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National and Regional Contests
North Macedonia Contests
JBMO TST - Macedonia
2019 Macedonia Junior BMO TST
4
4
Part of
2019 Macedonia Junior BMO TST
Problems
(1)
2019 JBMO TST- North Macedonia, problem 4
Source: 2019 JBMO TST- North Macedonia
5/27/2019
Let the real numbers
a
a
a
,
b
b
b
, and
c
c
c
satisfy the equations
(
a
+
b
)
(
b
+
c
)
(
c
+
a
)
=
a
b
c
(a+b)(b+c)(c+a)=abc
(
a
+
b
)
(
b
+
c
)
(
c
+
a
)
=
ab
c
and
(
a
9
+
b
9
)
(
b
9
+
c
9
)
(
c
9
+
a
9
)
=
(
a
b
c
)
9
(a^9+b^9)(b^9+c^9)(c^9+a^9)=(abc)^9
(
a
9
+
b
9
)
(
b
9
+
c
9
)
(
c
9
+
a
9
)
=
(
ab
c
)
9
. Prove that at least one of
a
a
a
,
b
b
b
, and
c
c
c
equals
0
0
0
.
JMMO
Macedonia
2019
Junior
algebra