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Baltic Way
2007 Baltic Way
17
17
Part of
2007 Baltic Way
Problems
(1)
Inequality with the gcd of the integers x,y,z
Source: Baltic Way 2007
11/30/2010
Let
x
,
y
,
z
x,y,z
x
,
y
,
z
be positive integers such that
x
+
1
y
+
y
+
1
z
+
z
+
1
x
\frac{x+1}{y}+\frac{y+1}{z}+\frac{z+1}{x}
y
x
+
1
+
z
y
+
1
+
x
z
+
1
is an integer. Let
d
d
d
be the greatest common divisor of
x
,
y
x,y
x
,
y
and
z
z
z
. Prove that
d
≤
x
y
+
y
z
+
z
x
3
d\le \sqrt[3]{xy+yz+zx}
d
≤
3
x
y
+
yz
+
z
x
.
inequalities
number theory
greatest common divisor
number theory proposed