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Math Open At Andover problems
2021 MOAA
2021 MOAA
Part of
Math Open At Andover problems
Subcontests
(24)
24
1
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Gunga P24
Freddy the Frog is situated at 1 on an infinitely long number line. On day
n
n
n
, where
n
≥
1
n\ge 1
n
≥
1
, Freddy can choose to hop 1 step to the right, stay where he is, or hop
k
k
k
steps to the left, where
k
k
k
is an integer at most
n
+
1
n+1
n
+
1
. After day 5, how many sequences of moves are there such that Freddy has landed on at least one negative number?Proposed by Andy Xu
23
1
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Gunga P23
Let
P
P
P
be a point chosen on the interior of side
B
C
‾
\overline{BC}
BC
of triangle
△
A
B
C
\triangle ABC
△
A
BC
with side lengths
A
B
‾
=
10
,
B
C
‾
=
10
,
A
C
‾
=
12
\overline{AB} = 10, \overline{BC} = 10, \overline{AC} = 12
A
B
=
10
,
BC
=
10
,
A
C
=
12
. If
X
X
X
and
Y
Y
Y
are the feet of the perpendiculars from
P
P
P
to the sides
A
B
AB
A
B
and
A
C
AC
A
C
, then the minimum possible value of
P
X
2
+
P
Y
2
PX^2 + PY^2
P
X
2
+
P
Y
2
can be expressed as
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
.Proposed by Andrew Wen
22
1
Hide problems
Gunga P22
Let
p
p
p
and
q
q
q
be positive integers such that
p
p
p
is a prime,
p
p
p
divides
q
−
1
q-1
q
−
1
, and
p
+
q
p+q
p
+
q
divides
p
2
+
2020
q
2
p^2+2020q^2
p
2
+
2020
q
2
. Find the sum of the possible values of
p
p
p
.Proposed by Andy Xu
21
1
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Gunga P21
King William is located at
(
1
,
1
)
(1, 1)
(
1
,
1
)
on the coordinate plane. Every day, he chooses one of the eight lattice points closest to him and moves to one of them with equal probability. When he exits the region bounded by the
x
,
y
x, y
x
,
y
axes and
x
+
y
=
4
x+y = 4
x
+
y
=
4
, he stops moving and remains there forever. Given that after an arbitrarily large amount of time he must exit the region, the probability he ends up on
x
+
y
=
4
x+y = 4
x
+
y
=
4
can be expressed as
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
.Proposed by Andrew Wen
20
2
Hide problems
Team Round P20
Compute the sum of all integers
x
x
x
for which there exists an integer
y
y
y
such that
x
3
+
x
y
+
y
3
=
503.
x^3+xy+y^3=503.
x
3
+
x
y
+
y
3
=
503.
Proposed by Nathan Xiong
Gunga P20
In the interior of square
A
B
C
D
ABCD
A
BC
D
with side length
1
1
1
, a point
P
P
P
is chosen such that the lines
ℓ
1
,
ℓ
2
\ell_1, \ell_2
ℓ
1
,
ℓ
2
through
P
P
P
