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National and Regional Contests
Poland Contests
Poland - Second Round
2011 Poland - Second Round
3
Perfect square
Perfect square
Source: Polish MO second round 2011
February 19, 2012
modular arithmetic
number theory unsolved
number theory
Problem Statement
Prove that
∀
x
1
,
x
2
,
…
,
x
2011
,
y
1
,
y
2
,
…
,
y
2011
∈
Z
+
\forall x_{1},x_{2},\ldots,x_{2011},y_{1},y_{2},\ldots,y_{2011}\in\mathbb{Z_{+}}
∀
x
1
,
x
2
,
…
,
x
2011
,
y
1
,
y
2
,
…
,
y
2011
∈
Z
+
product:
(
2
x
1
2
+
3
y
1
2
)
(
2
x
2
2
+
3
y
2
2
)
…
(
2
x
2011
2
+
3
y
2011
2
)
(2x_{1}^{2}+3y_{1}^{2})(2x_{2}^{2}+3y_{2}^{2})\ldots(2x_{2011}^{2}+3y_{2011}^{2})
(
2
x
1
2
+
3
y
1
2
)
(
2
x
2
2
+
3
y
2
2
)
…
(
2
x
2011
2
+
3
y
2011
2
)
is not a perfect square.
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