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Contests
National and Regional Contests
Netherlands Contests
Dutch BxMO/EGMO TST
2022 Dutch BxMO TST
2022 Dutch BxMO TST
Part of
Dutch BxMO/EGMO TST
Subcontests
(4)
3
1
Hide problems
p(p^2 -p - 1) = q(2q + 3), prime only diophantine
Find all pairs
(
p
,
q
)
(p, q)
(
p
,
q
)
of prime numbers such that
p
(
p
2
−
p
−
1
)
=
q
(
2
q
+
3
)
.
p(p^2 -p - 1) = q(2q + 3).
p
(
p
2
−
p
−
1
)
=
q
(
2
q
+
3
)
.
2
1
Hide problems
<BXD = <CY D if AXBCY is cyclic
Let
A
B
C
ABC
A
BC
be an acute triangle, and let
D
D
D
be the foot of the altitude from
A
A
A
. The circle with centre
A
A
A
passing through
D
D
D
intersects the circumcircle of triangle
A
B
C
ABC
A
BC
in
X
X
X
and
Y
Y
Y
, in such a way that the order of the points on this circumcircle is:
A
,
X
,
B
,
C
,
Y
A, X, B, C, Y
A
,
X
,
B
,
C
,
Y
. Show that
∠
B
X
D
=
∠
C
Y
D
\angle BXD = \angle CYD
∠
BX
D
=
∠
C
Y
D
.
1
1
Hide problems
f(n) | f(m) - n iff n | m for all n,m
Find all functions
f
:
Z
>
0
→
Z
>
0
f : Z_{>0} \to Z_{>0}
f
:
Z
>
0
→
Z
>
0
for which
f
(
n
)
∣
f
(
m
)
−
n
f(n) | f(m) - n
f
(
n
)
∣
f
(
m
)
−
n
if and only if
n
∣
m
n | m
n
∣
m
for all natural numbers
m
m
m
and
n
n
n
.
5
1
Hide problems
Fisherman and some kinds of fish
In a fish shop with 28 kinds of fish, there are 28 fish sellers. In every seller, there exists only one type of each fish kind, depending on where it comes, Mediterranean or Black Sea. Each of the
k
k
k
people gets exactly one fish from each seller and exactly one fish of each kind. For any two people, there exists a fish kind which they have different types of it (one Mediterranean, one Black Sea). What is the maximum possible number of
k
k
k
?