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National and Regional Contests
China Contests
China National Olympiad
2006 China National Olympiad
1
1
Part of
2006 China National Olympiad
Problems
(1)
2006 chinese mathematical olympiad 1
Source:
1/12/2006
Let
a
1
,
a
2
,
…
,
a
k
a_1,a_2,\ldots,a_k
a
1
,
a
2
,
…
,
a
k
be real numbers and
a
1
+
a
2
+
…
+
a
k
=
0
a_1+a_2+\ldots+a_k=0
a
1
+
a
2
+
…
+
a
k
=
0
. Prove that
max
1
≤
i
≤
k
a
i
2
≤
k
3
(
(
a
1
−
a
2
)
2
+
(
a
2
−
a
3
)
2
+
⋯
+
(
a
k
−
1
−
a
k
)
2
)
.
\max_{1\leq i \leq k} a_i^2 \leq \frac{k}{3} \left( (a_1-a_2)^2+(a_2-a_3)^2+\cdots +(a_{k-1}-a_k)^2\right).
1
≤
i
≤
k
max
a
i
2
≤
3
k
(
(
a
1
−
a
2
)
2
+
(
a
2
−
a
3
)
2
+
⋯
+
(
a
k
−
1
−
a
k
)
2
)
.
induction
inequalities
inequalities unsolved