MathDB
Problems
Contests
National and Regional Contests
Switzerland Contests
Switzerland - Final Round
2007 Switzerland - Final Round
7
7
Part of
2007 Switzerland - Final Round
Problems
(1)
sum \sqrt{a+\sqrt{b + \sqrt{c}}} <= 3\sqrt{m+\sqrt{m + \sqrt{m}}}
Source: Switzerland - 2007 Swiss MO Final Round p7
12/26/2022
Let
a
,
b
,
c
a, b, c
a
,
b
,
c
be nonnegative real numbers with arithmetic mean
m
=
a
+
b
+
c
3
m =\frac{a+b+c}{3}
m
=
3
a
+
b
+
c
. Provethat
a
+
b
+
c
+
b
+
c
+
a
+
c
+
a
+
b
≤
3
m
+
m
+
m
.
\sqrt{a+\sqrt{b + \sqrt{c}}} +\sqrt{b+\sqrt{c + \sqrt{a}}} +\sqrt{c +\sqrt{a + \sqrt{b}}}\le 3\sqrt{m+\sqrt{m + \sqrt{m}}}.
a
+
b
+
c
+
b
+
c
+
a
+
c
+
a
+
b
≤
3
m
+
m
+
m
.
inequalities
algebra
radical