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Contests
National and Regional Contests
Cuba Contests
Cuba MO
2015 Cuba MO
9
9
Part of
2015 Cuba MO
Problems
(1)
sum x/(1+yz/x) >= M if xy + yz + zx = 1
Source: 2015 Cuba 2.9
9/20/2024
Determine the largest possible value of
M
M
M
for which it holds that:
x
1
+
y
z
x
+
y
1
+
z
x
y
+
z
1
+
x
y
z
≥
M
,
\frac{x}{1 +\dfrac{yz}{x}}+ \frac{y}{1 + \dfrac{zx}{y}}+ \frac{z}{1 + \dfrac{xy}{z}} \ge M,
1
+
x
yz
x
+
1
+
y
z
x
y
+
1
+
z
x
y
z
≥
M
,
for all real numbers
x
,
y
,
z
>
0
x, y, z > 0
x
,
y
,
z
>
0
that satisfy the equation
x
y
+
y
z
+
z
x
=
1
xy + yz + zx = 1
x
y
+
yz
+
z
x
=
1
.
algebra
inequalities