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National and Regional Contests
France Contests
France Team Selection Test
2004 France Team Selection Test
1
A trivial inequality
A trivial inequality
Source: French TST 2004, pb.4
May 25, 2004
inequalities
calculus
derivative
function
AMC
USA(J)MO
USAMO
Problem Statement
Let
n
n
n
be a positive integer, and
a
1
,
.
.
.
,
a
n
,
b
1
,
.
.
.
,
b
n
a_1,...,a_n, b_1,..., b_n
a
1
,
...
,
a
n
,
b
1
,
...
,
b
n
be
2
n
2n
2
n
positive real numbers such that
a
1
+
.
.
.
+
a
n
=
b
1
+
.
.
.
+
b
n
=
1
a_1 + ... + a_n = b_1 + ... + b_n = 1
a
1
+
...
+
a
n
=
b
1
+
...
+
b
n
=
1
. Find the minimal value of
a
1
2
a
1
+
b
1
+
a
2
2
a
2
+
b
2
+
.
.
.
+
a
n
2
a
n
+
b
n
\frac {a_1^2} {a_1 + b_1} + \frac {a_2^2} {a_2 + b_2} + ...+ \frac {a_n^2} {a_n + b_n}
a
1
+
b
1
a
1
2
+
a
2
+
b
2
a
2
2
+
...
+
a
n
+
b
n
a
n
2
.
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