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2020 BAMO
2020 BAMO
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Subcontests
(7)
5
1
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2020 BAMO-12: 5
Let
S
S
S
be a set of
a
+
b
+
3
a+b+3
a
+
b
+
3
points on a sphere, where
a
a
a
,
b
b
b
are nonnegative integers and no four points of
S
S
S
are coplanar. Determine how many planes pass through three points of
S
S
S
and separate the remaining points into
a
a
a
points on one side of the plane and
b
b
b
points on the other side.
4
1
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2020 BAMO-12: 4
Consider
△
A
B
C
\triangle ABC
△
A
BC
. Choose a point
M
M
M
on side
B
C
BC
BC
and let
O
O
O
be the center of the circle passing through the vertices of
△
A
B
M
\triangle ABM
△
A
BM
. Let
k
k
k
be the circle that passes through
A
A
A
and
M
M
M
and whose center lies on
B
C
BC
BC
. Let line
M
O
MO
MO
intersect
K
K
K
again in point
K
K
K
. Prove that the line
B
K
BK
B
K
is the same for any point
M
M
M
on segment
B
C
BC
BC
, so long as all of these constructions are well-defined.Proposed by Evan Chen
D/2
1
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2020 BAMO-8: D
Here’s a screenshot of the problem. If someone could LaTEX a diagram, that would be great!
A
1
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2020 BAMO-8: A
A trapezoid is divided into seven strips of equal width as shown. What fraction of the trapezoid’s area is shaded?
E/3
1
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2020 BAMO 8: E, BAMO 12: 3 Inserting Digits
The integer
202020
202020
202020
is a multiple of
91
91
91
. For each positive integer
n
n
n
, show how
n
n
n
additional
2
2
2
's may be inserted into the digits of
202020
202020
202020
such that the resulting
(
n
+
6
)
(n+6)
(
n
+
6
)
-digit number is also a multiple of
91
91
91
. For example, a possible way to do this when
n
=
5
n=5
n
=
5
is 22020220222 (the inserted
2
2
2
's are underlined).
C/1
1
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2020 BAMO-8: C, BAMO-12: 1 Equation
Find all real numbers
x
x
x
that satisfy the equation
x
−
2020
1
+
x
−
2019
2
+
⋯
+
x
−
2000
21
=
x
−
1
2020
+
x
−
2
2019
+
⋯
+
x
−
21
2000
,
\frac{x-2020}{1}+\frac{x-2019}{2}+\cdots+\frac{x-2000}{21}=\frac{x-1}{2020}+\frac{x-2}{2019}+\cdots+\frac{x-21}{2000},
1
x
−
2020
+
2
x
−
2019
+
⋯
+
21
x
−
2000
=
2020
x
−
1
+
2019
x
−
2
+
⋯
+
2000
x
−
21
,
and simplify your answer(s) as much as possible. Justify your solution.
B
1
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2020 BAMO-8: B Exchanging Money
Four friends, Anna, Bob, Celia, and David, exchanged some money. For any two of these friends, exactly one gave money to the other. For example, Celia could have given some money to David but then David would not have given money to Celia. In the end, each person broke even (meaning that no one made or lost any money).(a) Is it possible that the amounts of money given were
10
10
10
,
20
20
20
,
30
30
30
,
40
40
40
,
50
50
50
,
60
60
60
? (b) Is it possible that the amounts of money given were
20
20
20
,
30
30
30
,
40
40
40
,
50
50
50
,
60
60
60
,
70
70
70
?For each part, if your answer is yes, show that the situation is possible by describing who could have given what amounts to whom. If your answer is no, prove that the situation is not possible.