MathDB
Problems
Contests
International Contests
Benelux
2016 Benelux
2016 Benelux
Part of
Benelux
Subcontests
(4)
4
1
Hide problems
Benelux Mathematical Olympiad 2016, Problem 4
A circle
ω
\omega
ω
passes through the two vertices
B
B
B
and
C
C
C
of a triangle
A
B
C
.
ABC.
A
BC
.
Furthermore,
ω
\omega
ω
intersects segment
A
C
AC
A
C
in
D
≠
C
D\ne C
D
=
C
and segment
A
B
AB
A
B
in
E
≠
B
.
E\ne B.
E
=
B
.
On the ray from
B
B
B
through
D
D
D
lies a point
K
K
K
such that
∣
B
K
∣
=
∣
A
C
∣
,
|BK| = |AC|,
∣
B
K
∣
=
∣
A
C
∣
,
and on the ray from
C
C
C
through
E
E
E
lies a point
L
L
L
such that
∣
C
L
∣
=
∣
A
B
∣
.
|CL| = |AB|.
∣
C
L
∣
=
∣
A
B
∣.
Show that the circumcentre
O
O
O
of triangle
A
K
L
AKL
A
K
L
lies on
ω
\omega
ω
.
3
1
Hide problems
Benelux Mathematical Olympiad 2016, Problem 3
Find all functions
f
:
R
→
Z
f :\Bbb{ R}\to \Bbb{Z}
f
:
R
→
Z
such that
(
f
(
f
(
y
)
−
x
)
)
2
+
f
(
x
)
2
+
f
(
y
)
2
=
f
(
y
)
⋅
(
1
+
2
f
(
f
(
y
)
)
)
,
\left( f(f(y) - x) \right)^2+ f(x)^2 + f(y)^2 = f(y) \cdot \left( 1 + 2f(f(y)) \right),
(
f
(
f
(
y
)
−
x
)
)
2
+
f
(
x
)
2
+
f
(
y
)
2
=
f
(
y
)
⋅
(
1
+
2
f
(
f
(
y
))
)
,
for all
x
,
y
∈
R
.
x, y \in \Bbb{R}.
x
,
y
∈
R
.
2
1
Hide problems
Benelux Mathematical Olympiad 2016, Problem 2
Let
n
n
n
be a positive integer. Suppose that its positive divisors can be partitioned into pairs (i.e. can be split in groups of two) in such a way that the sum of each pair is a prime number. Prove that these prime numbers are distinct and that none of these are a divisor of
n
.
n.
n
.
1
1
Hide problems
Benelux Mathematical Olympiad 2016, Problem 1
Find the greatest positive integer
N
N
N
with the following property: there exist integers
x
1
,
.
.
.
,
x
N
x_1, . . . , x_N
x
1
,
...
,
x
N
such that
x
i
2
−
x
i
x
j
x^2_i - x_ix_j
x
i
2
−
x
i
x
j
is not divisible by
1111
1111
1111
for any
i
≠
j
.
i\ne j.
i
=
j
.