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Problems
Contests
National and Regional Contests
Netherlands Contests
Dutch BxMO/EGMO TST
2016 Dutch BxMO TST
2016 Dutch BxMO TST
Part of
Dutch BxMO/EGMO TST
Subcontests
(5)
2
1
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system x^2 - y = (z - 1)^2, y^2 - z = (x - 1)^2, z^2 - x = (y -1)^2
Determine all triples (x, y, z) of non-negative real numbers that satisfy the following system of equations
{
x
2
−
y
=
(
z
−
1
)
2
y
2
−
z
=
(
x
−
1
)
2
z
2
−
x
=
(
y
−
1
)
2
\begin{cases} x^2 - y = (z - 1)^2\\ y^2 - z = (x - 1)^2 \\ z^2 - x = (y -1)^2 \end{cases}
⎩
⎨
⎧
x
2
−
y
=
(
z
−
1
)
2
y
2
−
z
=
(
x
−
1
)
2
z
2
−
x
=
(
y
−
1
)
2
.
1
1
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n = 3t(n) + 5r(n), greatest odd divisor, smallest pos. divisor unequal to 1
For a positive integer
n
n
n
that is not a power of two, we de fine
t
(
n
)
t(n)
t
(
n
)
as the greatest odd divisor of
n
n
n
and
r
(
n
)
r(n)
r
(
n
)
as the smallest positive odd divisor of
n
n
n
unequal to
1
1
1
. Determine all positive integers
n
n
n
that are not a power of two and for which we have
n
=
3
t
(
n
)
+
5
r
(
n
)
n = 3t(n) + 5r(n)
n
=
3
t
(
n
)
+
5
r
(
n
)
.
5
1
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(m + n)^3 / ( 2n (3m^2 + n2) + 8 )
Determine all pairs
(
m
,
n
)
(m, n)
(
m
,
n
)
of positive integers for which
(
m
+
n
)
3
/
2
n
(
3
m
2
+
n
2
)
+
8
(m + n)^3 / 2n (3m^2 + n^2) + 8
(
m
+
n
)
3
/2
n
(
3
m
2
+
n
2
)
+
8
4
1
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Facebook group Olympiad training, friendships combo
The Facebook group Olympiad training has at least five members. There is a certain integer
k
k
k
with following property: for each
k
k
k
-tuple of members there is at least one member of this
k
k
k
-tuple friends with each of the other
k
−
1
k - 1
k
−
1
. (Friendship is mutual: if
A
A
A
is friends with
B
B
B
, then also
B
B
B
is friends with
A
A
A
.) (a) Suppose
k
=
4
k = 4
k
=
4
. Can you say with certainty that the Facebook group has a member that is friends with each of the other members? (b) Suppose
k
=
5
k = 5
k
=
5
. Can you say with certainty that the Facebook group has a member that is friends with each of the other members?
3
1
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EF goes through the points of tangency of the incircle to AB and AC
Let
△
A
B
C
\vartriangle ABC
△
A
BC
be a right-angled triangle with
∠
A
=
9
0
o
\angle A = 90^o
∠
A
=
9
0
o
and circumcircle
Γ
\Gamma
Γ
. The inscribed circle is tangent to
B
C
BC
BC
in point
D
D
D
. Let
E
E
E
be the midpoint of the arc
A
B
AB
A
B
of
Γ
\Gamma
Γ
not containing
C
C
C
and let
F
F
F
be the midpoint of the arc
A
C
AC
A
C
of
Γ
\Gamma
Γ
not containing
B
B
B
. (a) Prove that
△
A
B
C
∼
△
D
E
F
\vartriangle ABC \sim \vartriangle DEF
△
A
BC
∼
△
D
EF
. (b) Prove that
E
F
EF
EF
goes through the points of tangency of the incircle to
A
B
AB
A
B
and
A
C
AC
A
C
.