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Math Open At Andover problems
2022 MOAA
2022 MOAA
Part of
Math Open At Andover problems
Subcontests
(18)
Speed
1
Hide problems
2022 MOAA Speed Round - Math Open At Andover
p1. What is the value of the sum
2
+
20
+
202
+
2022
2 + 20 + 202 + 2022
2
+
20
+
202
+
2022
? p2. Find the smallest integer greater than
10000
10000
10000
that is divisible by
12
12
12
. p3. Valencia chooses a positive integer factor of
6
10
6^{10}
6
10
at random. The probability that it is odd can be expressed in the form
m
n
\frac{m}{n}
n
m
where
m
m
m
and
n
n
n
are relatively prime integers. Find
m
+
n
m + n
m
+
n
. p4. How many three digit positive integers are multiples of
4
4
4
but not
8
8
8
? p5. At the Jane Street store, Andy accidentally buys
5
5
5
dollars more worth of shirts than he had planned. Originally, including the tip to the cashier, he planned to spend all of the remaining
90
90
90
dollars on his giftcard. To compensate for his gluttony, Andy instead gives the cashier a smaller,
12.5
%
12.5\%
12.5%
tip so that he still spends
90
90
90
dollars total. How much percent tip was Andy originally planning on giving? p6. Let
A
,
B
,
C
,
D
A,B,C,D
A
,
B
,
C
,
D
be four coplanar points satisfying the conditions
A
B
=
16
AB = 16
A
B
=
16
,
A
C
=
B
C
=
10
AC = BC =10
A
C
=
BC
=
10
, and
A
D
=
B
D
=
17
AD = BD = 17
A
D
=
B
D
=
17
. What is the minimum possible area of quadrilateral
A
D
B
C
ADBC
A
D
BC
? p7. How many ways are there to select a set of three distinct points from the vertices of a regular hexagon so that the triangle they form has its smallest angle(s) equal to
3
0
o
30^o
3
0
o
? p8. Jaeyong rolls five fair
6
6
6
-sided die. The probability that the sum of some three rolls is exactly
8
8
8
times the sum of the other two rolls 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
. p9. Find the least positive integer n for there exists some positive integer
k
>
1
k > 1
k
>
1
for which
k
k
k
and
k
+
2
k + 2
k
+
2
both divide
11...1
⏟
n
1
′
s
\underbrace{11...1}_{n\,\,\,1's}
n
1
′
s
11...1
. p10. For some real constant
k
k
k
, line
y
=
k
y = k
y
=
k
intersects the curve
y
=
∣
x
4
−
1
∣
y = |x^4-1|
y
=
∣
x
4
−
1∣
four times: points
A
A
A
,
B
B
B
,
C
C
C
and
D
D
D
, labeled from left to right. If
B
C
=
2
A
B
=
2
C
D
BC = 2AB = 2CD
BC
=
2
A
B
=
2
C
D
, then the value of
k
k
k
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
. p11. Let a be a positive real number and
P
(
x
)
=
x
2
−
8
x
+
a
P(x) = x^2 -8x+a
P
(
x
)
=
x
2
−
8
x
+
a
and
Q
(
x
)
=
x
2
−
8
x
+
a
+
1
Q(x) = x^2 -8x+a+1
Q
(
x
)
=
x
2
−
8
x
+
a
+
1
be quadratics with real roots such that the positive difference of the roots of
P
(
x
)
P(x)
P
(
x
)
is exactly one more than the positive difference of the roots of
Q
(
x
)
Q(x)
Q
(
x
)
. The value of a can be written as a common fraction
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
. p12. Let
A
B
C
D
ABCD
A
BC
D
be a trapezoid satisfying
A
B
∥
C
D
AB \parallel CD
A
B
∥
C
D
,
A
B
=
3
AB = 3
A
B
=
3
,
C
D
=
4
CD = 4
C
D
=
4
, with area
35
35
35
. Given
A
C
AC
A
C
and
B
D
BD
B
D
intersect at
E
E
E
, and
M
M
M
,
N
N
N
,
P
P
P
,
Q
Q
Q
are the midpoints of segments
A
E
AE
A
E
,
B
E
BE
BE
,
C
E
CE
CE
,
D
E
DE
D
E
, respectively, the area of the intersection of quadrilaterals
A
B
P
Q
ABPQ
A
BPQ
and
C
D
M
N
CDMN
C
D
MN
can be expressed as
m
n
\frac{m}{n}
n
m
where
m
,
n
m, n
m
,
n
are relatively prime positive integers. Find
m
+
n
m + n
m
+
n
. p13. There are
8
8
8
distinct points
P
1
,
P
2
,
.
