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2
38th Austrian Mathematical Competition 2007
38th Austrian Mathematical Competition 2007
Source: 1st round
February 7, 2009
floor function
function
algebra
algebra unsolved
Problem Statement
Find all tuples
(
x
1
,
x
2
,
x
3
,
x
4
,
x
5
)
(x_1,x_2,x_3,x_4,x_5)
(
x
1
,
x
2
,
x
3
,
x
4
,
x
5
)
of positive integers with
x
1
>
x
2
>
x
3
>
x
4
>
x
5
>
0
x_1>x_2>x_3>x_4>x_5>0
x
1
>
x
2
>
x
3
>
x
4
>
x
5
>
0
and
⌊
x
1
+
x
2
3
⌋
2
+
⌊
x
2
+
x
3
3
⌋
2
+
⌊
x
3
+
x
4
3
⌋
2
+
⌊
x
4
+
x
5
3
⌋
2
=
38.
{\left \lfloor \frac{x_1+x_2}{3} \right \rfloor }^2 + {\left \lfloor \frac{x_2+x_3}{3} \right \rfloor }^2 + {\left \lfloor \frac{x_3+x_4}{3} \right \rfloor }^2 + {\left \lfloor \frac{x_4+x_5}{3} \right \rfloor }^2 = 38.
⌊
3
x
1
+
x
2
⌋
2
+
⌊
3
x
2
+
x
3
⌋
2
+
⌊
3
x
3
+
x
4
⌋
2
+
⌊
3
x
4
+
x
5
⌋
2
=
38.
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