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Swedish Mathematical Competition
2022 Swedish Mathematical Competition
5
5
Part of
2022 Swedish Mathematical Competition
Problems
(1)
x_1 + x^2_2+ x^3_3+...+ x^k_k= n
Source: 2022 Swedish Mathematical Competition p5
3/24/2024
Prove that for every pair of positive integers
k
k
k
and
n
n
n
, there exists integer
x
1
x_1
x
1
,
x
2
x_2
x
2
,
.
.
.
...
...
,
x
k
x_k
x
k
with
0
≤
x
j
≤
2
k
−
1
⋅
n
k
0 \le x_j \le 2^{k-1}\cdot \sqrt[k]{n}
0
≤
x
j
≤
2
k
−
1
⋅
k
n
for
j
=
1
j = 1
j
=
1
,
2
2
2
,
.
.
.
...
...
,
k
k
k
, and such that
x
1
+
x
2
2
+
x
3
3
+
.
.
.
+
x
k
k
=
n
.
x_1 + x^2_2+ x^3_3+...+ x^k_k= n.
x
1
+
x
2
2
+
x
3
3
+
...
+
x
k
k
=
n
.
number theory