.
.
,
P
8
P_1, P_2, ... , P_8
P
1
,
P
2
,
...
,
P
8
on a circle. How many ways are there to choose a set of three distinct chords such that every chord has to touch at least one other chord, and if any two chosen chords touch, they must touch at a shared endpoint? p14. For every positive integer
k
k
k
, let
f
(
k
)
>
1
f(k) > 1
f
(
k
)
>
1
be defined as the smallest positive integer for which
f
(
k
)
f(k)
f
(
k
)
and
f
(
k
)
2
f(k)^2
f
(
k
)
2
leave the same remainder when divided by
k
k
k
. The minimum possible value of
1
x
f
(
x
)
\frac{1}{x}f(x)
x
1
f
(
x
)
across all positive integers
x
≤
1000
x \le 1000
x
≤
1000
can be expressed as
m
n
\frac{m}{n}
n
m
for relatively prime positive integers
m
,
n
m, n
m
,
n
. Find
m
+
n
m + n
m
+
n
. p15. In triangle
A
B
C
ABC
A
BC
, let
I
I
I
be the incenter and
O
O
O
be the circumcenter. If
A
O
AO
A
O
bisects
∠
I
A
C
\angle IAC
∠
I
A
C
,
A
B
+
A
C
=
21
AB + AC = 21
A
B
+
A
C
=
21
, and
B
C
=
7
BC = 7
BC
=
7
, then the length of segment
A
I
AI
A
I
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
. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.
Accuracy
1
Hide problems
2022 MOAA Accuracy Round - Math Open At Andover
p1. Find the last digit of
202
2
2022
2022^{2022}
202
2
2022
. p2. Let
a
1
<
a
2
<
.
.
.
<
a
8
a_1 < a_2 <... < a_8
a
1
<
a
2
<
...
<
a
8
be eight real numbers in an increasing arithmetic progression. If
a
1
+
a
3
+
a
5
+
a
7
=
39
a_1 + a_3 + a_5 + a_7 = 39
a
1
+
a
3
+
a
5
+
a
7
=
39
and
a
2
+
a
4
+
a
6
+
a
8
=
40
a_2 + a_4 + a_6 + a_8 = 40
a
2
+
a
4
+
a
6
+
a
8
=
40
, determine the value of
a
1
a_1
a
1
. p3. Patrick tries to evaluate the sum of the first
2022
2022
2022
positive integers, but accidentally omits one of the numbers,
N
N
N
, while adding all of them manually, and incorrectly arrives at a multiple of
1000
1000
1000
. If adds correctly otherwise, find the sum of all possible values of
N
N
N
. p4. A machine picks a real number uniformly at random from
[
0
,
2022
]
[0, 2022]
[
0
,
2022
]
. Andrew randomly chooses a real number from
[
2020
,
2022
]
[2020, 2022]
[
2020
,
2022
]
. The probability that Andrew’s number is less than the machine’s number 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
. p5. Let
A
B
C
D
ABCD
A
BC
D
be a square and
P
P
P
be a point inside it such that the distances from
P
P
P
to sides
A
B
AB
A
B
and
A
D
AD
A
D
respectively are
2
2
2
and
4
4
4
, while
P
C
=
6
PC = 6
PC
=
6
. If the side length of the square can be expressed in the form
a
+
b
a +\sqrt{b}
a
+
b
for positive integers
a
,
b
a, b
a
,
b
, then determine
a
+
b
a + b
a
+
b
. p6. Positive integers
a
1
,
a
2
,
.
.
.
,
a
20
a_1, a_2, ..., a_{20}
a
1
,
a
2
,
...
,
a
20
sum to
57
57
57
. Given that
M
M
M
is the minimum possible value of the quantity
a
1
!
a
2
!
.
.
.
a
20
!
a_1!a_2!...a_{20}!
a
1
!
a
2
!
...
a
20
!
