MathDB
Problems
Contests
National and Regional Contests
Netherlands Contests
Dutch Mathematical Olympiad
2019 Dutch Mathematical Olympiad
2019 Dutch Mathematical Olympiad
Part of
Dutch Mathematical Olympiad
Subcontests
(5)
5
1
Hide problems
card game, difference divisible by 3, winning strategy
Thomas and Nils are playing a game. They have a number of cards, numbered
1
,
2
,
3
1, 2, 3
1
,
2
,
3
, et cetera. At the start, all cards are lying face up on the table. They take alternate turns. The person whose turn it is, chooses a card that is still lying on the table and decides to either keep the card himself or to give it to the other player. When all cards are gone, each of them calculates the sum of the numbers on his own cards. If the difference between these two outcomes is divisible by
3
3
3
, then Thomas wins. If not, then Nils wins. (a) Suppose they are playing with
2018
2018
2018
cards (numbered from
1
1
1
to
2018
2018
2018
) and that Thomas starts. Prove that Nils can play in such a way that he will win the game with certainty. (b) Suppose they are playing with
2020
2020
2020
cards (numbered from
1
1
1
to
2020
2020
2020
) and that Nils starts. Which of the two players can play in such a way that he wins with certainty?
4
1
Hide problems
Fibonacci related sum a_i< 1 for a_n =\frac{1}{F_nF_{n+2}}
The sequence of Fibonacci numbers
F
0
,
F
1
,
F
2
,
.
.
.
F_0, F_1, F_2, . . .
F
0
,
F
1
,
F
2
,
...
is defined by
F
0
=
F
1
=
1
F_0 = F_1 = 1
F
0
=
F
1
=
1
and
F
n
+
2
=
F
n
+
F
n
+
1
F_{n+2} = F_n+F_{n+1}
F
n
+
2
=
F
n
+
F
n
+
1
for all
n
>
0
n > 0
n
>
0
. For example, we have
F
2
=
F
0
+
F
1
=
2
,
F
3
=
F
1
+
F
2
=
3
,
F
4
=
F
2
+
F
3
=
5
F_2 = F_0 + F_1 = 2, F_3 = F_1 + F_2 = 3, F_4 = F_2 + F_3 = 5
F
2
=
F
0
+
F
1
=
2
,
F
3
=
F
1
+
F
2
=
3
,
F
4
=
F
2
+
F
3
=
5
, and
F
5
=
F
3
+
F
4
=
8
F_5 = F_3 + F_4 = 8
F
5
=
F
3
+
F
4
=
8
. The sequence
a
0
,
a
1
,
a
2
,
.
.
.
a_0, a_1, a_2, ...
a
0
,
a
1
,
a
2
,
...
is defined by
a
n
=
1
F
n
F
n
+
2
a_n =\frac{1}{F_nF_{n+2}}
a
n
=
F
n
F
n
+
2
1
for all
n
≥
0
n \ge 0
n
≥
0
. Prove that for all
m
≥
0
m \ge 0
m
≥
0
we have:
a
0
+
a
1
+
a
2
+
.
.
.
+
a
m
<
1
a_0 + a_1 + a_2 + ... + a_m < 1
a
0
+
a
1
+
a
2
+
...
+
a
m
<
1
.
3
1
Hide problems
|CA| |CD| = |AB| |AM| , angle bisectors, reflection of circumcenter related
Points
A
,
B
A, B
A
,
B
, and
C
C
C
lie on a circle with centre
M
M
M
. The reflection of point
M
M
M
in the line
A
B
AB
A
B
lies inside triangle
A
B
C
ABC
A
BC
and is the intersection of the angle bisectors of angles
A
A
A
and
B
B
B
. Line
A
M
AM
A
M
intersects the circle again in point
D
D
D
. Show that
∣
C
A
∣
⋅
∣
C
D
∣
=
∣
A
B
∣
⋅
∣
A
M
∣
|CA| \cdot |CD| = |AB| \cdot |AM|
∣
C
A
∣
⋅
∣
C
D
∣
=
∣
A
B
∣
⋅
∣
A
M
∣
.
2
1
Hide problems
n guests at a party, any 2 guests are either friends or not friends
There are
n
n
n
guests at a party. Any two guests are either friends or not friends. Every guest is friends with exactly four of the other guests. Whenever a guest is not friends with two other guests, those two other guests cannot be friends with each other either. What are the possible values of
n
n
n
?
1
1
Hide problems
complete and difference numbers (9digit)
A complete number is a
9
9
9
digit number that contains each of the digits
1
1
1
to
9
9
9
exactly once. The difference number of a number
N
N
N
is the number you get by taking the differences of consecutive digits in
N
N
N
and then stringing these digits together. For instance, the difference number of
25143
25143
25143
is equal to
3431
3431
3431
. The complete number
124356879
124356879
124356879
has the additional property that its difference number,
12121212
12121212
12121212
, consists of digits alternating between
1
1
1
and
2
2
2
. Determine all
a
a
a
with
3
≤
a
≤
9
3 \le a \le 9
3
≤
a
≤
9
for which there exists a complete number
N
N
N
with the additional property that the digits of its difference number alternate between
1
1
1
and
a
a
a
.