parallel to
A
C
AC
A
C
and
B
D
BD
B
D
, respectively, divide the square into four distinct regions, the smallest of which has area
R
\mathcal{R}
R
. The area of the region of all points
P
P
P
for which
R
≥
1
6
\mathcal{R} \geq \tfrac{1}{6}
R
≥
6
1
can be expressed as
a
−
b
c
d
\frac{a-b\sqrt{c}}{d}
d
a
−
b
c
where
gcd
(
a
,
b
,
d
)
=
1
\gcd(a,b,d)=1
g
cd
(
a
,
b
,
d
)
=
1
and
c
c
c
is not divisible by the square of any prime. Compute
a
+
b
+
c
+
d
a+b+c+d
a
+
b
+
c
+
d
.Proposed by Andrew Wen
19
2
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Team Round P19
Consider the
5
5
5
by
5
5
5
by
5
5
5
equilateral triangular grid as shown:[asy] size(5cm); real n = 5; for (int i = 0; i < n; ++i) { draw((0.5*i,0.866*i)--(n-0.5*i,0.866*i)); } for (int i = 0; i < n; ++i) { draw((n-i,0)--((n-i)/2,(n-i)*0.866)); } for (int i = 0; i < n; ++i) { draw((i,0)--((n+i)/2,(n-i)*0.866)); } [/asy]Ethan chooses two distinct upward-oriented equilateral triangles bounded by the gridlines. The probability that Ethan chooses two triangles that share exactly one vertex can be expressed 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
.Proposed by Andrew Wen
Gunga P19
Let
S
S
S
be the set of triples
(
a
,
b
,
c
)
(a,b,c)
(
a
,
b
,
c
)
of non-negative integers with
a
+
b
+
c
a+b+c
a
+
b
+
c
even. The value of the sum
∑
(
a
,
b
,
c
)
∈
S
1
2
a
3
b
5
c
\sum_{(a,b,c)\in S}\frac{1}{2^a3^b5^c}
(
a
,
b
,
c
)
∈
S
∑
2
a
3
b
5
c
1
can be expressed as
m
n
\frac{m}{n}
n
m
for relative prime positive integers
m
m
m
and
n
n
n
. Compute
m
+
n
m+n
m
+
n
.Proposed by Nathan Xiong
18
2
Hide problems
Team Round P18
Let
△
A
B
C
\triangle ABC
△
A
BC
be a triangle with side length
B
C
=
4
6
BC= 4\sqrt{6}
BC
=
4
6
. Denote
ω
\omega
ω
as the circumcircle of
△
A
B
C
\triangle{ABC}
△
A
BC
. Point
D
D
D
lies on
ω
\omega
ω
such that
A
D
AD
A
D
is the diameter of
ω
\omega
ω
. Let
N
N
N
be the midpoint of arc
B
C
BC
BC
that contains
A
A
A
.
H
H
H
is the intersection of the altitudes in
△
A
B
C
\triangle{ABC}
△
A
BC
and it is given that
H
N
=
H
D
=
6
HN = HD= 6
H
N
=
HD
=
6
. If the area of
△
A
B
C
\triangle{ABC}
△
A
BC
can be expressed as
a
b
c
\frac{a\sqrt{b}}{c}
c
a
b
, where
a
,
b
,
c
a,b,c
a
,
b
,
c
are positive integers with
a
a
a
and
c
c
c
relatively prime and
b
b
b
not divisible by the square of any prime, compute
a
+
b
+
c
a+b+c
a
+
b
+
c
.Proposed by Andy Xu
Gunga P18
Find the largest positive integer
n
n
n
such that the number
(
2
n
)
!
(2n)!
(
2
n
)!
ends with
10
10
10
more zeroes than the number
n
!
n!
n
!
