MathDB
Problems
Contests
National and Regional Contests
Canada Contests
Canada National Olympiad
2017 Canada National Olympiad
3
3
Part of
2017 Canada National Olympiad
Problems
(1)
CMO 2017 P3
Source: Canadian Mathematical Olympiad 2017
3/31/2017
Define
S
n
S_n
S
n
as the set
1
,
2
,
⋯
,
n
{1,2,\cdots,n}
1
,
2
,
⋯
,
n
. A non-empty subset
T
n
T_n
T
n
of
S
n
S_n
S
n
is called
b
a
l
a
n
c
e
d
balanced
ba
l
an
ce
d
if the average of the elements of
T
n
T_n
T
n
is equal to the median of
T
n
T_n
T
n
. Prove that, for all
n
n
n
, the number of balanced subsets
T
n
T_n
T
n
is odd.
combinatorics
Sets
Average
median