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National and Regional Contests
Bosnia Herzegovina Contests
JBMO TST - Bosnia and Herzegovina
2018 Bosnia and Herzegovina Junior BMO TST
4
4
Part of
2018 Bosnia and Herzegovina Junior BMO TST
Problems
(1)
P4 Bosnia and Herzegovina JBMO TST
Source: Bosnia and Herzegovina JBMO TST 2018
7/14/2018
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be real numbers which satisfy:
a
+
b
+
c
=
2
a+b+c=2
a
+
b
+
c
=
2
a
2
+
b
2
+
c
2
=
2
a^2+b^2+c^2=2
a
2
+
b
2
+
c
2
=
2
Prove that at least one of numbers
∣
a
−
b
∣
,
∣
b
−
c
∣
,
∣
c
−
a
∣
|a-b|, |b-c|, |c-a|
∣
a
−
b
∣
,
∣
b
−
c
∣
,
∣
c
−
a
∣
is greater or equal than
1
1
1
.
algebra