjacent vertices have the same color? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.
2018 Ind
1
Hide problems
2018 MOAA Individual Round - Math Open At Andover
p1. Find
20
⋅
18
+
20
+
18
+
1
20 \cdot 18 + 20 + 18 + 1
20
⋅
18
+
20
+
18
+
1
. p2. Suzie’s Ice Cream has
10
10
10
flavors of ice cream,
5
5
5
types of cones, and
5
5
5
toppings to choose from. An ice cream cone consists of one flavor, one cone, and one topping. How many ways are there for Sebastian to order an ice cream cone from Suzie’s? p3. Let
a
=
7
a = 7
a
=
7
and
b
=
77
b = 77
b
=
77
. Find
(
2
a
b
)
2
(
a
+
b
)
2
−
(
a
−
b
)
2
\frac{(2ab)^2}{(a+b)^2-(a-b)^2}
(
a
+
b
)
2
−
(
a
−
b
)
2
(
2
ab
)
2
. p4. Sebastian invests
100
,
000
100,000
100
,
000
dollars. On the first day, the value of his investment falls by
20
20
20
percent. On the second day, it increases by
25
25
25
percent. On the third day, it falls by
25
25
25
percent. On the fourth day, it increases by
60
60
60
percent. How many dollars is his investment worth by the end of the fourth day? p5. Square
A
B
C
D
ABCD
A
BC
D
has side length
5
5
5
. Points
K
,
L
,
M
,
N
K,L,M,N
K
,
L
,
M
,
N
are on segments
A
B
AB
A
B
,
B
C
BC
BC
,
C
D
CD
C
D
,
D
A
DA
D
A
respectively,such that
M
C
=
C
L
=
2
MC = CL = 2
MC
=
C
L
=
2
and
N
A
=
A
K
=
1
NA = AK = 1
N
A
=
A
K
=
1
. The area of trapezoid
K
L
M
N
KLMN
K
L
MN
can be expressed 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. Suppose that
p
p
p
and
q
q
q
are prime numbers. If
p
+
q
=
30
p + q = 30
p
+
q
=
30
, find the sum of all possible values of
p
q
pq
pq
. p7. Tori receives a
15
−
20
−
25
15 - 20 - 25
15
−
20
−
25
right triangle. She cuts the triangle into two pieces along the altitude to the side of length
25
25
25
. What is the difference between the areas of the two pieces? p8. The factorial of a positive integer
n
n
n
, denoted
n
!
n!
n
!
, is the product of all the positive integers less than or equal to
n
n
n
. For example,
1
!
=
1
1! = 1
1
!
=
1
and
5
!
=
120
5! = 120
5
!
=
120
. Let
m
!
m!
m
!
and
n
!
n!
n
!
be the smallest and largest factorial ending in exactly
3
3
3
zeroes, respectively. Find
m
+
n
m + n
m
+
n
. p9. Sam is late to class, which is located at point
B
B
B
. He begins his walk at point
A
A
A
and is only allowed to walk on the grid lines. He wants to get to his destination quickly; how many paths are there that minimize his walking distance? https://cdn.artofproblemsolving.com/attachments/a/5/764e64ac315c950367357a1a8658b08abd635b.pngp10. Mr. Iyer owns a set of
6
6
6
antique marbles, where
1
1
1
is red,
2
2
2
are yellow, and
3
3
3
are blue. Unfortunately, he has randomly lost two of the marbles. His granddaughter starts drawing the remaining
4
4
4
out of a bag without replacement. She draws a yellow marble, then the red marble. Suppose that the probability that the next marble she draws is blue is equal to
m
n
\frac{m}{n}
n
m
, where
m
m
m
and
n
n
n
are relatively prime positiveintegers. What is
m
+
n
m + n
m
+
n
? p11. If
a
a
a
is a positive integer, what is the largest integer that will always be a factor of
