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Problems
Contests
National and Regional Contests
PEN Problems
PEN A Problems
24
24
Part of
PEN A Problems
Problems
(1)
A 24
Source:
5/25/2007
Let
p
>
3
p>3
p
>
3
is a prime number and
k
=
⌊
2
p
3
⌋
k=\lfloor\frac{2p}{3}\rfloor
k
=
⌊
3
2
p
⌋
. Prove that
(
p
1
)
+
(
p
2
)
+
⋯
+
(
p
k
)
{p \choose 1}+{p \choose 2}+\cdots+{p \choose k}
(
1
p
)
+
(
2
p
)
+
⋯
+
(
k
p
)
is divisible by
p
2
p^{2}
p
2
.
floor function
modular arithmetic
Divisibility Theory