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Poland - Second Round
1973 Poland - Second Round
1
1
Part of
1973 Poland - Second Round
Problems
(1)
\frac{x^2+y^2-z^2}{2xy} + \frac{y^2+z^2-x^2}{2yz} + \frac{z^2+x^2-y ^2}{2xz} > 1
Source: Polish MO Second Round 1973 p1
9/8/2024
Prove that if positive numbers
x
,
y
,
z
x, y, z
x
,
y
,
z
satisfy the inequality
x
2
+
y
2
−
z
2
2
x
y
+
y
2
+
z
2
−
x
2
2
y
z
+
z
2
+
x
2
−
y
2
2
x
z
>
1
,
\frac{x^2+y^2-z^2}{2xy} + \frac{y^2+z^2-x^2}{2yz} + \frac{z^2+x^2-y ^2}{2xz} > 1,
2
x
y
x
2
+
y
2
−
z
2
+
2
yz
y
2
+
z
2
−
x
2
+
2
x
z
z
2
+
x
2
−
y
2
>
1
,
then they are the lengths of the sides of a certain triangle.
algebra
inequalities
geometry
Geometric Inequalities