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Problems
Contests
National and Regional Contests
Netherlands Contests
Dutch Mathematical Olympiad
2009 Dutch Mathematical Olympiad
2009 Dutch Mathematical Olympiad
Part of
Dutch Mathematical Olympiad
Subcontests
(5)
4
1
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dutch parallelogram, similar triangles on triangle sides
Let
A
B
C
ABC
A
BC
be an arbitrary triangle. On the perpendicular bisector of
A
B
AB
A
B
, there is a point
P
P
P
inside of triangle
A
B
C
ABC
A
BC
. On the sides
B
C
BC
BC
and
C
A
CA
C
A
, triangles
B
Q
C
BQC
BQC
and
C
R
A
CRA
CR
A
are placed externally. These triangles satisfy
△
B
P
A
∼
△
B
Q
C
∼
△
C
R
A
\vartriangle BPA \sim \vartriangle BQC \sim \vartriangle CRA
△
BP
A
∼
△
BQC
∼
△
CR
A
. (So
Q
Q
Q
and
A
A
A
lie on opposite sides of
B
C
BC
BC
, and
R
R
R
and
B
B
B
lie on opposite sides of
A
C
AC
A
C
.) Show that the points
P
,
Q
,
C
P, Q, C
P
,
Q
,
C
and
R
R
R
form a parallelogram.
5
1
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game with 100 numbered cards, 1-100
We number a hundred blank cards on both sides with the numbers
1
1
1
to
100
100
100
. The cards are then stacked in order, with the card with the number
1
1
1
on top. The order of the cards is changed step by step as follows: at the
1
1
1
st step the top card is turned around, and is put back on top of the stack (nothing changes, of course), at the
2
2
2
nd step the topmost
2
2
2
cards are turned around, and put back on top of the stack, up to the
100
100
100
th step, in which the entire stack of
100
100
100
cards is turned around. At the
101
101
101
st step, again only the top card is turned around, at the
102
102
102
nd step, the top most
2
2
2
cards are turned around, and so on. Show that after a finite number of steps, the cards return to their original positions.
3
1
Hide problems
A wins against B, B wins against C, and C wins against A, in tennis tournament
A tennis tournament has at least three participants. Every participant plays exactly one match against every other participant. Moreover, every participant wins at least one of the matches he plays. (Draws do not occur in tennis matches.) Show that there are three participants
A
,
B
A, B
A
,
B
and
C
C
C
for which the following holds:
A
A
A
wins against
B
,
B
B, B
B
,
B
wins against
C
C
C
, and
C
C
C
wins against
A
A
A
.
2
1
Hide problems
a_{2n-1} = a_{2n-2} + n$ , $a_{2n} = a_{2n-1} + n , perfect square
Consider the sequence of integers
0
,
1
,
2
,
4
,
6
,
9
,
12
,
.
.
.
0, 1, 2, 4, 6, 9, 12,...
0
,
1
,
2
,
4
,
6
,
9
,
12
,
...
obtained by starting with zero, adding
1
1
1
, then adding
1
1
1
again, then adding
2
2
2
, and adding
2
2
2
again, then adding
3
3
3
, and adding
3
3
3
again, and so on. If we call the subsequent terms of this sequence
a
0
,
a
1
,
a
2
,
.
.
.
a_0, a_1, a_2, ...
a
0
,
a
1
,
a
2
,
...
, then we have
a
0
=
0
a_0 = 0
a
0
=
0
, and
a
2
n
−
1
=
a
2
n
−
2
+
n
a_{2n-1} = a_{2n-2} + n
a
2
n
−
1
=
a
2
n
−
2
+
n
,
a
2
n
=
a
2
n
−
1
+
n
a_{2n} = a_{2n-1} + n
a
2
n
=
a
2
n
−
1
+
n
for all integers
n
≥
1
n \ge 1
n
≥
1
. Find all integers
k
≥
0
k \ge 0
k
≥
0
for which
a
k
a_k
a
k
is the square of an integer.
1
1
Hide problems
5-digit panlindrome product numbers
In this problem, we consider integers consisting of
5
5
5
digits, of which the rst and last one are nonzero. We say that such an integer is a palindromic product if it satis es the following two conditions: - the integer is a palindrome, (i.e. it doesn't matter if you read it from left to right, or the other way around); - the integer is a product of two positive integers, of which the fi rst, when read from left to right, is equal to the second, when read from right to left, like
4831
4831
4831
and
1384
1384
1384
. For example,
20502
20502
20502
is a palindromic product, since
102
⋅
201
=
20502
102 \cdot 201 = 20502
102
⋅
201
=
20502
, and
20502
20502
20502
itself is a palindrome. Determine all palindromic products of
5
5
5
digits.