MathDB
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Contests
National and Regional Contests
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USA - Middle School Tournaments
LMT
2016 LMT
23
23
Part of
2016 LMT
Problems
(1)
2016 LMT Individual #23
Source:
4/10/2016
Call a positive integer
n
≥
2
n\geq 2
n
≥
2
junk if there exist two distinct
n
n
n
digit binary strings
a
1
a
2
⋯
a
n
a_1a_2\cdots a_n
a
1
a
2
⋯
a
n
and
b
1
b
2
⋯
b
n
b_1b_2\cdots b_n
b
1
b
2
⋯
b
n
such that[*]
a
1
+
a
2
=
b
1
+
b
2
,
a_1+a_2=b_1+b_2,
a
1
+
a
2
=
b
1
+
b
2
,
[*]
a
i
−
1
+
a
i
+
a
i
+
1
=
b
i
−
1
+
b
i
+
b
i
+
1
a_{i-1}+a_i+a_{i+1}=b_{i-1}+b_i+b_{i+1}
a
i
−
1
+
a
i
+
a
i
+
1
=
b
i
−
1
+
b
i
+
b
i
+
1
for all
2
≤
i
≤
n
−
1
,
2\leq i\leq n-1,
2
≤
i
≤
n
−
1
,
and [*]
a
n
−
1
+
a
n
=
b
n
−
1
+
b
n
a_{n-1}+a_n=b_{n-1}+b_n
a
n
−
1
+
a
n
=
b
n
−
1
+
b
n
.Find the number of junk positive integers less than or equal to
2016
2016
2016
.Proposed by Nathan Ramesh