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Sweden Contests
Swedish Mathematical Competition
1985 Swedish Mathematical Competition
1
1
Part of
1985 Swedish Mathematical Competition
Problems
(1)
(a-b)^2/8a < (a+b)/2 - \sqrt{ab} < (a-b)^2/8b for a > b > 0,
Source: 1985 Swedish Mathematical Competition p1
3/28/2021
If
a
>
b
>
0
a > b > 0
a
>
b
>
0
, prove the inequality
(
a
−
b
)
2
8
a
<
a
+
b
2
−
a
b
<
(
a
−
b
)
2
8
b
.
\frac{(a-b)^2}{8a}< \frac{a+b}{2}- \sqrt{ab} < \frac{(a-b)^2}{8b}.
8
a
(
a
−
b
)
2
<
2
a
+
b
−
ab
<
8
b
(
a
−
b
)
2
.
inequalities
algebra
radical