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VMEO = Vietnam Mathematical E-Olympiad
VMEO III 2006 Shortlist
A7
A problems related to floor function
A problems related to floor function
Source: VMEO III
October 15, 2015
number theory
floor function
algebra
function
Problem Statement
Prove that for all
n
∈
Z
+
n\in\mathbb{Z}^+
n
∈
Z
+
, we have
∑
p
=
1
n
∑
q
=
1
p
⌊
−
1
+
8
q
+
(
2
p
−
1
)
2
2
⌋
=
−
n
(
n
+
1
)
(
n
+
2
)
3
\sum\limits_{p=1}^n\sum\limits_{q=1}^p\left\lfloor -\frac{1+\sqrt{8q+(2p-1)^2}}{2}\right\rfloor =-\frac{n(n+1)(n+2)}{3}
p
=
1
∑
n
q
=
1
∑
p
⌊
−
2
1
+
8
q
+
(
2
p
−
1
)
2
⌋
=
−
3
n
(
n
+
1
)
(
n
+
2
)
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