MathDB
Problems
Contests
National and Regional Contests
Azerbaijan Contests
Azerbaijan BMO TST
2018 Azerbaijan BMO TST
2018 Azerbaijan BMO TST
Part of
Azerbaijan BMO TST
Subcontests
(2)
1
1
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Typical number theory problem with case work
Find all positive integers
(
x
,
y
)
(x,y)
(
x
,
y
)
such that
x
2
+
y
2
=
2017
(
x
−
y
)
x^2+y^2=2017(x-y)
x
2
+
y
2
=
2017
(
x
−
y
)
3
1
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Combinatorics problem
Prove that it is possible to color each positive integers with one of three colors so that the following conditions are satisfied:
i
)
i)
i
)
For each
n
∈
N
0
n\in\mathbb{N}_{0}
n
∈
N
0
all positive integers
x
x
x
such that
2
n
≤
x
<
2
n
+
1
2^n\le x<2^{n+1}
2
n
≤
x
<
2
n
+
1
have the same color.
i
i
)
ii)
ii
)
There are no positive integers
x
,
y
,
z
x,y,z
x
,
y
,
z
of the same color (except
x
=
y
=
z
=
2
x=y=z=2
x
=
y
=
z
=
2
) such that
x
+
y
=
z
2
.
x+y=z^2.
x
+
y
=
z
2
.