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Vietnam Team Selection Test
2012 Vietnam Team Selection Test
2
Best constant c for the positive reals a_i
Best constant c for the positive reals a_i
Source: VN TST 2012 problem 2 day 2
April 17, 2012
inequalities
inequalities proposed
Problem Statement
Prove that
c
=
10
24
c=10\sqrt{24}
c
=
10
24
is the largest constant such that if there exist positive numbers
a
1
,
a
2
,
…
,
a
17
a_1,a_2,\ldots ,a_{17}
a
1
,
a
2
,
…
,
a
17
satisfying:
∑
i
=
1
17
a
i
2
=
24
,
∑
i
=
1
17
a
i
3
+
∑
i
=
1
17
a
i
<
c
\sum_{i=1}^{17}a_i^2=24,\ \sum_{i=1}^{17}a_i^3+\sum_{i=1}^{17}a_i<c
i
=
1
∑
17
a
i
2
=
24
,
i
=
1
∑
17
a
i
3
+
i
=
1
∑
17
a
i
<
c
then for every
i
,
j
,
k
i,j,k
i
,
j
,
k
such that
1
≤
1
<
j
<
k
≤
17
1\le 1<j<k\le 17
1
≤
1
<
j
<
k
≤
17
, we have that
x
i
,
x
j
,
x
k
x_i,x_j,x_k
x
i
,
x
j
,
x
k
are sides of a triangle.
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