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Contests
National and Regional Contests
Ukraine Contests
Official Ukraine Selection Cycle
Ukraine Team Selection Test
2009 Ukraine Team Selection Test
12
12
Part of
2009 Ukraine Team Selection Test
Problems
(1)
sum h_a^{2}/(a ^2-CH_a ^2) >= 3, altitudes
Source: Ukraine TST 2009 p12
5/4/2020
Denote an acute-angle
△
A
B
C
\vartriangle ABC
△
A
BC
with sides
a
,
b
,
c
a, b, c
a
,
b
,
c
respectively by
H
a
,
H
b
,
H
c
{{H}_{a}}, {{H}_{b}}, {{H}_{c}}
H
a
,
H
b
,
H
c
the feet of altitudes
h
a
,
h
b
,
h
c
{{h}_{a}}, {{h}_{b}}, {{h}_{c}}
h
a
,
h
b
,
h
c
. Prove the inequality:
h
a
2
a
2
−
C
H
a
2
+
h
b
2
b
2
−
A
H
b
2
+
h
c
2
c
2
−
B
H
c
2
≥
3
\frac {h_ {a} ^{2}} {{{a} ^{2}} - CH_ {a} ^{2}} + \frac{h_{b} ^{2}} {{{ b}^{2}} - AH_{b} ^{2}} + \frac{h_{c}^{2}}{{{c}^{2}} - BH_{c}^{2}} \ge 3
a
2
−
C
H
a
2
h
a
2
+
b
2
−
A
H
b
2
h
b
2
+
c
2
−
B
H
c
2
h
c
2
≥
3
(Dmitry Petrovsky)
altitudes
geometric inequality
geometry