MathDB
Problems
Contests
National and Regional Contests
Netherlands Contests
Dutch Mathematical Olympiad
2008 Dutch Mathematical Olympiad
2008 Dutch Mathematical Olympiad
Part of
Dutch Mathematical Olympiad
Subcontests
(5)
3
1
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set S 756 of arbitrary integers between 1 and 2008 , exist a,b in S=> 8 \ (a+b)
Suppose that we have a set
S
S
S
of
756
756
756
arbitrary integers between
1
1
1
and
2008
2008
2008
(
1
1
1
and
2008
2008
2008
included). Prove that there are two distinct integers
a
a
a
and
b
b
b
in
S
S
S
such that their sum
a
+
b
a + b
a
+
b
is divisible by
8
8
8
.
2
1
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scent of a classic, diophantine 3 \cdot 2^n + 1 = m^2.
Find all positive integers
(
m
,
n
)
(m, n)
(
m
,
n
)
such that
3
⋅
2
n
+
1
=
m
2
3 \cdot 2^n + 1 = m^2
3
⋅
2
n
+
1
=
m
2
.
5
1
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game with sequence of 2008 non-negative integers (a,b,c)->(a-1,b+7,c-1)
We’re playing a game with a sequence of
2008
2008
2008
non-negative integers. A move consists of picking a integer
b
b
b
from that sequence, of which the neighbours
a
a
a
and
c
c
c
are positive. We then replace
a
,
b
a, b
a
,
b
and
c
c
c
by
a
−
1
,
b
+
7
a - 1, b + 7
a
−
1
,
b
+
7
and
c
−
1
c - 1
c
−
1
respectively. It is not allowed to pick the first or the last integer in the sequence, since they only have one neighbour. If there is no integer left such that both of its neighbours are positive, then there is no move left, and the game ends. Prove that the game always ends, regardless of the sequence of integers we begin with, and regardless of the moves we make.
4
1
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radius of circle ext. tangent to 3 circles, externally tangent also
Three circles
C
1
,
C
2
,
C
3
C_1,C_2,C_3
C
1
,
C
2
,
C
3
, with radii
1
,
2
,
3
1, 2, 3
1
,
2
,
3
respectively, are externally tangent. In the area enclosed by these circles, there is a circle
C
4
C_4
C
4
which is externally tangent to all three circles. Find the radius of
C
4
C_4
C
4
.[asy] unitsize(0.4 cm);pair[] O; real[] r;O[1] = (-12/5,16/5); r[1] = 1; O[2] = (0,5); r[2] = 2; O[3] = (0,0); r[3] = 3; O[4] = (-132/115, 351/115); r[4] = 6/23;draw(Circle(O[1],r[1])); draw(Circle(O[2],r[2])); draw(Circle(O[3],r[3])); draw(Circle(O[4],r[4]));label("
C
1
C_1
C
1
", O[1]); label("
C
2
C_2
C
2
", O[2]); label("
C
3
C_3
C
3
", O[3]); [/asy]
1
1
Hide problems
equal sum of areas, 2 squares related to homothety
Suppose we have a square
A
B
C
D
ABCD
A
BC
D
and a point
S
S
S
in the interior of this square. Under homothety with centre
S
S
S
and ratio of magnification
k
>
1
k > 1
k
>
1
, this square becomes another square
A
′
B
′
C
′
D
′
A'B'C'D'
A
′
B
′
C
′
D
′
. Prove that the sum of the areas of the two quadrilaterals
A
′
A
B
B
′
A'ABB'
A
′
A
B
B
′
and
C
′
C
D
D
′
C'CDD'
C
′
C
D
D
′
are equal to the sum of the areas of the two quadrilaterals
B
′
B
C
C
′
B'BCC'
B
′
BC
C
′
and
D
′
D
A
A
′
D'DAA'
D
′
D
A
A
′
.[asy] unitsize(3 cm);pair[] A, B, C, D; pair S;A[1] = (0,1); B[1] = (0,0); C[1] = (1,0); D[1] = (1,1); S = (0.3,0.6); A[0] = interp(S,A[1],2/3); B[0] = interp(S,B[1],2/3); C[0] = interp(S,C[1],2/3); D[0] = interp(S,D[1],2/3);draw(A[0]--B[0]--C[0]--D[0]--cycle); draw(A[1]--B[1]--C[1]--D[1]--cycle); draw(A[1]--S, dashed); draw(B[1]--S, dashed); draw(C[1]--S, dashed); draw(D[1]--S, dashed);dot("
A
A
A
", A[0], N); dot("
B
B
B
", B[0], SE); dot("
C
C
C
", C[0], SW); dot("
D
D
D
", D[0], SE); dot("
A
′
A'
A
′
", A[1], NW); dot("
B
′
B'
B
′
", B[1], SW); dot("
C
′
C'
C
′
", C[1], SE); dot("
D
′
D'
D
′
", D[1], NE); dot("
S
S
S
", S, dir(270)); [/asy]