MathDB
Problems
Contests
National and Regional Contests
Argentina Contests
Argentina National Olympiad
2022 Argentina National Olympiad
2
2
Part of
2022 Argentina National Olympiad
Problems
(1)
x_1+x_2+...+x_k is divided by k for all k with 1<=k<=n
Source: 2022 Argentina OMA Finals L3 p2
3/25/2024
Determine all positive integers
n
n
n
such that numbers from
1
1
1
to
n
n
n
can be sorted in some order
x
1
,
x
2
,
.
.
.
,
x
n
x_1,x_2,...,x_n
x
1
,
x
2
,
...
,
x
n
with the property that the number
x
1
+
x
2
+
.
.
.
+
x
k
x_1+x_2+...+x_k
x
1
+
x
2
+
...
+
x
k
is divisible by
k
k
k
, for all
1
≤
k
≤
n
1\le k\le n
1
≤
k
≤
n
., that is
1
1
1
is divides
x
1
x_1
x
1
,
2
2
2
divides
x
1
+
x
2
x_1+x_2
x
1
+
x
2
,
3
3
3
divides
x
1
+
x
2
+
x
3
x_1+x_2+x_3
x
1
+
x
2
+
x
3
, and so on until
n
n
n
divides
x
1
+
x
2
+
.
.
.
+
x
n
x_1+x_2+...+x_n
x
1
+
x
2
+
...
+
x
n
.
number theory