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Miklós Schweitzer
2000 Miklós Schweitzer
4
4
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2000 Miklós Schweitzer
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Miklós Schweitzer 2000, Problem 4
Source: Miklós Schweitzer 2000
7/30/2016
Let
a
1
<
a
2
<
a
3
a_1<a_2<a_3
a
1
<
a
2
<
a
3
be positive integers. Prove that there are integers
x
1
,
x
2
,
x
3
x_1,x_2,x_3
x
1
,
x
2
,
x
3
such that
∑
i
=
1
3
∣
x
i
∣
>
0
\sum_{i=1}^3 |x_i | >0
∑
i
=
1
3
∣
x
i
∣
>
0
,
∑
i
=
1
3
a
i
x
i
=
0
\sum_{i=1}^3 a_ix_i= 0
∑
i
=
1
3
a
i
x
i
=
0
and
max
1
≤
i
≤
3
∣
x
i
∣
<
2
3
a
3
+
1
\max_{1\le i\le 3} | x_i|<\frac{2}{\sqrt{3}}\sqrt{a_3}+1
1
≤
i
≤
3
max
∣
x
i
∣
<
3
2
a
3
+
1
.
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