MathDB
Problems
Contests
International Contests
IMO Longlists
1982 IMO Longlists
1
1
Part of
1982 IMO Longlists
Problems
(1)
Integer for all n ≥ k
Source: IMO LongList 1982 - P1
3/16/2011
(a) Prove that
1
n
+
1
⋅
(
2
n
n
)
\frac{1}{n+1} \cdot \binom{2n}{n}
n
+
1
1
⋅
(
n
2
n
)
is an integer for
n
≥
0.
n \geq 0.
n
≥
0.
(b) Given a positive integer
k
k
k
, determine the smallest integer
C
k
C_k
C
k
with the property that
C
k
n
+
k
+
1
⋅
(
2
n
n
)
\frac{C_k}{n+k+1} \cdot \binom{2n}{n}
n
+
k
+
1
C
k
⋅
(
n
2
n
)
is an integer for all
n
≥
k
.
n \geq k.
n
≥
k
.
induction
least common multiple
number theory proposed
number theory