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National and Regional Contests
Colombia Contests
Colombia Team Selection Test
2005 Colombia Team Selection Test
2005 Colombia Team Selection Test
Part of
Colombia Team Selection Test
Subcontests
(2)
4
1
Hide problems
Nice inequality with means
1. Prove the following inequality for positive reals
a
1
,
a
2
.
.
.
,
a
n
a_1,a_2...,a_n
a
1
,
a
2
...
,
a
n
and
b
1
,
b
2
.
.
.
,
b
n
b_1,b_2...,b_n
b
1
,
b
2
...
,
b
n
:
(
∑
a
i
)
(
∑
b
i
)
≥
(
∑
a
i
+
b
i
)
(
∑
a
i
b
i
a
i
+
b
i
)
(\sum a_i)(\sum b_i)\geq (\sum a_i+b_i)(\sum\frac{a_ib_i}{a_i+b_i})
(
∑
a
i
)
(
∑
b
i
)
≥
(
∑
a
i
+
b
i
)
(
∑
a
i
+
b
i
a
i
b
i
)
1
1
Hide problems
Colombia TST
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be integers such that
a
b
+
b
c
+
c
a
=
3
\frac{a}{b}+\frac{b}{c}+\frac{c}{a}=3
b
a
+
c
b
+
a
c
=
3
prove that
a
b
c
abc
ab
c
is a perfect cube!