MathDB
Problems
Contests
National and Regional Contests
Netherlands Contests
Dutch BxMO/EGMO TST
2010 Dutch BxMO TST
2010 Dutch BxMO TST
Part of
Dutch BxMO/EGMO TST
Subcontests
(5)
2
1
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fuctional in R, f(x)f(y) = f(x + y) + xy
Find all functions
f
:
R
→
R
f : R \to R
f
:
R
→
R
satisfying
f
(
x
)
f
(
y
)
=
f
(
x
+
y
)
+
x
y
f(x)f(y) = f(x + y) + xy
f
(
x
)
f
(
y
)
=
f
(
x
+
y
)
+
x
y
for all
x
,
y
∈
R
x, y \in R
x
,
y
∈
R
.
5
1
Hide problems
permutation is quadratic if k+a_k is square, existence for every n
For any non-negative integer
n
n
n
, we say that a permutation
(
a
0
,
a
1
,
.
.
.
,
a
n
)
(a_0,a_1,...,a_n)
(
a
0
,
a
1
,
...
,
a
n
)
of
{
0
,
1
,
.
.
.
,
n
}
\{0,1,..., n\}
{
0
,
1
,
...
,
n
}
is quadratic if
k
+
a
k
k + a_k
k
+
a
k
is a square for
k
=
0
,
1
,
.
.
.
,
n
k = 0, 1,...,n
k
=
0
,
1
,
...
,
n
. Show that for any non-negative integer
n
n
n
, there exists a quadratic permutation of
{
0
,
1
,
.
.
.
,
n
}
\{0,1,..., n\}
{
0
,
1
,
...
,
n
}
.
1
1
Hide problems
rhombus wanted, DN passes through the midpoint of BE, trapezoid
Let
A
B
C
D
ABCD
A
BC
D
be a trapezoid with
A
B
/
/
C
D
AB // CD
A
B
//
C
D
,
2
∣
A
B
∣
=
∣
C
D
∣
2|AB| = |CD|
2∣
A
B
∣
=
∣
C
D
∣
and
B
D
⊥
B
C
BD \perp BC
B
D
⊥
BC
. Let
M
M
M
be the midpoint of
C
D
CD
C
D
and let
E
E
E
be the intersection
B
C
BC
BC
and
A
D
AD
A
D
. Let
O
O
O
be the intersection of
A
M
AM
A
M
and
B
D
BD
B
D
. Let
N
N
N
be the intersection of
O
E
OE
OE
and
A
B
AB
A
B
. (a) Prove that
A
B
M
D
ABMD
A
BM
D
is a rhombus. (b) Prove that the line
D
N
DN
D
N
passes through the midpoint of the line segment
B
E
BE
BE
.
4
1
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cyclic hexagon wanted, starting wih intersecting circles
The two circles
Γ
1
\Gamma_1
Γ
1
and
Γ
2
\Gamma_2
Γ
2
intersect at
P
P
P
and
Q
Q
Q
. The common tangent that's on the same side as
P
P
P
, intersects the circles at
A
A
A
and
B
B
B
,respectively. Let
C
C
C
be the second intersection with
Γ
2
\Gamma_2
Γ
2
of the tangent to
Γ
1
\Gamma_1
Γ
1
at
P
P
P
, and let
D
D
D
be the second intersection with
Γ
1
\Gamma_1
Γ
1
of the tangent to
Γ
2
\Gamma_2
Γ
2
at
Q
Q
Q
. Let
E
E
E
be the intersection of
A
P
AP
A
P
and
B
C
BC
BC
, and let
F
F
F
be the intersection of
B
P
BP
BP
and
A
D
AD
A
D
. Let
M
M
M
be the image of
P
P
P
under point reflection with respect to the midpoint of
A
B
AB
A
B
. Prove that
A
M
B
E
Q
F
AMBEQF
A
MBEQF
is a cyclic hexagon.
3
1
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Number of 5-tuples.
Let
N
N
N
be the number of ordered 5-tuples
(
a
1
,
a
2
,
a
3
,
a
4
,
a
5
)
(a_{1}, a_{2}, a_{3}, a_{4}, a_{5})
(
a
1
,
a
2
,
a
3
,
a
4
,
a
5
)
of positive integers satisfying
1
a
1
+
1
a
2
+
1
a
3
+
1
a
4
+
1
a
5
=
1
\frac{1}{a_{1}}+\frac{1}{a_{2}}+\frac{1}{a_{3}}+\frac{1}{a_{4}}+\frac{1}{a_{5}}=1
a
1
1
+
a
2
1
+
a
3
1
+
a
4
1
+
a
5
1
=
1
Is
N
N
N
even or odd?Oh and HINTS ONLY, please do not give full solutions. Thanks.