MathDB
Problems
Contests
National and Regional Contests
Russia Contests
All-Russian Olympiad Regional Round
1999 All-Russian Olympiad Regional Round
9.3
9.3
Part of
1999 All-Russian Olympiad Regional Round
Problems
(1)
sum 1/x^k >= sum x^k - All-Russian MO 1999 Regional (R4) 9.3
Source:
9/25/2024
The product of positive numbers
x
,
y
x, y
x
,
y
and
z
z
z
is equal to
1
1
1
. Prove that if it holds that
1
x
+
1
y
+
1
z
≥
x
+
y
+
z
,
\frac1x +\frac1y + \frac1z \ge x + y + z,
x
1
+
y
1
+
z
1
≥
x
+
y
+
z
,
then for any natural
k
k
k
, holds the inequality
1
x
k
+
1
y
k
+
1
z
k
≥
x
k
+
y
k
+
z
k
.
\frac{1}{x^k} +\frac{1}{y^k} + \frac{1}{z^k} \ge x^k + y^k + z^k.
x
k
1
+
y
k
1
+
z
k
1
≥
x
k
+
y
k
+
z
k
.
algebra
inequalities