, find the number of positive integer divisors of
M
M
M
. p7. Jessica has
16
16
16
balls in a box, where
15
15
15
of them are red and one is blue. Jessica draws balls out the box three at a time until one of the three is blue. If she ever draws three red marbles, she discards one of them and shuffles the remaining two back into the box. The expected number of draws it takes for Jessica to draw the blue ball can be written as a common fraction
m
n
\frac{m}{n}
n
m
where
m
,
n
m, n
m
,
n
are relatively prime positive integers. Find
m
+
n
m + n
m
+
n
. p8. The Lucas sequence is defined by these conditions:
L
0
=
2
L_0 = 2
L
0
=
2
,
L
1
=
1
L_1 = 1
L
1
=
1
, and
L
n
+
2
=
L
n
+
1
+
L
n
L_{n+2} =L_{n+1} +L_n
L
n
+
2
=
L
n
+
1
+
L
n
for all
n
≥
0
n \ge 0
n
≥
0
. Determine the remainder when
L
2019
2
+
L
2020
2
L^2_{2019} +L^2_{2020}
L
2019
2
+
L
2020
2
is divided by
L
2023
L_{2023}
L
2023
. p9. Let
A
B
C
D
ABCD
A
BC
D
be a parallelogram. Point
P
P
P
is selected in its interior such that the distance from
P
P
P
to
B
C
BC
BC
is exactly
6
6
6
times the distance from
P
P
P
to
A
D
AD
A
D
, and
∠
A
P
B
=
∠
C
P
D
=
9
0
o
\angle APB = \angle CPD = 90^o
∠
A
PB
=
∠
CP
D
=
9
0
o
. Given that
A
P
=
2
AP = 2
A
P
=
2
and
C
P
=
9
CP = 9
CP
=
9
, the area of
A
B
C
D
ABCD
A
BC
D
can be expressed as
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
. p10. Consider the polynomial
P
(
x
)
=
x
35
+
.
.
.
+
x
+
1
P(x) = x^{35} + ... + x + 1
P
(
x
)
=
x
35
+
...
+
x
+
1
. How many pairs
(
i
,
j
)
(i, j)
(
i
,
j
)
of integers are there with
0
≤
i
<
j
≤
35
0 \le i < j \le 35
0
≤
i
<
j
≤
35
such that if we flip the signs of the
x
i
x^i
x
i
and
x
j
x^j
x
j
terms in
P
(
x
)
P(x)
P
(
x
)
to form a new polynomial
Q
(
x
)
Q(x)
Q
(
x
)
, then there exists a nonconstant polynomial
R
(
x
)
R(x)
R
(
x
)
with integer coefficients dividing both
P
(
x
)
P(x)
P
(
x
)
and
Q
(
x
)
Q(x)
Q
(
x
)
? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.
15
1
Hide problems
MOAA 2022 Team #15
Let
I
B
,
I
C
I_B, I_C
I
B
,
I
C
be the
B
,
C
B, C
B
,
C
-excenters of triangle
A
B
C
ABC
A
BC
, respectively. Let
O
O
O
be the circumcenter of
A
B
C
ABC
A
BC
. If
B
I
B
BI_B
B
I
B
is perpendicular to
A
O
AO
A
O
,
A
I
C
=
3
AI_C = 3
A
I
C
=
3
and
A
C
=
4
2
AC = 4\sqrt2
A
C
=
4
2
, then
A
B
2
AB^2
A
B
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
.Note: In triangle
△
A
B
C
\vartriangle ABC
△
A
BC
, the
A
A
A
-excenter is the intersection of the exterior angle bisectors of
∠
A
B
C
\angle ABC
∠
A
BC
and
∠
A
C
B
\angle ACB
∠
A
CB
. The
B
B
B
-excenter and
C
C
C
-excenter are defined similarly.
14
1
Hide problems
MOAA 2022 Team #14
Find the greatest prime number
p
p
p
for which there exists a prime number
q
q
q
such that
p
p
p
divides
4
q
+
1
4^q + 1
4
q
+
1
and
q
q
q
divides
4
p
+
1
4^p + 1
4
p
+
1
.
13
1
Hide problems
MOAA 2022 Team #13
Determine the number of distinct positive real solutions to
⌊
x
⌋
{
x
}
=
1
2022
x
2
\lfloor x \rfloor ^{\{x\}} = \frac{1}{2022}x^2
⌊
x
⌋
{
x
}
=
2022
1
x
2
. Note:
⌊
x
⌋
\lfloor x \rfloor
⌊
x
⌋
is known as the floor function, which returns the greatest integer less than or equal to
x
x
x
. Furthermore,
{
x
}
\{x\}
{
x
}
is defined as
x
−
⌊
x
⌋
x - \lfloor x \rfloor
x
−
⌊
x
⌋
.
