MathDB
Problems
Contests
National and Regional Contests
Netherlands Contests
Dutch BxMO/EGMO TST
2017 Dutch BxMO TST
2017 Dutch BxMO TST
Part of
Dutch BxMO/EGMO TST
Subcontests
(5)
4
1
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combinatorics and number theory beautiful problem
A quadruple
(
a
;
b
;
c
;
d
)
(a; b; c; d)
(
a
;
b
;
c
;
d
)
of positive integers with
a
≤
b
≤
c
≤
d
a \leq b \leq c \leq d
a
≤
b
≤
c
≤
d
is called good if we can colour each integer red, blue, green or purple, in such a way that
i
i
i
of each
a
a
a
consecutive integers at least one is coloured red;
i
i
ii
ii
of each
b
b
b
consecutive integers at least one is coloured blue;
i
i
i
iii
iii
of each
c
c
c
consecutive integers at least one is coloured green;
i
i
i
i
iiii
iiii
of each
d
d
d
consecutive integers at least one is coloured purple. Determine all good quadruples with
a
=
2.
a = 2.
a
=
2.
5
1
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square of an integer
Determine all pairs of prime numbers
(
p
;
q
)
(p; q)
(
p
;
q
)
such that
p
2
+
5
p
q
+
4
q
2
p^2 + 5pq + 4q^2
p
2
+
5
pq
+
4
q
2
is the square of an integer.
3
1
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parallelism
Let
A
B
C
ABC
A
BC
be a triangle with
∠
A
=
90
\angle A = 90
∠
A
=
90
and let
D
D
D
be the orthogonal projection of
A
A
A
onto
B
C
BC
BC
. The midpoints of
A
D
AD
A
D
and
A
C
AC
A
C
are called
E
E
E
and
F
F
F
, respectively. Let
M
M
M
be the circumcentre of
B
E
F
BEF
BEF
. Prove that
A
C
AC
A
C
and
B
M
BM
BM
are parallel.
2
1
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beautiful functional equation problem
Let define a function
f
:
N
→
Z
f: \mathbb{N} \rightarrow \mathbb{Z}
f
:
N
→
Z
such that :
i
)
i)
i
)
f
(
p
)
=
1
f(p)=1
f
(
p
)
=
1
for all prime numbers
p
p
p
.
i
i
)
ii)
ii
)
f
(
x
y
)
=
x
f
(
y
)
+
y
f
(
x
)
f(xy)=xf(y)+yf(x)
f
(
x
y
)
=
x
f
(
y
)
+
y
f
(
x
)
for all positive integers
x
,
y
x,y
x
,
y
find the smallest
n
≥
2016
n \geq 2016
n
≥
2016
such that
f
(
n
)
=
n
f(n)=n
f
(
n
)
=
n
1
1
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complete integral values
Let
n
n
n
be an even positive integer. A sequence of
n
n
n
real numbers is called complete if for every integer
m
m
m
with
1
≤
m
≤
n
1 \leq m \leq n
1
≤
m
≤
n
either the sum of the first
m
m
m
terms of the sum or the sum of the last
m
m
m
terms is integral. Determine the minimum number of integers in a complete sequence of
n
n
n
numbers.