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National and Regional Contests
Korea Contests
Korea National Olympiad
1997 Korea National Olympiad
5
5
Part of
1997 Korea National Olympiad
Problems
(1)
Ineq in Geo again
Source: 1997 Korea National Olympiad #5
3/18/2018
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be the side lengths of any triangle
△
A
B
C
\triangle ABC
△
A
BC
opposite to
A
,
B
A,B
A
,
B
and
C
,
C,
C
,
respectively. Let
x
,
y
,
z
x,y,z
x
,
y
,
z
be the length of medians from
A
,
B
A,B
A
,
B
and
C
,
C,
C
,
respectively. If
T
T
T
is the area of
△
A
B
C
\triangle ABC
△
A
BC
, prove that
a
2
x
+
b
2
y
+
c
2
z
≥
3
T
\frac{a^2}{x}+\frac{b^2}{y}+\frac{c^2}{z}\ge\sqrt{\sqrt{3}T}
x
a
2
+
y
b
2
+
z
c
2
≥
3
T
geometry
inequalities