.Proposed by Andy Xu
17
2
Hide problems
Team Round P17
Compute the remainder when
1
0
2021
10^{2021}
1
0
2021
is divided by
10101
10101
10101
.Proposed by Nathan Xiong
Gunga P17
Isosceles trapezoid
A
B
C
D
ABCD
A
BC
D
has side lengths
A
B
=
6
AB = 6
A
B
=
6
and
C
D
=
12
CD = 12
C
D
=
12
, while
A
D
=
B
C
AD = BC
A
D
=
BC
. It is given that
O
O
O
, the circumcenter of
A
B
C
D
ABCD
A
BC
D
, lies in the interior of the trapezoid. The extensions of lines
A
D
AD
A
D
and
B
C
BC
BC
intersect at
T
T
T
. Given that
O
T
=
18
OT = 18
OT
=
18
, the area of
A
B
C
D
ABCD
A
BC
D
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 where
c
c
c
is not divisible by the square of any prime. Compute
a
+
b
+
c
a+b+c
a
+
b
+
c
.Proposed by Andrew Wen
16
2
Hide problems
Team Round P16
Let
△
A
B
C
\triangle ABC
△
A
BC
have
∠
A
B
C
=
6
7
∘
\angle ABC=67^{\circ}
∠
A
BC
=
6
7
∘
. Point
X
X
X
is chosen such that
A
B
=
X
C
AB = XC
A
B
=
XC
,
∠
X
A
C
=
3
2
∘
\angle{XAC}=32^\circ
∠
X
A
C
=
3
2
∘
, and
∠
X
C
A
=
3
5
∘
\angle{XCA}=35^\circ
∠
XC
A
=
3
5
∘
. Compute
∠
B
A
C
\angle{BAC}
∠
B
A
C
in degrees.Proposed by Raina Yang
Gunga P16
Let
1
,
7
,
19
,
…
1,7,19,\ldots
1
,
7
,
19
,
…
be the sequence of numbers such that for all integers
n
≥
1
n\ge 1
n
≥
1
, the average of the first
n
n
n
terms is equal to the
n
n
n
th perfect square. Compute the last three digits of the
2021
2021
2021
st term in the sequence.Proposed by Nathan Xiong
15
2
Hide problems
Team Round P15
Consider the polynomial
P
(
x
)
=
x
3
+
3
x
2
+
6
x
+
10.
P(x)=x^3+3x^2+6x+10.
P
(
x
)
=
x
3
+
3
x
2
+
6
x
+
10.
Let its three roots be
a
a
a
,
b
b
b
,
c
c
c
. Define
Q
(
x
)
Q(x)
Q
(
x
)
to be the monic cubic polynomial with roots
a
b
ab
ab
,
b
c
bc
b
c
,
c
a
ca
c
a
. Compute
∣
Q
(
1
)
∣
|Q(1)|
∣
Q
(
1
)
∣
.Proposed by Nathan Xiong
Gunga P15
Let
a
,
b
,
c
,
d
a,b,c,d
a
,
b
,
c
,
d
be the four roots of the polynomial
x
4
+
3
x
3
−
x
2
+
x
−
2.
x^4+3x^3-x^2+x-2.
x
4
+
3
x
3
−
x
2
+
x
−
2.
Given that
1
a
+
1
b
+
1
c
+
1
d
=
1
2
\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}=\frac{1}{2}
a
1
+
b
1
+
c
1
+
d
1
=
2
1
and
1
a
2
+
1
b
2
+
1
c
2
+
1
d
2
=
−
3
4
\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}+\frac{1}{d^2}=-\frac{3}{4}
a
2
1
+
b
2
1
+
c
2
1
+
d
2
1
=
−
4
3
, the value of
1
a
3
+
1
b
3
+
1
c
3
+
1
d
3
\frac{1}{a^3}+\frac{1}{b^3}+\frac{1}{c^3}+\frac{1}{d^3}
a
3
1
+
b
3
1
+
c
3
1
+
d
3
1
can be expressed 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
.Proposed by Nathan Xiong
14
2
Hide problems
Team Round P14
Evaluate
⌊
1
×
5
7
⌋
+
⌊
2
×
5
7
⌋
+
⌊
3
×
5
7
⌋
+
⋯
+
⌊
100
×
5
7
⌋
.
\left\lfloor\frac{1\times 5}{7}\right\rfloor + \left\lfloor\frac{2\times 5}{7}\right\rfloor + \left\lfloor\frac{3\times 5}{7}\right\rfloor+\cdots+\left\lfloor\frac{100\times 5}{7}\right\rfloor.
⌊
7
1
×
5
⌋
+
⌊
7
2
×
5
⌋
+
⌊
7
3
×
5
⌋
+
⋯
+
⌊
7
100
×
5
⌋
.
