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Problems
Contests
National and Regional Contests
Netherlands Contests
Dutch Mathematical Olympiad
1965 Dutch Mathematical Olympiad
2
2
Part of
1965 Dutch Mathematical Olympiad
Problems
(1)
5|(n + 6)^2+(n + 7)^2+ ... +(n + 10)^2 - [(n + 1)^2 + (n + 2)^2 +...+ (n + 5)]
Source: Netherlands - Dutch NMO 1965 p2
1/31/2023
Prove that
S
1
=
(
n
+
1
)
2
+
(
n
+
2
)
2
+
.
.
.
+
(
n
+
5
)
2
S_1 = (n + 1)^2 + (n + 2)^2 +...+ (n + 5)^2
S
1
=
(
n
+
1
)
2
+
(
n
+
2
)
2
+
...
+
(
n
+
5
)
2
is divisible by
5
5
5
for every
n
n
n
. Prove that for no
n
n
n
:
∑
ℓ
=
1
5
(
n
+
ℓ
)
2
\sum_{\ell=1}^5 (n+\ell)^2
∑
ℓ
=
1
5
(
n
+
ℓ
)
2
is a perfect square. Let
S
2
=
(
n
+
6
)
2
+
(
n
+
7
)
2
+
.
.
.
+
(
n
+
10
)
2
S_2=(n + 6)^2 + (n + 7)^2 + ... + (n + 10)^2
S
2
=
(
n
+
6
)
2
+
(
n
+
7
)
2
+
...
+
(
n
+
10
)
2
. Prove that
S
1
⋅
S
2
S_1 \cdot S_2
S
1
⋅
S
2
is divisible by
150
150
150
.
number theory
Perfect Square
divides
divisible