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Bosnia Herzegovina Contests
JBMO TST - Bosnia and Herzegovina
2022 Bosnia and Herzegovina Junior BMO TST
2
2
Part of
2022 Bosnia and Herzegovina Junior BMO TST
Problems
(1)
JBMO TST Bosnia and Herzegovina 2022 P2
Source: JBMO TST Bosnia and Herzegovina 2022
5/21/2022
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be positive integers greater than
1
1
1
such that
p
=
a
b
+
b
c
+
a
c
p=ab+bc+ac
p
=
ab
+
b
c
+
a
c
is prime. A) Prove that
a
2
,
b
2
,
c
2
a^2, b^2, c^2
a
2
,
b
2
,
c
2
all have different reminder
m
o
d
p
mod\ p
m
o
d
p
. B) Prove that
a
3
,
b
3
,
c
3
a^3, b^3, c^3
a
3
,
b
3
,
c
3
all have different reminder
m
o
d
p
mod\ p
m
o
d
p
.
JBMO TST
number theory
national olympiad