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1999 Harvard-MIT Mathematics Tournament
1999 Harvard-MIT Mathematics Tournament
Part of
Harvard-MIT Mathematics Tournament
Subcontests
(12)
12
1
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1999 HMMT Team #12
A fair coin is flipped every second and the results are recorded with
1
1
1
meaning heads and
0
0
0
meaning tails. What is the probability that the sequence
10101
10101
10101
occurs before the first occurance of the sequence
010101
010101
010101
?
11
1
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angle wanted, 3 angles and 3 circles related 1999 HMMT Team 11
Circles
C
1
C_1
C
1
,
C
2
C_2
C
2
,
C
3
C_3
C
3
have radius
1
1
1
and centers
O
,
P
,
Q
O, P, Q
O
,
P
,
Q
respectively.
C
1
C_1
C
1
and
C
2
C_2
C
2
intersect at
A
A
A
,
C
2
C_2
C
2
and
C
3
C_3
C
3
intersect at
B
B
B
,
C
3
C_3
C
3
and
C
1
C_1
C
1
intersect at
C
C
C
, in such a way that
∠
A
P
B
=
6
0
o
\angle APB = 60^o
∠
A
PB
=
6
0
o
,
∠
B
Q
C
=
3
6
o
\angle BQC = 36^o
∠
BQC
=
3
6
o
, and
∠
C
O
A
=
7
2
o
\angle COA = 72^o
∠
CO
A
=
7
2
o
. Find angle
∠
A
B
C
\angle ABC
∠
A
BC
(degrees).
10
6
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9
6
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8
6
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6
5
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5
6
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4
6
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3
6
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2
6
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1
6
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7
6
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