MathDB
Problems
Contests
National and Regional Contests
Netherlands Contests
Dutch Mathematical Olympiad
2001 Dutch Mathematical Olympiad
2001 Dutch Mathematical Olympiad
Part of
Dutch Mathematical Olympiad
Subcontests
(5)
3
1
Hide problems
parallelepiped cuts of a wooden beam
A wooden beam
E
F
G
H
EFGH
EFG
H
A
B
C
D
ABCD
A
BC
D
is with three cuts in
8
8
8
smaller ones sawn beams. Each cut is parallel to one of the three pair of opposit sides. Each pair of saw cuts is shown perpendicular to each other. The smaller bars at the corners
A
,
C
,
F
A, C, F
A
,
C
,
F
and
H
H
H
have a capacity of
9
,
12
,
8
,
24
9, 12, 8, 24
9
,
12
,
8
,
24
respectively.(The proportions in the picture are not correct!!). Calculate content of the entire bar.[asy] unitsize (0.5 cm);pair A, B, C, D, E, F, G, H; pair x, y, z;x = (1,0.5); y = (-0.8,0.8); z = (0,1);B = (0,0); C = 5*x; A = 3*y; F = 4*z; E = A + F; G = C + F; H = A + C + F;fill(y--3*y--(3*y + z)--(y + z)--cycle, gray(0.8)); fill(2*x--5*x--(5*x + z)--(2*x + z)--cycle, gray(0.8)); fill((y + z)--(y + 4*z)--(y + 4*z + 2*x)--(4*z + 2*x)--(2*x + z)--z--cycle, gray(0.8)); fill((2*x + y + 4*z)--(2*x + 3*y + 4*z)--(5*x + 3*y + 4*z)--(5*x + y + 4*z)--cycle, gray(0.8)); draw(B--C--G--H--E--A--cycle); draw(B--F); draw(E--F); draw(G--F); draw(y--(y + 4*z)--(y + 4*z + 5*x)); draw(2*x--(2*x + 4*z)--(2*x + 4*z + 3*y)); draw((3*y + z)--z--(5*x + z));label("
A
A
A
", A, SW); label("
B
B
B
", B, S); label("
C
C
C
", C, SE); label("
E
E
E
", E, NW); label("
F
F
F
", F, SE); label("
G
G
G
", G, NE); label("
H
H
H
", H, N); [/asy]
5
1
Hide problems
property of subset of 2001 no.s from set of 4002 no.s from 1 to 6003
If you take a subset of
4002
4002
4002
numbers from the whole numbers
1
1
1
to
6003
6003
6003
, then there is always a subset of
2001
2001
2001
numbers within that subset with the following property: If you order the
2001
2001
2001
numbers from small to large, the numbers are alternately even and odd (or odd and even). Prove this.
4
1
Hide problems
(2x^3 -6x^2 + 13x + 10)/(2x^2 - 9x) is an integer for pos.integer x
The function is given
f
(
x
)
=
2
x
3
−
6
x
2
+
13
x
+
10
2
x
2
−
9
x
f(x) = \frac{2x^3 -6x^2 + 13x + 10}{2x^2 - 9x}
f
(
x
)
=
2
x
2
−
9
x
2
x
3
−
6
x
2
+
13
x
+
10
.Determine all positive integers
x
x
x
for which
f
(
x
)
f(x)
f
(
x
)
is an integer
2
1
Hide problems
f(x + y) = f(x) + f(y) + xy, f(4) = 10, f(2001)=?
The function f has the following properties :
f
(
x
+
y
)
=
f
(
x
)
+
f
(
y
)
+
x
y
f(x + y) = f(x) + f(y) + xy
f
(
x
+
y
)
=
f
(
x
)
+
f
(
y
)
+
x
y
for all real
x
x
x
and
y
y
y
f
(
4
)
=
10
f(4) = 10
f
(
4
)
=
10
Calculate
f
(
2001
)
f(2001)
f
(
2001
)
.
1
1
Hide problems
tournament 3-1-0 points, total 15 points, last 1 point, 2nd to las undefeated
In a tournament, every team plays exactly once against every other team. One won match earns
3
3
3
points for the winner and
0
0
0
for the loser. With a draw both teams receive
1
1
1
point each. At the end of the tournament it appears that all teams together have achieved
15
15
15
points. The last team on the final list scored exactly
1
1
1
point. The second to last team has not lost a match. a) How many teams participated in the tournament? b) How many points did the team score in second place in the final ranking?