Proposed by Nathan Xiong
Gunga P14
Sinclair starts with the number
1
1
1
. Every minute, he either squares his number or adds
1
1
1
to his number, both with equal probability. What is the expected number of minutes until his number is divisible by
3
3
3
?Proposed by Nathan Xiong
13
2
Hide problems
Team Round P13
Bob has
30
30
30
identical unit cubes. He can join two cubes together by gluing a face on one cube to a face on the other cube. He must join all the cubes together into one connected solid. Over all possible solids that Bob can build, what is the largest possible surface area of the solid?Proposed by Nathan Xiong
Gunga P13
Determine the greatest power of
2
2
2
that is a factor of
3
15
+
3
11
+
3
6
+
1
3^{15}+3^{11}+3^{6}+1
3
15
+
3
11
+
3
6
+
1
.Proposed by Nathan Xiong
12
2
Hide problems
Team Round P12
Let
△
A
B
C
\triangle ABC
△
A
BC
have
A
B
=
9
AB=9
A
B
=
9
and
A
C
=
10
AC=10
A
C
=
10
. A semicircle is inscribed in
△
A
B
C
\triangle ABC
△
A
BC
with its center on segment
B
C
BC
BC
such that it is tangent
A
B
AB
A
B
at point
D
D
D
and
A
C
AC
A
C
at point
E
E
E
. If
A
D
=
2
D
B
AD=2DB
A
D
=
2
D
B
and
r
r
r
is the radius of the semicircle,
r
2
r^2
r
2
can be expressed 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
.Proposed by Andy Xu
Gunga P12
Andy wishes to open an electronic lock with a keypad containing all digits from
0
0
0
to
9
9
9
. He knows that the password registered in the system is
2469
2469
2469
. Unfortunately, he is also aware that exactly two different buttons (but he does not know which ones)
a
‾
\underline{a}
a
and
b
‾
\underline{b}
b
on the keypad are broken
−
-
−
when
a
‾
\underline{a}
a
is pressed the digit
b
b
b
is registered in the system, and when
b
‾
\underline{b}
b
is pressed the digit
a
a
a
is registered in the system. Find the least number of attempts Andy needs to surely be able to open the lock.Proposed by Andrew Wen
11
2
Hide problems
Team Round P11
Find the product of all possible real values for
k
k
k
such that the system of equations
x
2
+
y
2
=
80
x^2+y^2= 80
x
2
+
y
2
=
80
x
2
+
y
2
=
k
+
2
x
−
8
y
x^2+y^2= k+2x-8y
x
2
+
y
2
=
k
+
2
x
−
8
y
has exactly one real solution
(
x
,
y
)
(x,y)
(
x
,
y
)
.Proposed by Nathan Xiong
Gunga P11
Let
A
B
C
D
ABCD
A
BC
D
be a rectangle with
A
B
=
10
AB=10
A
B
=
10
and
B
C
=
26
BC=26
BC
=
26
. Let
ω
1
\omega_1
ω
1
be the circle with diameter
A
B
‾
\overline{AB}
A
B
and
ω
2
\omega_2
ω
2
be the circle with diameter
C
D
‾
\overline{CD}
C
D
. Suppose
ℓ
\ell
ℓ
is a common internal tangent to
ω
1
\omega_1
ω
1
and
ω
2
\omega_2
ω
2
and that
ℓ
\ell
ℓ
intersects
A
D
AD
A
D
and
B
C
BC
BC
at
E
E
E
and
F
F
F
respectively. What is
E
F
EF
EF
?[asy] size(10cm); draw((0,0)--(26,0)--(26,10)--(0,10)--cycle); draw((1,0)--(25,10)); draw(circle((0,5),5)); draw(circle((26,5),5)); dot((1,0)); dot((25,10)); label("
E
E
E
",(1,0),SE); label("
F
F
F
",(25,10),NW); label("
A
A
A
", (0,0), SW); label("
B
B
B
", (0,10), NW); label("
C
C
C
", (26,10), NE); label("
D
D
D
", (26,0), SE); dot((0,0)); dot((0,10)); dot((26,0)); dot((26,10)); [/asy]Proposed by Nathan Xiong
10
4
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9
4
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8
4
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7
4
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6
4
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5
4
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4
4
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3
4
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2
4
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1
4
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