MathDB
Problems
Contests
National and Regional Contests
India Contests
India Pre-Regional Mathematical Olympiad
2019 India PRMO
8
8
Part of
2019 India PRMO
Problems
(2)
Algebra Problem
Source:
8/11/2019
How many positive integers
n
n
n
are there such that
3
≤
n
≤
100
3 \leq n \leq 100
3
≤
n
≤
100
and
x
2
n
+
x
+
1
x^{2^{n}} + x + 1
x
2
n
+
x
+
1
is divisible by
x
2
+
x
+
1
x^2 + x + 1
x
2
+
x
+
1
?
algebra
(a+b)^k-a^k-b^k
Source: PRMO 2019 Leg 2 P8
8/25/2019
Let
F
k
(
a
,
b
)
=
(
a
+
b
)
k
−
a
k
−
b
k
F_k(a,b)=(a+b)^k-a^k-b^k
F
k
(
a
,
b
)
=
(
a
+
b
)
k
−
a
k
−
b
k
and let
S
=
1
,
2
,
3
,
4
,
5
,
6
,
7
,
8
,
9
,
10
S={1,2,3,4,5,6,7,8,9,10}
S
=
1
,
2
,
3
,
4
,
5
,
6
,
7
,
8
,
9
,
10
. For how many ordered pairs
(
a
,
b
)
(a,b)
(
a
,
b
)
with
a
,
b
∈
S
a,b\in S
a
,
b
∈
S
and
a
≤
b
a\leq b
a
≤
b
is
F
5
(
a
,
b
)
F
3
(
a
,
b
)
\frac{F_5(a,b)}{F_3(a,b)}
F
3
(
a
,
b
)
F
5
(
a
,
b
)
an integer?
algebra
number theory
PRMO
Divisibility