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National and Regional Contests
Vietnam Contests
Vietnam National Olympiad
1965 Vietnam National Olympiad
3
3
Part of
1965 Vietnam National Olympiad
Problems
(1)
min of Σx_i^m from i=1 to n, where Σx_i=k, x_i nonnegative real, m,n positive
Source: Vietnamese MO (VMO) 1965
8/22/2018
1) Two nonnegative real numbers
x
,
y
x, y
x
,
y
have constant sum
a
a
a
. Find the minimum value of
x
m
+
y
m
x^m + y^m
x
m
+
y
m
, where m is a given positive integer. 2) Let
m
,
n
m, n
m
,
n
be positive integers and
k
k
k
a positive real number. Consider nonnegative real numbers
x
1
,
x
2
,
.
.
.
,
x
n
x_1, x_2, . . . , x_n
x
1
,
x
2
,
...
,
x
n
having constant sum
k
k
k
. Prove that the minimum value of the quantity
x
1
m
+
.
.
.
+
x
n
m
x^m_1+ ... + x^m_n
x
1
m
+
...
+
x
n
m
occurs when
x
1
=
x
2
=
.
.
.
=
x
n
x_1 = x_2 = ... = x_n
x
1
=
x
2
=
...
=
x
n
.
algebra
Inequality