MathDB
Problems
Contests
International Contests
APMO
1991 APMO
3
3
Part of
1991 APMO
Problems
(1)
Sum a_n = sum b_n
Source: APMO 1991
3/11/2006
Let
a
1
a_1
a
1
,
a
2
a_2
a
2
,
⋯
\cdots
⋯
,
a
n
a_n
a
n
,
b
1
b_1
b
1
,
b
2
b_2
b
2
,
⋯
\cdots
⋯
,
b
n
b_n
b
n
be positive real numbers such that
a
1
+
a
2
+
⋯
+
a
n
=
b
1
+
b
2
+
⋯
+
b
n
a_1 + a_2 + \cdots + a_n = b_1 + b_2 + \cdots + b_n
a
1
+
a
2
+
⋯
+
a
n
=
b
1
+
b
2
+
⋯
+
b
n
. Show that
a
1
2
a
1
+
b
1
+
a
2
2
a
2
+
b
2
+
⋯
+
a
n
2
a
n
+
b
n
≥
a
1
+
a
2
+
⋯
+
a
n
2
\frac{a_1^2}{a_1 + b_1} + \frac{a_2^2}{a_2 + b_2} + \cdots + \frac{a_n^2}{a_n + b_n} \geq \frac{a_1 + a_2 + \cdots + a_n}{2}
a
1
+
b
1
a
1
2
+
a
2
+
b
2
a
2
2
+
⋯
+
a
n
+
b
n
a
n
2
≥
2
a
1
+
a
2
+
⋯
+
a
n
inequalities
n-variable inequality