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Costa Rica - Final Round
2020 Costa Rica - Final Round
3
sum x/(x +\sqrt{(x + y)(x + z)) )<=1
sum x/(x +\sqrt{(x + y)(x + z)) )<=1
Source: OLCOMA Costa Rica National Olympiad, Final Round, 2020 p3
March 26, 2024
algebra
inequalities
Problem Statement
Let
x
,
y
,
z
∈
R
+
x, y, z \in R^+
x
,
y
,
z
∈
R
+
. Prove that
x
x
+
(
x
+
y
)
(
x
+
z
)
+
y
y
+
(
y
+
z
)
(
y
+
x
)
+
z
z
+
(
x
+
z
)
(
z
+
y
)
≤
1
\frac{x}{x +\sqrt{(x + y)(x + z)}}+\frac{y}{y +\sqrt{(y + z)(y + x)}}+\frac{z}{z +\sqrt{(x + z)(z + y)}} \le 1
x
+
(
x
+
y
)
(
x
+
z
)
x
+
y
+
(
y
+
z
)
(
y
+
x
)
y
+
z
+
(
x
+
z
)
(
z
+
y
)
z
≤
1
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