MathDB
Problems
Contests
International Contests
IMO Longlists
1972 IMO Longlists
32
32
Part of
1972 IMO Longlists
Problems
(1)
Maximum product of natural numbers if their sum is constant
Source:
12/6/2010
If
n
1
,
n
2
,
⋯
,
n
k
n_1, n_2, \cdots, n_k
n
1
,
n
2
,
⋯
,
n
k
are natural numbers and
n
1
+
n
2
+
⋯
+
n
k
=
n
n_1+n_2+\cdots+n_k = n
n
1
+
n
2
+
⋯
+
n
k
=
n
, show that
m
a
x
(
n
1
n
2
⋯
n
k
)
=
(
t
+
1
)
r
t
k
−
r
,
max(n_1n_2\cdots n_k)=(t + 1)^rt^{k-r},
ma
x
(
n
1
n
2
⋯
n
k
)
=
(
t
+
1
)
r
t
k
−
r
,
where
t
=
[
n
k
]
t =\left[\frac{n}{k}\right]
t
=
[
k
n
]
and
r
r
r
is the remainder of
n
n
n
upon division by
k
k
k
; i.e.,
n
=
t
k
+
r
,
0
≤
r
≤
k
−
1
n = tk + r, 0 \le r \le k- 1
n
=
t
k
+
r
,
0
≤
r
≤
k
−
1
.
inequalities
number theory unsolved
number theory