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2013 Greece Junior Math Olympiad
1
1
Part of
2013 Greece Junior Math Olympiad
Problems
(1)
Algebra expression
Source: Archimedes Junior 2013
3/16/2020
(a) Write
A
=
k
4
+
4
A = k^4 + 4
A
=
k
4
+
4
, where
k
k
k
is a positive integer, as a product of two factors each of them is sum of two squares of integers. (b) Simplify the expression
K
=
(
2
4
+
1
4
)
(
4
4
+
1
4
)
.
.
.
(
(
2
n
)
4
+
1
4
)
(
1
4
+
1
4
)
(
3
4
+
1
4
)
.
.
.
(
(
2
n
−
1
)
4
+
1
4
)
K=\frac{(2^4+\frac14)(4^4+\frac14)...((2n)^4+\frac14)}{(1^4+\frac14)(3^4+\frac14)...((2n-1)^4+\frac14)}
K
=
(
1
4
+
4
1
)
(
3
4
+
4
1
)
...
((
2
n
−
1
)
4
+
4
1
)
(
2
4
+
4
1
)
(
4
4
+
4
1
)
...
((
2
n
)
4
+
4
1
)
and write it as sum of squares of two consecutive positive integers
algebra