12
1
Hide problems
MOAA 2022 Team #12
Triangle
A
B
C
ABC
A
BC
has circumcircle
ω
\omega
ω
where
B
′
B'
B
′
is the point diametrically opposite
B
B
B
and
C
′
C'
C
′
is the point diametrically opposite
C
C
C
. Given
B
′
C
′
B'C'
B
′
C
′
passes through the midpoint of
A
B
AB
A
B
, if
A
C
′
=
3
AC' = 3
A
C
′
=
3
and
B
C
=
7
BC = 7
BC
=
7
, find
A
B
′
2
AB'^2
A
B
′2
..
11
1
Hide problems
MOAA 2022 Team #11
Let a triplet be some set of three distinct pairwise parallel lines.
20
20
20
triplets are drawn on a plane. Find the maximum number of regions these
60
60
60
lines can divide the plane into.
10
1
Hide problems
MOAA 2022 Team #10
Three integers
A
,
B
,
C
A, B, C
A
,
B
,
C
are written on a whiteboard. Every move, Mr. Doba can either subtract
1
1
1
from all numbers on the board, or choose two numbers on the board and subtract
1
1
1
from both of them whilst leaving the third untouched. For how many ordered triples
(
A
,
B
,
C
)
(A, B, C)
(
A
,
B
,
C
)
with
1
≤
A
<
B
<
C
≤
20
1 \le A < B < C\le 20
1
≤
A
<
B
<
C
≤
20
is it possible for Mr. Doba to turn all three of the numbers on the board to
0
0
0
?
9
1
Hide problems
MOAA 2022 Team #9
Emily has two cups
A
A
A
and
B
B
B
, each of which can hold
400
400
400
mL, A initially with
200
200
200
mL of water and
B
B
B
initially with
300
300
300
mL of water. During a round, she chooses the cup with more water (randomly picking if they have the same amount), drinks half of the water in the chosen cup, then pours the remaining half into the other cup and refills the chosen cup to back to half full. If Emily goes for
20
20
20
rounds, how much water does she drink, to the nearest integer?
8
1
Hide problems
MOAA 2022 Team #8
Raina the frog is playing a game in a circular pond with six lilypads around its perimeter numbered clockwise from
1
1
1
to
6
6
6
(so that pad
1
1
1
is adjacent to pad
6
6
6
). She starts at pad
1
1
1
, and when she is on pad i, she may jump to one of its two adjacent pads, or any pad labeled with
j
j
j
for which
j
−
i
j - i
j
−
i
is even. How many jump sequences enable Raina to hop to each pad exactly once?
7
1
Hide problems
MOAA 2022 Team #7
A point
P
P
P
is chosen uniformly at random in the interior of triangle
A
B
C
ABC
A
BC
with side lengths
A
B
=
5
AB = 5
A
B
=
5
,
B
C
=
12
BC = 12
BC
=
12
,
C
A
=
13
CA = 13
C
A
=
13
. The probability that a circle with radius
1
3
\frac13
3
1
centered at
P
P
P
does not intersect the perimeter of
A
B
C
ABC
A
BC
can be written as
m
n
\frac{m}{n}
n
m
where
m
,
n
m, n
m
,
n
are relatively prime positive integers. Find
m
+
n
m + n
m
+
n
.
6
1
Hide problems
MOAA 2022 Team #6
Define a positive integer
n
n
n
to be almost-cubic if it becomes a perfect cube upon concatenating the digit
5
5
5
. For example,
12
12
12
is almost-cubic because
125
=
5
3
125 = 5^3
125
=
5
3
. Find the remainder when the sum of all almost-cubic
n
<
1
0
8
n < 10^8
n
<
1
0
8
is divided by
1000
1000
1000
.
5
1
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MOAA 2022 Team #5
Find the smallest positive integer that is equal to the sum of the product of its digits and the sum of its digits.
4
1
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MOAA 2022 Team #4
Angeline flips three fair coins, and if there are any tails, she then flips all coins that landed tails each one more time. The probability that all coins are now heads 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
.
