MathDB
Problems
Contests
National and Regional Contests
Netherlands Contests
Dutch IMO TST
2009 Dutch IMO TST
2009 Dutch IMO TST
Part of
Dutch IMO TST
Subcontests
(4)
4
1
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f : Z \to Z, f(m + n) + f(mn - 1) = f(m)f(n) + 2
Find all functions
f
:
Z
→
Z
f : Z \to Z
f
:
Z
→
Z
satisfying
f
(
m
+
n
)
+
f
(
m
n
−
1
)
=
f
(
m
)
f
(
n
)
+
2
f(m + n) + f(mn -1) = f(m)f(n) + 2
f
(
m
+
n
)
+
f
(
mn
−
1
)
=
f
(
m
)
f
(
n
)
+
2
for all
m
,
n
∈
Z
m, n \in Z
m
,
n
∈
Z
.
3
1
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in R+: a + b + c >= abc =>a^2 + b^2 + c^2 >= \sqrt3 abc
Let
a
,
b
a, b
a
,
b
and
c
c
c
be positive reals such that a + b + c \ge abc. Prove that a^2 + b^2 + c^2 \ge \sqrt3 abc.
2
1
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double segment given, equal segments wanted
Let
A
B
C
ABC
A
BC
be a triangle,
P
P
P
the midpoint of
B
C
BC
BC
, and
Q
Q
Q
a point on segment
C
A
CA
C
A
such that
∣
C
Q
∣
=
2
∣
Q
A
∣
|CQ| = 2|QA|
∣
CQ
∣
=
2∣
Q
A
∣
. Let
S
S
S
be the intersection of
B
Q
BQ
BQ
and
A
P
AP
A
P
. Prove that
∣
A
S
∣
=
∣
S
P
∣
|AS| = |SP|
∣
A
S
∣
=
∣
SP
∣
.
5
1
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polygon with equal sides and vertices with rational coordinates
Suppose that we are given an
n
n
n
-gon of which all sides have the same length, and of which all the vertices have rational coordinates. Prove that
n
n
n
is even.