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Problems
Contests
National and Regional Contests
Netherlands Contests
Dutch BxMO/EGMO TST
2012 Dutch BxMO/EGMO TST
2012 Dutch BxMO/EGMO TST
Part of
Dutch BxMO/EGMO TST
Subcontests
(3)
1
1
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P(Q(x)) has only the zeros at x= 2,3,5,7 wher P,Q trinomials
Do there exist quadratic polynomials
P
(
x
)
P(x)
P
(
x
)
and
Q
(
x
)
Q(x)
Q
(
x
)
with real coeffcients such that the polynomial
P
(
Q
(
x
)
)
P(Q(x))
P
(
Q
(
x
))
has precisely the zeros
x
=
2
,
x
=
3
,
x
=
5
x = 2, x = 3, x =5
x
=
2
,
x
=
3
,
x
=
5
and
x
=
7
x = 7
x
=
7
?
5
1
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for each n, exactly 1 of n,2n,3n is element of A, 2 \in A, prove 13824 \notin A
Let
A
A
A
be a set of positive integers having the following property: for each positive integer
n
n
n
exactly one of the three numbers
n
,
2
n
n, 2n
n
,
2
n
and
3
n
3n
3
n
is an element of
A
A
A
. Furthermore, it is given that
2
∈
A
2 \in A
2
∈
A
. Prove that
13824
∉
A
13824 \notin A
13824
∈
/
A
.
4
1
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2 lines as area bisectors given , segment bisector wanted, in a convex ABCD
Let
A
B
C
D
ABCD
A
BC
D
a convex quadrilateral (this means that all interior angles are smaller than
18
0
o
180^o
18
0
o
), such that there exist a point
M
M
M
on line segment
A
B
AB
A
B
and a point
N
N
N
on line segment
B
C
BC
BC
having the property that
A
N
AN
A
N
cuts the quadrilateral in two parts of equal area, and such that the same property holds for
C
M
CM
CM
. Prove that
M
N
MN
MN
cuts the diagonal
B
D
BD
B
D
in two segments of equal length.