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2012 Argentina National Olympiad
2
2
Part of
2012 Argentina National Olympiad
Problems
(1)
product (11x_i^2 +12y_i^2) is perfect square
Source: 2012 Argentina OMA Finals L3 p2
1/15/2023
Determine all natural numbers
n
n
n
for which there are
2
n
2n
2
n
distinct positive integers
x
1
,
…
,
x
n
,
y
1
,
…
,
y
n
x_1,…,x_n,y_1,…,y_n
x
1
,
…
,
x
n
,
y
1
,
…
,
y
n
such that the product
(
11
x
1
2
+
12
y
1
2
)
(
11
x
2
2
+
12
y
2
2
)
…
(
11
x
n
2
+
12
y
n
2
)
(11x^2_1+12y^2_1)(11x^2_2+12y^2_2)…(11x^2_n+12y^2_n)
(
11
x
1
2
+
12
y
1
2
)
(
11
x
2
2
+
12
y
2
2
)
…
(
11
x
n
2
+
12
y
n
2
)
is a perfect square.
number theory
Perfect Square