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Problems
Contests
National and Regional Contests
Belgium Contests
Flanders Math Olympiad
2005 Flanders Math Olympiad
2005 Flanders Math Olympiad
Part of
Flanders Math Olympiad
Subcontests
(4)
2
1
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balls in box
We can obviously put 100 unit balls in a
10
×
10
×
1
10\times10\times1
10
×
10
×
1
box. How can one put
105
105
105
unit balls in? How can we put
106
106
106
unit balls in?
1
1
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Remainder - dividy by 7
For all positive integers
n
n
n
, find the remainder of
(
7
n
)
!
7
n
⋅
n
!
\dfrac{(7n)!}{7^n \cdot n!}
7
n
⋅
n
!
(
7
n
)!
upon division by 7.
3
1
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2005^2
Prove that
200
5
2
2005^2
200
5
2
can be written in at least
4
4
4
ways as the sum of 2 perfect (non-zero) squares.
4
1
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Integer problem
If
n
n
n
is an integer, then find all values of
n
n
n
for which
n
+
n
+
2005
\sqrt{n}+\sqrt{n+2005}
n
+
n
+
2005
is an integer as well.