MathDB
Problems
Contests
Undergraduate contests
Vojtěch Jarník IMC
2002 VJIMC
Problem 3
inequality in n variables, equality condition
inequality in n variables, equality condition
Source: VJIMC 2002 1.3
July 16, 2021
inequalities
Problem Statement
Positive numbers
x
1
,
…
,
x
n
x_1,\ldots,x_n
x
1
,
…
,
x
n
satisfy
1
1
+
x
1
+
1
1
+
x
2
+
…
+
1
1
+
x
n
=
1.
\frac1{1+x_1}+\frac1{1+x_2}+\ldots+\frac1{1+x_n}=1.
1
+
x
1
1
+
1
+
x
2
1
+
…
+
1
+
x
n
1
=
1.
Prove that
x
1
+
x
2
+
…
+
x
n
≥
(
n
−
1
)
(
1
x
1
+
1
x
2
+
…
+
1
x
n
)
.
\sqrt{x_1}+\sqrt{x_2}+\ldots+\sqrt{x_n}\ge(n-1)\left(\frac1{\sqrt{x_1}}+\frac1{\sqrt{x_2}}+\ldots+\frac1{\sqrt{x_n}}\right).
x
1
+
x
2
+
…
+
x
n
≥
(
n
−
1
)
(
x
1
1
+
x
2
1
+
…
+
x
n
1
)
.
Back to Problems
View on AoPS