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Bulgaria National Olympiad
1991 Bulgaria National Olympiad
Problem 3
p^3|(kp choose k)-k
p^3|(kp choose k)-k
Source: Bulgaria 1991 P3
June 2, 2021
number theory
Problem Statement
Prove that for every prime number
p
≥
5
p\ge5
p
≥
5
,(a)
p
3
p^3
p
3
divides
(
2
p
p
)
−
2
\binom{2p}p-2
(
p
2
p
)
−
2
; (b)
p
3
p^3
p
3
divides
(
k
p
p
)
−
k
\binom{kp}p-k
(
p
k
p
)
−
k
for every natural number
k
k
k
.
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