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Problems
Contests
National and Regional Contests
Netherlands Contests
Dutch BxMO/EGMO TST
2020 Dutch BxMO TST
2020 Dutch BxMO TST
Part of
Dutch BxMO/EGMO TST
Subcontests
(5)
5
1
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product of 100 pos. integers equal to product of 1.919 pos. integers
A set S consisting of
2019
2019
2019
(different) positive integers has the following property: the product of every 100 elements of
S
S
S
is a divisor of the product of the remaining
1919
1919
1919
elements. What is the maximum number of prime numbers that
S
S
S
can contain?
3
1
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f (x^2y) + 2f (y^2) =(x^2 + f (y)) f (y)
Find all functions
f
:
R
→
R
f: R \to R
f
:
R
→
R
that satisfy
f
(
x
2
y
)
+
2
f
(
y
2
)
=
(
x
2
+
f
(
y
)
)
⋅
f
(
y
)
f (x^2y) + 2f (y^2) =(x^2 + f (y)) \cdot f (y)
f
(
x
2
y
)
+
2
f
(
y
2
)
=
(
x
2
+
f
(
y
))
⋅
f
(
y
)
for all
x
,
y
∈
R
x, y \in R
x
,
y
∈
R
1
1
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a circle with n points on it, a positive integer at each point
For an integer
n
≥
3
n \ge 3
n
≥
3
we consider a circle with
n
n
n
points on it. We place a positive integer at each point, where the numbers are not necessary need to be different. Such placement of numbers is called stable as three numbers next to always have product
n
n
n
each other. For how many values of
n
n
n
with
3
≤
n
≤
2020
3 \le n \le 2020
3
≤
n
≤
2020
is it possible to place numbers in a stable way?
4
1
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F D =FM wanted , AB=BC, equilateral BCD
Three different points
A
,
B
A,B
A
,
B
and
C
C
C
lie on a circle with center
M
M
M
so that
∣
A
B
∣
=
∣
B
C
∣
| AB | = | BC |
∣
A
B
∣
=
∣
BC
∣
. Point
D
D
D
is inside the circle in such a way that
△
B
C
D
\vartriangle BCD
△
BC
D
is equilateral. Let
F
F
F
be the second intersection of
A
D
AD
A
D
with the circle . Prove that
∣
F
D
∣
=
∣
F
M
∣
| F D | = | FM |
∣
F
D
∣
=
∣
FM
∣
.
2
1
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concyclic points whose center lies on circumcircle wanted
In an acute-angled triangle
A
B
C
,
D
ABC, D
A
BC
,
D
is the foot of the altitude from
A
A
A
. Let
D
1
D_1
D
1
and
D
2
D_2
D
2
be the symmetric points of
D
D
D
wrt
A
B
AB
A
B
and
A
C
AC
A
C
, respectively. Let
E
1
E_1
E
1
be the intersection of
B
C
BC
BC
and the line through
D
1
D_1
D
1
parallel to
A
B
AB
A
B
. Let
E
2
E_2
E
2
be the intersection of
B
C
BC
BC
and the line through
D
2
D_2
D
2
parallel to
A
C
AC
A
C
. Prove that
D
1
,
D
2
,
E
1
D_1, D_2, E_1
D
1
,
D
2
,
E
1
and
E
2
E_2
E
2
on one circle whose center lies on the circumscribed circle of
△
A
B
C
\vartriangle ABC
△
A
BC
.