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Rioplatense Mathematical Olympiad, Level 3
1998 Rioplatense Mathematical Olympiad, Level 3
4
4
Part of
1998 Rioplatense Mathematical Olympiad, Level 3
Problems
(1)
a + b is a power of 2 for subset of {1,2,..., 1998} with 1000 elements
Source: Rioplatense 1998 L3 P4
9/19/2022
Let
M
M
M
be a subset of
{
1
,
2
,
.
.
.
,
1998
}
\{1,2,..., 1998\}
{
1
,
2
,
...
,
1998
}
with
1000
1000
1000
elements. Prove that it is always possible to find two elements
a
a
a
and
b
b
b
in
M
M
M
, not necessarily distinct, such that
a
+
b
a + b
a
+
b
is a power of
2
2
2
.
number theory
power of 2