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National and Regional Contests
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All-Russian Olympiad
1978 All Soviet Union Mathematical Olympiad
264
ASU 264 All Soviet Union MO 1978 Sx_i &Sigma;1/x_i <= ((a+b)^2/4ab)n^2
ASU 264 All Soviet Union MO 1978 Sx_i &Sigma;1/x_i <= ((a+b)^2/4ab)n^2
Source:
July 6, 2019
algebra
inequalities
Problem Statement
Given
0
<
a
≤
x
1
≤
x
2
≤
.
.
.
≤
x
n
≤
b
0 < a \le x_1\le x_2\le ... \le x_n \le b
0
<
a
≤
x
1
≤
x
2
≤
...
≤
x
n
≤
b
. Prove that
(
x
1
+
x
2
+
.
.
.
+
x
n
)
(
1
x
1
+
1
x
2
+
.
.
.
+
1
x
n
)
≤
(
a
+
b
)
2
4
a
b
n
2
(x_1+x_2+...+x_n)\left ( \frac{1}{x_1}+ \frac{1}{x_2}+...+ \frac{1}{x_n}\right)\le \frac{(a+b)^2}{4ab}n^2
(
x
1
+
x
2
+
...
+
x
n
)
(
x
1
1
+
x
2
1
+
...
+
x
n
1
)
≤
4
ab
(
a
+
b
)
2
n
2
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