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Problems
Contests
National and Regional Contests
New Zealand Contests
NZMOC Camp Selection Problems
2016 NZMOC Camp Selection Problems
4
4
Part of
2016 NZMOC Camp Selection Problems
Problems
(1)
a karaka quadruple (p, a, b, c) exists with p + 2 = (a + b + c)/3
Source: New Zealand NZMOC Camp Selection Problems 2016 p4
9/19/2021
A quadruple
(
p
,
a
,
b
,
c
)
(p, a, b, c)
(
p
,
a
,
b
,
c
)
of positive integers is a karaka quadruple if
∙
\bullet
∙
p
p
p
is an odd prime number
∙
\bullet
∙
a
,
b
a, b
a
,
b
and
c
c
c
are distinct, and
∙
\bullet
∙
a
b
+
1
ab + 1
ab
+
1
,
b
c
+
1
bc + 1
b
c
+
1
and
c
a
+
1
ca + 1
c
a
+
1
are divisible by
p
p
p
. (a) Prove that for every karaka quadruple
(
p
,
a
,
b
,
c
)
(p, a, b, c)
(
p
,
a
,
b
,
c
)
we have
p
+
2
≤
a
+
b
+
c
3
p + 2 \le\frac{a + b + c}{3}
p
+
2
≤
3
a
+
b
+
c
. (b) Determine all numbers
p
p
p
for which a karaka quadruple
(
p
,
a
,
b
,
c
)
(p, a, b, c)
(
p
,
a
,
b
,
c
)
exists with
p
+
2
=
a
+
b
+
c
3
p + 2 =\frac{a + b + c}{3}
p
+
2
=
3
a
+
b
+
c
number theory
divisible