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Hungary Contests
Kürschák Math Competition
1964 Kurschak Competition
3
3
Part of
1964 Kurschak Competition
Problems
(1)
\sqrt{(w^2 + x^2 + y^2 + z^2)/4} >=\sqrt[3]{wxy + wxz + wyz + xyz)/4}
Source: 1964 Hungary - Kürschák Competition p3
10/11/2022
Show that for any positive reals
w
,
x
,
y
,
z
w, x, y, z
w
,
x
,
y
,
z
we have
w
2
+
x
2
+
y
2
+
z
2
4
≥
w
x
y
+
w
x
z
+
w
y
z
+
x
y
z
4
3
\sqrt{\frac{w^2 + x^2 + y^2 + z^2}{4}}\ge \sqrt[3]{ \frac{wxy + wxz + wyz + xyz}{4}}
4
w
2
+
x
2
+
y
2
+
z
2
≥
3
4
w
x
y
+
w
x
z
+
w
yz
+
x
yz
algebra
inequalities