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Moldova Team Selection Test
1998 Moldova Team Selection Test
4
\frac{a+z}{a+x}\cdot x+\frac{a+x}{a+y}\cdot y+\frac{a+y}{a+z}\cdot z \leq x+y+z
\frac{a+z}{a+x}\cdot x+\frac{a+x}{a+y}\cdot y+\frac{a+y}{a+z}\cdot z \leq x+y+z
Source: Moldova TST 1998
August 7, 2023
inequalities
Problem Statement
Show that for any positive real numbers
a
,
x
,
y
,
z
a, x, y, z
a
,
x
,
y
,
z
the following inequalities are true
a
+
z
a
+
x
⋅
x
+
a
+
x
a
+
y
⋅
y
+
a
+
y
a
+
z
⋅
z
≤
x
+
y
+
z
≤
a
+
y
a
+
z
⋅
x
+
a
+
z
a
+
x
⋅
y
+
a
+
x
a
+
y
⋅
z
.
\frac{a+z}{a+x}\cdot x+\frac{a+x}{a+y}\cdot y+\frac{a+y}{a+z}\cdot z \leq x+y+z \leq \frac{a+y}{a+z}\cdot x+\frac{a+z}{a+x}\cdot y+\frac{a+x}{a+y}\cdot z.
a
+
x
a
+
z
⋅
x
+
a
+
y
a
+
x
⋅
y
+
a
+
z
a
+
y
⋅
z
≤
x
+
y
+
z
≤
a
+
z
a
+
y
⋅
x
+
a
+
x
a
+
z
⋅
y
+
a
+
y
a
+
x
⋅
z
.
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