3
1
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MOAA 2022 Team #3
The area of the figure enclosed by the
x
x
x
-axis,
y
y
y
-axis, and line
7
x
+
8
y
=
15
7x + 8y = 15
7
x
+
8
y
=
15
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
.
2
1
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MOAA 2022 Team #2
While doing her homework for a Momentum Learning class, Valencia draws two intersecting segments
A
B
=
10
AB = 10
A
B
=
10
and
C
D
=
7
CD = 7
C
D
=
7
on a plane. Across all possible configurations of those two segments, determine the maximum possible area of quadrilateral
A
C
B
D
ACBD
A
CB
D
.
1
1
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MOAA 2022 Team #1
Consider the
5
5
5
by
5
5
5
equilateral triangular grid as shown: https://cdn.artofproblemsolving.com/attachments/1/2/cac43ae24fd4464682a7992e62c99af4acaf8f.png How many equilateral triangles are there with sides along the gridlines?
3
Hide problems
2022 MOAA Gunga Bowl - Math Open At Andover - first 3 sets - 9 problems
Set 1G1. The Daily Challenge office has a machine that outputs the number
2.75
2.75
2.75
when operated. If it is operated
12
12
12
times, then what is the sum of all
12
12
12
of the machine outputs? G2. A car traveling at a constant velocity
v
v
v
takes
30
30
30
minutes to travel a distance of
d
d
d
. How long does it take, in minutes, for it travel
10
d
10d
10
d
with a constant velocity of
2.5
v
2.5v
2.5
v
? G3. Andy originally has
3
3
3
times as many jelly beans as Andrew. After Andrew steals 15 of Andy’s jelly beans, Andy now only has
2
2
2
times as many jelly beans as Andrew. Find the number of jelly beans Andy originally had.Set 2 G4. A coin is weighted so that it is
3
3
3
times more likely to come up as heads than tails. How many times more likely is it for the coin to come up heads twice consecutively than tails twice consecutively? G5. There are
n
n
n
students in an Areteem class. When 1 student is absent, the students can be evenly divided into groups of
5
5
5
. When
8
8
8
students are absent, the students can evenly be divided into groups of
7
7
7
. Find the minimum possible value of
n
n
n
. G6. Trapezoid
A
B
C
D
ABCD
A
BC
D
has
A
B
∥
C
D
AB \parallel CD
A
B
∥
C
D
such that
A
B
=
5
AB = 5
A
B
=
5
,
B
C
=
4
BC = 4
BC
=
4
and
D
A
=
2
DA = 2
D
A
=
2
. If there exists a point
M
M
M
on
C
D
CD
C
D
such that
A
M
=
A
D
AM = AD
A
M
=
A
D
and
B
M
=
B
C
BM = BC
BM
=
BC
, find
C
D
CD
C
D
. Set 3 G7. Angeline has
10
10
10
coins (either pennies, nickels, or dimes) in her pocket. She has twice as many nickels as pennies. If she has
62
62
62
cents in total, then how many dimes does she have? G8. Equilateral triangle
A
B
C
ABC
A
BC
has side length
6
6
6
. There exists point
D
D
D
on side
B
C
BC
BC
such that the area of
A
B
D
ABD
A
B
D
is twice the area of
A
C
D
ACD
A
C
D
. There also exists point
E
E
E
on segment
A
D
AD
A
D
such that the area of
A
B
E
ABE
A
BE
is twice the area of
B
D
E
BDE
B
D
E
. If
k
k
k
is the area of triangle
A
C
E
ACE
A
CE
, then find
k
2
k^2
k
2
. G9. A number
n
n
n
can be represented in base
6
6
6
as
a
b
a
‾
6
\underline{aba}_6
aba
6
and base
15
15
15
as
b
a
‾
15
\underline{ba}_{15}
ba
15
, where
a
a
a
and
b
b
b
are not necessarily distinct digits. Find
n
n
n
. PS. You should use hide for answers. Sets 4-6 have been posted [url=https://artofproblemsolving.com/community/c3h3131305p28367080]here and 7-9 [url=https://artofproblemsolving.com/community/c3h3131308p28367095]here.Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.
