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Problems
Contests
National and Regional Contests
Netherlands Contests
Dutch BxMO/EGMO TST
2023 Dutch BxMO TST
2023 Dutch BxMO TST
Part of
Dutch BxMO/EGMO TST
Subcontests
(5)
3
1
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Playing a game of musical chairs
We play a game of musical chairs with
n
n
n
chairs numbered
1
1
1
to
n
n
n
. You attach
n
n
n
leaves, numbered
1
1
1
to
n
n
n
, to the chairs in such a way that the number on a leaf does not match the number on the chair it is attached to. One player sits on each chair. Every time you clap, each player looks at the number on the leaf attached to his current seat and moves to sit on the seat with that number. Prove that, for any
m
m
m
that is not a prime power with
1
<
m
≤
n
1 < m \leq n
1
<
m
≤
n
, it is possible to attach the leaves to the seats in such a way that after
m
m
m
claps everyone has returned to the chair they started on for the first time.
2
1
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Asymmetric functional inequality from the Netherworld
Find all functions
f
:
R
→
R
f : \mathbb R \to \mathbb R
f
:
R
→
R
for which
f
(
a
−
b
)
f
(
c
−
d
)
+
f
(
a
−
d
)
f
(
b
−
c
)
≤
(
a
−
c
)
f
(
b
−
d
)
,
f(a - b) f(c - d) + f(a - d) f(b - c) \leq (a - c) f(b - d),
f
(
a
−
b
)
f
(
c
−
d
)
+
f
(
a
−
d
)
f
(
b
−
c
)
≤
(
a
−
c
)
f
(
b
−
d
)
,
for all real numbers
a
,
b
,
c
a, b, c
a
,
b
,
c
and
d
d
d
. Note that there is only one occurrence of
f
f
f
on the right hand side!
1
1
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Taking a test of n questions with increasing worth
Let
n
≥
1
n \geq 1
n
≥
1
be an integer. Ruben takes a test with
n
n
n
questions. Each question on this test is worth a different number of points. The first question is worth
1
1
1
point, the second question
2
2
2
, the third
3
3
3
and so on until the last question which is worth
n
n
n
points. Each question can be answered either correctly or incorrectly. So an answer for a question can either be awarded all, or none of the points the question is worth. Let
f
(
n
)
f(n)
f
(
n
)
be the number of ways he can take the test so that the number of points awarded equals the number of questions he answered incorrectly. Do there exist in finitely many pairs
(
a
;
b
)
(a; b)
(
a
;
b
)
with
a
<
b
a < b
a
<
b
and
f
(
a
)
=
f
(
b
)
f(a) = f(b)
f
(
a
)
=
f
(
b
)
?
4
1
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Reciprocal geometric equation with excenter and circumcenter
In a triangle
△
A
B
C
\triangle ABC
△
A
BC
with
∠
A
B
C
<
∠
B
C
A
\angle ABC < \angle BCA
∠
A
BC
<
∠
BC
A
, we define
K
K
K
as the excenter with respect to
A
A
A
. The lines
A
K
AK
A
K
and
B
C
BC
BC
intersect in a point
D
D
D
. Let
E
E
E
be the circumcenter of
△
B
K
C
\triangle BKC
△
B
K
C
. Prove that
1
∣
K
A
∣
=
1
∣
K
D
∣
+
1
∣
K
E
∣
.
\frac{1}{|KA|} = \frac{1}{|KD|} + \frac{1}{|KE|}.
∣
K
A
∣
1
=
∣
KD
∣
1
+
∣
K
E
∣
1
.
5
1
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Classical NT using modular arithmetic
Find all pairs of prime numbers
(
p
,
q
)
(p,q)
(
p
,
q
)
for which
2
p
=
2
q
−
2
+
q
!
.
2^p = 2^{q-2} + q!.
2
p
=
2
q
−
2
+
q
!
.