(
a
3
+
1
)
(
a
3
+
2
)
(
a
3
+
3
)
(a^3+1)(a^3+2)(a^3+3)
(
a
3
+
1
)
(
a
3
+
2
)
(
a
3
+
3
)
? p12. What is the largest prime number that is a factor of
160
,
401
160,401
160
,
401
? p13. For how many integers
m
m
m
does the equation
x
2
+
m
x
+
2018
=
0
x^2 + mx + 2018 = 0
x
2
+
m
x
+
2018
=
0
have no real solutions in
x
x
x
? p14. What is the largest palindrome that can be expressed as the product of two two-digit numbers? A palindrome is a positive integer that has the same value when its digits are reversed. An example of a palindrome is
7887887
7887887
7887887
. p15. In circle
ω
\omega
ω
inscribe quadrilateral
A
D
B
C
ADBC
A
D
BC
such that
A
B
⊥
C
D
AB \perp CD
A
B
⊥
C
D
. Let
E
E
E
be the intersection of diagonals
A
B
AB
A
B
and
C
D
CD
C
D
, and suppose that
E
C
=
3
EC = 3
EC
=
3
,
E
D
=
4
ED = 4
E
D
=
4
, and
E
B
=
2
EB = 2
EB
=
2
. If the radius of
ω
\omega
ω
is
r
r
r
, then
r
2
=
m
n
r^2 =\frac{m}{n}
r
2
=
n
m
for relatively prime positive integers
m
m
m
and
n
n
n
. Determine
m
+
n
m + n
m
+
n
. p16. Suppose that
a
,
b
,
c
a, b, c
a
,
b
,
c
are nonzero real numbers such that
2
a
2
+
5
b
2
+
45
c
2
=
4
a
b
+
6
b
c
+
12
c
a
2a^2 + 5b^2 + 45c^2 = 4ab + 6bc + 12ca
2
a
2
+
5
b
2
+
45
c
2
=
4
ab
+
6
b
c
+
12
c
a
. Find the value of
9
(
a
+
b
+
c
)
3
5
a
b
c
\frac{9(a + b + c)^3}{5abc}
5
ab
c
9
(
a
+
b
+
c
)
3
. p17. Call a positive integer n spicy if there exist n distinct integers
k
1
,
k
2
,
.
.
.
,
k
n
k_1, k_2, ... , k_n
k
1
,
k
2
,
...
,
k
n
such that the following two conditions hold:
∙
\bullet
∙
∣
k
1
∣
+
∣
k
2
∣
+
.
.
.
+
∣
k
n
∣
=
n
2
|k_1| + |k_2| +... + |k_n| = n2
∣
k
1
∣
+
∣
k
2
∣
+
...
+
∣
k
n
∣
=
n
2
,
∙
\bullet
∙
k
1
+
k
2
+
.
.
.
+
k
n
=
0
k_1 + k_2 + ...+ k_n = 0
k
1
+
k
2
+
...
+
k
n
=
0
. Determine the number of spicy integers less than
1
0
6
10^6
1
0
6
. p18. Consider the system of equations
∣
x
2
−
y
2
−
4
x
+
4
y
∣
=
4
|x^2 - y^2 - 4x + 4y| = 4
∣
x
2
−
y
2
−
4
x
+
4
y
∣
=
4
∣
x
2
+
y
2
−
4
x
−
4
y
∣
=
4.
|x^2 + y^2 - 4x - 4y| = 4.
∣
x
2
+
y
2
−
4
x
−
4
y
∣
=
4.
Find the sum of all
x
x
x
and
y
y
y
that satisfy the system. p19. Determine the number of
8
8
8
letter sequences, consisting only of the letters
W
,
Q
,
N
W,Q,N
W
,
Q
,
N
, in which none of the sequences
W
W
WW
WW
,
Q
Q
Q
QQQ
QQQ
, or
N
N
N
N
NNNN
NNNN
appear. For example,
W
Q
Q
N
N
N
Q
Q
WQQNNNQQ
W
QQNNNQQ
is a valid sequence, while
W
W
W
Q
N
Q
N
Q
WWWQNQNQ
WWW
QNQNQ
is not. p20. Triangle
△
A
B
C
\vartriangle ABC
△
A
BC
has
A
B
=
7
AB = 7
A
B
=
7
,
C
A
=
8
CA = 8
C
A
=
8
, and
B
C
=
9
BC = 9
BC
=
9
. Let the reflections of
A
,
B
,
C
A,B,C
A
,
B
,
C
over the orthocenter H be
A
′
A'
A
′
,
B
′
B'
B
′
,
C
′
C'
C
′
. The area of the intersection of triangles
A
B
C
ABC
A
BC
and
A
′
B
′
C
′
A'B'C'
A
′
B
′
C
′
can be expressed in the form
a
b
c
\frac{a\sqrt{b}}{c}
c
a
b
, where
b
b
b
is squarefree and
a
a
a
and
c
c
c
are relatively prime. determine
a
+
b
+
c
a+b+c
a
+
b
+
c
. (The orthocenter of a triangle is the intersection of its three altitudes.) PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.