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National and Regional Contests
Azerbaijan Contests
Azerbaijan Junior National Olympiad
2016 Azerbaijan Junior Mathematical Olympiad
5
number theory
number theory
Source:
February 7, 2016
number theory
Problem Statement
Positive integers
(
p
,
a
,
b
,
c
)
(p,a,b,c)
(
p
,
a
,
b
,
c
)
called good quadruple if a)
p
p
p
is odd prime,b)
a
,
b
,
c
a,b,c
a
,
b
,
c
are distinct ,c)
a
b
+
1
,
b
c
+
1
ab+1,bc+1
ab
+
1
,
b
c
+
1
and
c
a
+
1
ca+1
c
a
+
1
are divisible by
p
p
p
.Prove that for all good quadruple
p
+
2
≤
a
+
b
+
c
3
p+2\le \frac {a+b+c}{3}
p
+
2
≤
3
a
+
b
+
c
, and show the equality case.
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