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Problems
Contests
National and Regional Contests
Netherlands Contests
Dutch Mathematical Olympiad
2006 Dutch Mathematical Olympiad
2006 Dutch Mathematical Olympiad
Part of
Dutch Mathematical Olympiad
Subcontests
(5)
5
1
Hide problems
2 colour game in a 8x8 chessboard, B can always prevent A from winning
Player
A
A
A
and player
B
B
B
play the next game on an
8
8
8
by
8
8
8
square chessboard. They in turn color a field that is not yet colored. One player uses red and the other blue. Player
A
A
A
starts. The winner is the first person to color the four squares of a square of
2
2
2
by
2
2
2
squares with his color somewhere on the board. Prove that player
B
B
B
can always prevent player
A
A
A
from winning.
4
1
Hide problems
r_A + r_B + r_C = r, 4 incircles related
Given is triangle
A
B
C
ABC
A
BC
with an inscribed circle with center
M
M
M
and radius
r
r
r
. The tangent to this circle parallel to
B
C
BC
BC
intersects
A
C
AC
A
C
in
D
D
D
and
A
B
AB
A
B
in
E
E
E
. The tangent to this circle parallel to
A
C
AC
A
C
intersects
A
B
AB
A
B
in
F
F
F
and
B
C
BC
BC
in
G
G
G
. The tangent to this circle parallel to
A
B
AB
A
B
intersects
B
C
BC
BC
in
H
H
H
and
A
C
AC
A
C
in
K
K
K
. Name the centers of the inscribed circles of triangle
A
E
D
AED
A
E
D
, triangle
F
B
G
FBG
FBG
and triangle
K
H
C
KHC
KH
C
successively
M
A
,
M
B
,
M
C
M_A, M_B, M_C
M
A
,
M
B
,
M
C
and the rays successively
r
A
,
r
B
r_A, r_B
r
A
,
r
B
and
r
C
r_C
r
C
. Prove that
r
A
+
r
B
+
r
C
=
r
r_A + r_B + r_C = r
r
A
+
r
B
+
r
C
=
r
.
3
1
Hide problems
smallest k>6 such 1 + 2 +...+ k = k + (k+1) +...+ n for n>k
1
+
2
+
3
+
4
+
5
+
6
=
6
+
7
+
8
1+2+3+4+5+6=6+7+8
1
+
2
+
3
+
4
+
5
+
6
=
6
+
7
+
8
. What is the smallest number
k
k
k
greater than
6
6
6
for which:
1
+
2
+
.
.
.
+
k
=
k
+
(
k
+
1
)
+
.
.
.
+
n
1 + 2 +...+ k = k + (k+1) +...+ n
1
+
2
+
...
+
k
=
k
+
(
k
+
1
)
+
...
+
n
, with
n
n
n
an integer greater than
k
k
k
?
2
1
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computational with distances of a points from triangle sides in
Given is a acute angled triangle
A
B
C
ABC
A
BC
. The lengths of the altitudes from
A
,
B
A, B
A
,
B
and
C
C
C
are successively
h
A
,
h
B
h_A, h_B
h
A
,
h
B
and
h
C
h_C
h
C
. Inside the triangle is a point
P
P
P
. The distance from
P
P
P
to
B
C
BC
BC
is
1
/
3
h
A
1/3 h_A
1/3
h
A
and the distance from
P
P
P
to
A
C
AC
A
C
is
1
/
4
h
B
1/4 h_B
1/4
h
B
. Express the distance from
P
P
P
to
A
B
AB
A
B
in terms of
h
C
h_C
h
C
.
1
1
Hide problems
palindromes using 5 diff. letters under conditions
A palindrome is a word that doesn't matter if you read it from left to right or from right to left. Examples: OMO, lepel and parterretrap. How many palindromes can you make with the five letters
a
,
b
,
c
,
d
a, b, c, d
a
,
b
,
c
,
d
and
e
e
e
under the conditions: - each letter may appear no more than twice in each palindrome, - the length of each palindrome is at least
3
3
3
letters. (Any possible combination of letters is considered a word.)