2022 MOAA Gunga Bowl - Math Open At Andover - last 3 sets - 9 problems
Set 7 G19. How many ordered triples
(
x
,
y
,
z
)
(x, y, z)
(
x
,
y
,
z
)
with
1
≤
x
,
y
,
z
≤
50
1 \le x, y, z \le 50
1
≤
x
,
y
,
z
≤
50
are there such that both
x
+
y
+
z
x + y + z
x
+
y
+
z
and
x
y
+
y
z
+
z
x
xy + yz + zx
x
y
+
yz
+
z
x
are divisible by
6
6
6
? G20. Triangle
A
B
C
ABC
A
BC
has orthocenter
H
H
H
and circumcenter
O
O
O
. If
D
D
D
is the foot of the perpendicular from
A
A
A
to
B
C
BC
BC
, then
A
H
=
8
AH = 8
A
H
=
8
and
H
D
=
3
HD = 3
HD
=
3
. If
∠
A
O
H
=
9
0
o
\angle AOH = 90^o
∠
A
O
H
=
9
0
o
, find
B
C
2
BC^2
B
C
2
. G21. Nate flips a fair coin until he gets two heads in a row, immediately followed by a tails. The probability that he flips the coin exactly
12
12
12
times 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
. Set 8 G22. Let
f
f
f
be a function defined by
f
(
1
)
=
1
f(1) = 1
f
(
1
)
=
1
and
f
(
n
)
=
1
p
f
(
n
p
)
f
(
p
)
+
2
p
−
2
,
f(n) = \frac{1}{p}f\left(\frac{n}{p}\right)f(p) + 2p - 2,
f
(
n
)
=
p
1
f
(
p
n
)
f
(
p
)
+
2
p
−
2
,
where
p
p
p
is the least prime dividing
n
n
n
, for all integers
n
≥
2
n \ge 2
n
≥
2
. Find
f
(
2022
)
f(2022)
f
(
2022
)
. G23. Jessica has
15
15
15
balls numbered
1
1
1
through
15
15
15
. With her left hand, she scoops up
2
2
2
of the balls. With her right hand, she scoops up
2
2
2
of the remaining balls. The probability that the sum of the balls in her left hand is equal to the sum of the balls in her right hand 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
. G24. Let
A
B
C
D
ABCD
A
BC
D
be a cyclic quadrilateral such that its diagonal
B
D
=
17
BD = 17
B
D
=
17
is the diameter of its circumcircle. Given
A
B
=
8
AB = 8
A
B
=
8
,
B
C
=
C
D
BC = CD
BC
=
C
D
, and that a line
ℓ
\ell
ℓ
through A intersects the incircle of
A
B
D
ABD
A
B
D
at two points
P
P
P
and
Q
Q
Q
, the maximum area of
C
P
Q
CP Q
CPQ
can be expressed as a fraction
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
. Set 9This set consists of three estimation problems, with scoring schemes described.G25. Estimate
N
N
N
, the total number of participants (in person and online) at MOAA this year. An estimate of
e
e
e
gets a total of max
(
0
,
⌊
150
(
1
−
∣
N
−
e
∣
N
)
⌋
−
120
)
\left( 0, \lfloor 150 \left( 1- \frac{|N-e|}{N}\right) \rfloor -120 \right)
(
0
,
⌊
150
(
1
−
N
∣
N
−
e
∣
)
⌋
−
120
)
points. G26. If
A
A
A
is the the total number of in person participants at MOAA this year, and
B
B
B
is the total number of online participants at MOAA this year, estimate
N
N
N
, the product
A
B
AB
A
B
. An estimate of
e
e
e
gets a total of max
(
0
,
30
−
⌈
log
10
(
8
∣
N
−
e
∣
+
1
)
⌉
)
(0, 30 - \lceil \log10(8|N - e| + 1)\rceil )
(
0
,
30
−
⌈
lo
g
10
(
8∣
N
−
e
∣
+
1
)⌉)
points. G27. Estimate
N
N
N
, the total number of letters in all the teams that signed up for MOAA this year, both in person and online. An estimate of e gets a total of max
(
0
,
30
−
⌈
7
l
o
g
5
(
∣
N
−
E
∣
)
⌉
)
(0, 30 - \lceil 7 log5(|N - E|)\rceil )
(
0
,
30
−
⌈
7
l
o
g
5
(
∣
N
−
E
∣
)⌉)
points. PS. You should use hide for answers. Sets 1-3 have been posted [url=https://artofproblemsolving.com/community/c3h3131303p28367061]here and 4-6 [url=https://artofproblemsolving.com/community/c3h3131305p28367080]here. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.
