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National and Regional Contests
Argentina Contests
Argentina National Olympiad
2019 Argentina National Olympiad
2
2
Part of
2019 Argentina National Olympiad
Problems
(1)
min sum a_i^2/(a_i+b_i) when sum a_i=1, sum b_i=1
Source: 2019 Argentina OMA Finals L3 p2
1/16/2023
Let
n
≥
1
n\geq1
n
≥
1
be an integer. We have two sequences, each of
n
n
n
positive real numbers
a
1
,
a
2
,
…
,
a
n
a_1,a_2,\ldots ,a_n
a
1
,
a
2
,
…
,
a
n
and
b
1
,
b
2
,
…
,
b
n
b_1,b_2,\ldots ,b_n
b
1
,
b
2
,
…
,
b
n
such that
a
1
+
a
2
+
…
+
a
n
=
1
a_1+a_2+\ldots +a_n=1
a
1
+
a
2
+
…
+
a
n
=
1
and
b
1
+
b
2
+
…
+
b
n
=
1
b_1+b_2+\ldots +b_n=1
b
1
+
b
2
+
…
+
b
n
=
1
. Find the smallest possible value that the sum can take
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}+\ldots +\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
.
algebra
Sum
inequalities