MathDB
Problems
Contests
National and Regional Contests
Netherlands Contests
Dutch Mathematical Olympiad
2003 Dutch Mathematical Olympiad
2003 Dutch Mathematical Olympiad
Part of
Dutch Mathematical Olympiad
Subcontests
(5)
4
1
Hide problems
rectangle wanted, 4 circles inside a circle given
In a circle with center
M
M
M
, two chords
A
C
AC
A
C
and
B
D
BD
B
D
intersect perpendicularly. The circle of diameter
A
M
AM
A
M
intersects the circle of diameter
B
M
BM
BM
besides
M
M
M
also in point
P
P
P
. The circle of diameter
B
M
BM
BM
intersects the circle with diameter
C
M
CM
CM
besides
M
M
M
also in point
Q
Q
Q
. The circle of diameter
C
M
CM
CM
intersects the circle of diameter
D
M
DM
D
M
besides
M
M
M
also in point
R
R
R
. The circle of diameter
D
M
DM
D
M
intersects the circle of diameter
A
M
AM
A
M
besides
M
M
M
also in point
S
S
S
. Prove that quadrilateral
P
Q
R
S
PQRS
PQRS
is a rectangle.[asy] unitsize (3 cm);pair A, B, C, D, M, P, Q, R, S;M = (0,0); A = dir(170); C = dir(10); B = dir(120); D = dir(240);draw(Circle(M,1)); draw(A--C); draw(B--D); draw(Circle(A/2,1/2)); draw(Circle(B/2,1/2)); draw(Circle(C/2,1/2)); draw(Circle(D/2,1/2)); P = (A + B)/2; Q = (B + C)/2; R = (C + D)/2; S = (D + A)/2;dot("
A
A
A
", A, A); dot("
B
B
B
", B, B); dot("
C
C
C
", C, C); dot("
D
D
D
", D, D); dot("
M
M
M
", M, E); dot("
P
P
P
", P, SE); dot("
Q
Q
Q
", Q, SE); dot("
R
R
R
", R, NE); dot("
S
S
S
", S, NE); [/asy]
2
1
Hide problems
common area of two equal squares
Two squares with side
12
12
12
lie exactly on top of each other. One square is rotated around a corner point through an angle of
30
30
30
degrees relative to the other square. Determine the area of the common piece of the two squares.[asy] unitsize (2 cm);pair A, B, C, D, Bp, Cp, Dp, P;A = (0,0); B = (-1,0); C = (-1,1); D = (0,1); Bp = rotate(-30)*(B); Cp = rotate(-30)*(C); Dp = rotate(-30)*(D); P = extension(C, D, Bp, Cp);fill(A--Bp--P--D--cycle, gray(0.8)); draw(A--B--C--D--cycle); draw(A--Bp--Cp--Dp--cycle);label("
3
0
∘
30^\circ
3
0
∘
", (-0.5,0.1), fontsize(10)); [/asy]
5
1
Hide problems
card game, replace a and b cards with (a+b+ab) card
There are a number of cards on a table. A number is written on each card. The "pick and replace" operation involves the following: two random cards are taken from the table and replaced by one new card. If the numbers
a
a
a
and
b
b
b
appear on the two packed cards, the number
a
+
b
+
a
b
a + b + ab
a
+
b
+
ab
is set on the new card. If we start with ten cards with the numbers
1
,
2
,
3
,
4
,
5
,
6
,
7
,
8
,
9
1, 2, 3, 4, 5, 6, 7, 8, 9
1
,
2
,
3
,
4
,
5
,
6
,
7
,
8
,
9
and
10
10
10
respectively, what value(s) can the number have that "grab and replace" nine times is on the only card still on the table? Prove your answer
3
1
Hide problems
n = a (a + 1) = b (b + 1) (b + 2) (b + 3) , diophantine
Determine all positive integers
n
n
n
that can be written as the product of two consecutive integers and as well as the product of four consecutive integers numbers. In the formula:
n
=
a
(
a
+
1
)
=
b
(
b
+
1
)
(
b
+
2
)
(
b
+
3
)
n = a (a + 1) = b (b + 1) (b + 2) (b + 3)
n
=
a
(
a
+
1
)
=
b
(
b
+
1
)
(
b
+
2
)
(
b
+
3
)
.
1
1
Hide problems
Pythagorean triangles whose area is twice the perimeter
A Pythagorean triangle is a right triangle whose three sides are integers. The best known example is the triangle with rectangular sides
3
3
3
and
4
4
4
and hypotenuse
5
5
5
. Determine all Pythagorean triangles whose area is twice the perimeter.