2022 MOAA Gunga Bowl - Math Open At Andover - second 3 sets - 9 problems
Set 4 G10. Let
A
B
C
D
ABCD
A
BC
D
be a square with side length
1
1
1
. It is folded along a line
ℓ
\ell
ℓ
that divides the square into two pieces with equal area. The minimum possible area of the resulting shape is
A
A
A
. Find the integer closest to
100
A
100A
100
A
. G11. The
10
10
10
-digit number
1
A
2
B
3
C
5
D
6
E
‾
\underline{1A2B3C5D6E}
1
A
2
B
3
C
5
D
6
E
is a multiple of
99
99
99
. Find
A
+
B
+
C
+
D
+
E
A + B + C + D + E
A
+
B
+
C
+
D
+
E
. G12. Let
A
,
B
,
C
,
D
A, B, C, D
A
,
B
,
C
,
D
be four points satisfying
A
B
=
10
AB = 10
A
B
=
10
and
A
C
=
B
C
=
A
D
=
B
D
=
C
D
=
6
AC = BC = AD = BD = CD = 6
A
C
=
BC
=
A
D
=
B
D
=
C
D
=
6
. If
V
V
V
is the volume of tetrahedron
A
B
C
D
ABCD
A
BC
D
, then find
V
2
V^2
V
2
. Set 5 G13. Nate the giant is running a
5000
5000
5000
meter long race. His first step is
4
4
4
meters, his next step is
6
6
6
meters, and in general, each step is
2
2
2
meters longer than the previous one. Given that his
n
n
n
th step will get him across the finish line, find
n
n
n
. G14. In square
A
B
C
D
ABCD
A
BC
D
with side length
2
2
2
, there exists a point
E
E
E
such that
D
A
=
D
E
DA = DE
D
A
=
D
E
. Let line
B
E
BE
BE
intersect side
A
D
AD
A
D
at
F
F
F
such that
B
E
=
E
F
BE = EF
BE
=
EF
. The area of
A
B
E
ABE
A
BE
can be expressed in the form
a
−
b
a -\sqrt{b}
a
−
b
where
a
a
a
is a positive integer and
b
b
b
is a square-free integer. Find
a
+
b
a + b
a
+
b
. G15. Patrick the Beetle is located at
1
1
1
on the number line. He then makes an infinite sequence of moves where each move is either moving
1
1
1
,
2
2
2
, or
3
3
3
units to the right. The probability that he does reach
6
6
6
at some point in his sequence of moves 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
. Set 6 G16. Find the smallest positive integer
c
c
c
greater than
1
1
1
for which there do not exist integers
0
≤
x
,
y
≤
9
0 \le x, y \le9
0
≤
x
,
y
≤
9
that satisfy
2
x
+
3
y
=
c
2x + 3y = c
2
x
+
3
y
=
c
. G17. Jaeyong is on the point
(
0
,
0
)
(0, 0)
(
0
,
0
)
on the coordinate plane. If Jaeyong is on point
(
x
,
y
)
(x, y)
(
x
,
y
)
, he can either walk to
(
x
+
2
,
y
)
(x + 2, y)
(
x
+
2
,
y
)
,
(
x
+
1
,
y
+
1
)
(x + 1, y + 1)
(
x
+
1
,
y
+
1
)
, or
(
x
,
y
+
2
)
(x, y + 2)
(
x
,
y
+
2
)
. Call a walk to
(
x
+
1
,
y
+
1
)
(x + 1, y + 1)
(
x
+
1
,
y
+
1
)
an Brilliant walk. If Jaeyong cannot have two Brilliant walks in a row, how many ways can he walk to the point
(
10
,
10
)
(10, 10)
(
10
,
10
)
? G18. Deja vu? Let
A
B
C
D
ABCD
A
BC
D
be a square with side length
1
1
1
. It is folded along a line
ℓ
\ell
ℓ
that divides the square into two pieces with equal area. The maximum possible area of the resulting shape is
B
B
B
. Find the integer closest to
100
B
100B
100
B
. PS. You should use hide for answers. Sets 1-3 have been posted [url=https://artofproblemsolving.com/community/c3h3131303p28367061]here and 7-9 [url=https://artofproblemsolving.com/community/c3h3131308p28367095]here. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.