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Problems
Contests
National and Regional Contests
Netherlands Contests
Dutch Mathematical Olympiad
1995 Dutch Mathematical Olympiad
1995 Dutch Mathematical Olympiad
Part of
Dutch Mathematical Olympiad
Subcontests
(5)
5
1
Hide problems
tame array
An array
(
a
1
,
a
2
,
.
.
.
,
a
13
)
(a_1,a_2,...,a_{13})
(
a
1
,
a
2
,
...
,
a
13
)
of
13
13
13
integers is called
t
a
m
e
tame
t
am
e
if for each
1
≤
i
≤
13
1 \le i \le 13
1
≤
i
≤
13
the following condition holds: If
a
i
a_i
a
i
is left out, the remaining twelve integers can be divided into two groups with the same sum of elements. A tame array is called
t
u
r
b
o
turbo
t
u
r
b
o
t
a
m
e
tame
t
am
e
if the remaining twelve numbers can always be divided in two groups of six numbers having the same sum.
(
a
)
(a)
(
a
)
Give an example of a tame array of
13
13
13
integers (not all equal).
(
b
)
(b)
(
b
)
Prove that in a tame array all numbers are of the same parity.
(
c
)
(c)
(
c
)
Prove that in a turbo tame array all numbers are equal.
4
1
Hide problems
compute the height of the pyramid
A number of spheres with radius
1
1
1
are being placed in the form of a square pyramid. First, there is a layer in the form of a square with
n
2
n^2
n
2
spheres. On top of that layer comes the next layer with (n\minus{}1)^2 spheres, and so on. The top layer consists of only one sphere. Compute the height of the pyramid.
3
1
Hide problems
marbles
Let
101
101
101
marbles be numbered from
1
1
1
to
101
101
101
. The marbles are divided over two baskets
A
A
A
and
B
B
B
. The marble numbered
40
40
40
is in basket
A
A
A
. When this marble is removed from basket
A
A
A
and put in
B
B
B
, the averages of the numbers
A
A
A
and
B
B
B
both increase by
1
4
\frac{1}{4}
4
1
. How many marbles were there originally in basket
A
?
A?
A
?
2
1
Hide problems
locus
For any point
P
P
P
on a segment
A
B
AB
A
B
, isosceles and right-angled triangles
A
Q
P
AQP
A
QP
and
P
R
B
PRB
PRB
are constructed on the same side of
A
B
AB
A
B
, with
A
P
AP
A
P
and
P
B
PB
PB
as the bases. Determine the locus of the midpoint
M
M
M
of
Q
R
QR
QR
when
P
P
P
describes the segment
A
B
AB
A
B
.
1
1
Hide problems
kangaroo
A kangaroo jumps from lattice poin to lattice point in the coordinate plane. It can make only two kinds of jumps:
(
A
)
(A)
(
A
)
1
1
1
to right and
3
3
3
up, and
(
B
)
(B)
(
B
)
2
2
2
to the left and
4
4
4
down.
(
a
)
(a)
(
a
)
The start position of the kangaroo is
(
0
,
0
)
(0,0)
(
0
,
0
)
. Show that it can jump to the point
(
19
,
95
)
(19,95)
(
19
,
95
)
and determine the number of jumps needed.
(
b
)
(b)
(
b
)
Show that if the start position is
(
1
,
0
)
(1,0)
(
1
,
0
)
, then it cannot reach
(
19
,
95
)
(19,95)
(
19
,
95
)
.
(
c
)
(c)
(
c
)
If the start position is
(
0
,
0
)
(0,0)
(
0
,
0
)
, find all points
(
m
,
n
)
(m,n)
(
m
,
n
)
with
m
,
n
≥
0
m,n \ge 0
m
,
n
≥
0
which the kangaroo can reach.