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Problems
Contests
National and Regional Contests
Netherlands Contests
Dutch BxMO/EGMO TST
2018 Dutch BxMO TST
2018 Dutch BxMO TST
Part of
Dutch BxMO/EGMO TST
Subcontests
(5)
5
1
Hide problems
nx^2 +2^2/(x + 1}+...+(n + 1)^2/(x + n)= nx + n(n+3)/2
Let
n
n
n
be a positive integer. Determine all positive real numbers
x
x
x
satisfying
n
x
2
+
2
2
x
+
1
+
3
2
x
+
2
+
.
.
.
+
(
n
+
1
)
2
x
+
n
=
n
x
+
n
(
n
+
3
)
2
nx^2 +\frac{2^2}{x + 1}+\frac{3^2}{x + 2}+...+\frac{(n + 1)^2}{x + n}= nx + \frac{n(n + 3)}{2}
n
x
2
+
x
+
1
2
2
+
x
+
2
3
2
+
...
+
x
+
n
(
n
+
1
)
2
=
n
x
+
2
n
(
n
+
3
)
3
1
Hide problems
a_1, a_1a_2, a_1a_2a_3,..., a_1a_2a_3...a_p different remainders divided by p
Let
p
p
p
be a prime number. Prove that it is possible to choose a permutation
a
1
,
a
2
,
.
.
.
,
a
p
a_1, a_2,...,a_p
a
1
,
a
2
,
...
,
a
p
of
1
,
2
,
.
.
.
,
p
1,2,...,p
1
,
2
,
...
,
p
such that the numbers
a
1
,
a
1
a
2
,
a
1
a
2
a
3
,
.
.
.
,
a
1
a
2
a
3
.
.
.
a
p
a_1, a_1a_2, a_1a_2a_3,..., a_1a_2a_3...a_p
a
1
,
a
1
a
2
,
a
1
a
2
a
3
,
...
,
a
1
a
2
a
3
...
a
p
all have different remainder upon division by
p
p
p
.
1
1
Hide problems
1000 balls in 40 different colours, 25 balls of each colour, in a circle
We have
1000
1000
1000
balls in
40
40
40
different colours,
25
25
25
balls of each colour. Determine the smallest
n
n
n
for which the following holds: if you place the
1000
1000
1000
balls in a circle, in any arbitrary way, then there are always
n
n
n
adjacent balls which have at least
20
20
20
different colours.
4
1
Hide problems
dutch angle chasing candidate, < AED + < ADO = 90^o wanted
In a non-isosceles triangle
△
A
B
C
\vartriangle ABC
△
A
BC
we have
∠
B
A
C
=
6
0
o
\angle BAC = 60^o
∠
B
A
C
=
6
0
o
. Let
D
D
D
be the intersection of the angular bisector of
∠
B
A
C
\angle BAC
∠
B
A
C
with side
B
C
,
O
BC, O
BC
,
O
the centre of the circumcircle of
△
A
B
C
\vartriangle ABC
△
A
BC
and
E
E
E
the intersection of
A
O
AO
A
O
and
B
C
BC
BC
. Prove that
∠
A
E
D
+
∠
A
D
O
=
9
0
o
\angle AED + \angle ADO = 90^o
∠
A
E
D
+
∠
A
D
O
=
9
0
o
.
2
1
Hide problems
triangle, pairwise coprime sidelengths, segment related to tangent integer
Let
△
A
B
C
\vartriangle ABC
△
A
BC
be a triangle of which the side lengths are positive integers which are pairwise coprime. The tangent in
A
A
A
to the circumcircle intersects line
B
C
BC
BC
in
D
D
D
. Prove that
B
D
BD
B
D
is not an integer.