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Problems
Contests
National and Regional Contests
Brazil Contests
Brazil National Olympiad
2011 Brazil National Olympiad
6
Hard inequality
Hard inequality
Source: Brazil MO 2011, problem 6
October 16, 2011
inequalities
algorithm
modular arithmetic
inequalities unsolved
Problem Statement
Let
a
1
,
a
2
,
a
3
,
.
.
.
a
2011
a_{1}, a_{2}, a_{3}, ... a_{2011}
a
1
,
a
2
,
a
3
,
...
a
2011
be nonnegative reals with sum
2011
2
\frac{2011}{2}
2
2011
, prove :
∣
∏
c
y
c
(
a
n
−
a
n
+
1
)
∣
=
∣
(
a
1
−
a
2
)
(
a
2
−
a
3
)
.
.
.
(
a
2011
−
a
1
)
∣
≤
3
3
16
.
|\prod_{cyc} (a_{n} - a_{n+1})| = |(a_{1} - a_{2})(a_{2} - a_{3})...(a_{2011}-a_{1})| \le \frac{3 \sqrt3}{16}.
∣
∏
cyc
(
a
n
−
a
n
+
1
)
∣
=
∣
(
a
1
−
a
2
)
(
a
2
−
a
3
)
...
(
a
2011
−
a
1
)
∣
≤
16
3
3
.
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