MathDB
Problems
Contests
National and Regional Contests
Albania Contests
Albania JBMO TST
2015 Albania JBMO TST
4
4
Part of
2015 Albania JBMO TST
Problems
(1)
$a_1=n^2-10n+23, a_1=n^2-9n+31, a_1=n^2-12n+46.$
Source: Albania JTST 2015
10/4/2023
For every positive integer
n
n{}
n
, consider the numbers
a
1
=
n
2
−
10
n
+
23
,
a
2
=
n
2
−
9
n
+
31
,
a
3
=
n
2
−
12
n
+
46.
a_1=n^2-10n+23, a_2=n^2-9n+31, a_3=n^2-12n+46.
a
1
=
n
2
−
10
n
+
23
,
a
2
=
n
2
−
9
n
+
31
,
a
3
=
n
2
−
12
n
+
46.
a) Prove that
a
1
+
a
2
+
a
3
a_1+a_2+a_3
a
1
+
a
2
+
a
3
is even. b) Find all positive integers
n
n
n
for which
a
1
,
a
2
a_1, a_2
a
1
,
a
2
and
a
3
a_3
a
3
are primes.
number theory