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Problems
Contests
National and Regional Contests
Netherlands Contests
Dutch Mathematical Olympiad
1964 Dutch Mathematical Olympiad
1964 Dutch Mathematical Olympiad
Part of
Dutch Mathematical Olympiad
Subcontests
(5)
2
1
Hide problems
quartets of points, OP = p, PQ = q, QR = r, RS = <OPQ = < PQR = < QRS = 135^o
Given is a flat plane
V
V
V
containing a rectangular coordinate system
x
O
y
xOy
x
O
y
. We consider quartets of numbers
(
p
,
q
,
r
,
s
)
(p,q,r,s)
(
p
,
q
,
r
,
s
)
;
p
≤
0
p\le 0
p
≤
0
,
q
≤
0
q \le 0
q
≤
0
,
r
≤
0
r \le 0
r
≤
0
,
s
≤
0
s \le 0
s
≤
0
. On every quartet we add a point
S
S
S
from
V
V
V
in a way that is in the accompanying figure is displayed. In this figure
O
P
=
p
OP = p
OP
=
p
,
P
Q
=
q
PQ = q
PQ
=
q
,
Q
R
=
r
QR = r
QR
=
r
,
R
S
=
s
RS = s
RS
=
s
,
∠
O
P
Q
=
∠
P
Q
R
=
∠
Q
R
S
=
13
5
o
\angle OPQ = \angle PQR = \angle QRS = 135^o
∠
OPQ
=
∠
PQR
=
∠
QRS
=
13
5
o
. (a) What is the set of the points of
V
V
V
, which are added to these quartets ? (b) Which of these points has been added to only one quartet? How many quartets have the other points been added? (c) What is the set of points added to the quartets for which
p
+
q
=
1
p + q = 1
p
+
q
=
1
and
r
=
s
=
0
r = s = 0
r
=
s
=
0
? (d) What is the set of points added to the quartets for which
p
+
1
=
p + 1 =
p
+
1
=
and
r
+
s
=
1
r + s = 1
r
+
s
=
1
?[asy] unitsize(0.6 cm);pair O, P, Q, R, S;O = (0,0); P = (2,0); Q = P + 2*dir(45); R = Q + (0,2.5); S = R + 3*dir(135);draw((-1,0)--(7,0)); draw((0,-1)--(0,8)); draw(P--Q--R--S);label("
O
O
O
", O, SW); label("
P
P
P
", P, dir(270)); label("
Q
Q
Q
", Q, E); label("
R
R
R
", R, E); label("
S
S
S
", S, N); label("
X
X
X
", (7,0), E); label("
Y
Y
Y
", (0,8), N); [/asy]
4
1
Hide problems
g {g(x) } = g(x)
The function
ƒ
ƒ
ƒ
is defined at
[
0
,
1
]
[0,1]
[
0
,
1
]
, and
f
{
f
(
x
)
}
=
ƒ
(
x
)
f\{f(x)\} = ƒ(x)
f
{
f
(
x
)}
=
ƒ
(
x
)
.
∃
c
∈
[
0
,
1
]
[
f
(
c
)
=
1
2
]
\exists _{c\in [0,1]} \left[f(c) =\frac12 \right]
∃
c
∈
[
0
,
1
]
[
f
(
c
)
=
2
1
]
Determine
f
(
1
2
)
.
f\left(\frac12 \right).
f
(
2
1
)
.
∀
t
∈
[
0
,
1
]
∃
s
∈
[
0
,
1
]
[
f
(
s
)
=
t
]
\forall _{t\in [0,1]}\exists _{s\in [0,1]}[f(s) = t]
∀
t
∈
[
0
,
1
]
∃
s
∈
[
0
,
1
]
[
f
(
s
)
=
t
]
. Determine
f
f
f
.Prove that the function
g
g
g
, with
g
(
x
)
=
x
g(x) = x
g
(
x
)
=
x
,
0
≤
x
≤
k
0 \le x \le k
0
≤
x
≤
k
,
g
(
x
)
=
k
g(x) = k
g
(
x
)
=
k
,
k
≤
x
≤
1
k \le x \le 1
k
≤
x
≤
1
satisfies the relation
g
{
g
(
x
)
}
=
g
(
x
)
g\{g(x)\} = g(x)
g
{
g
(
x
)}
=
g
(
x
)
.
5
1
Hide problems
product of digits sequence
Consider a sequence of non-negative integers g
1
,
g
2
,
g
3
,
.
.
.
_1,g_2,g_3,...
1
,
g
2
,
g
3
,
...
each consisting of three digits (numbers smaller than
100
100
100
are also written with three digits; the number
27
27
27
, for example, is written as
027
027
027
). Each number consists of the preceding by taking the product of the three digits that make up the preceding. The resulting sequence is of course dependent on the choice of
g
1
g_1
g
1
(e.g.
g
1
=
359
g_1 = 359
g
1
=
359
leads to
g
2
=
135
g_2= 135
g
2
=
135
,
g
3
=
015
g_3= 015
g
3
=
015
,
g
4
=
000
g_4 = 000
g
4
=
000
).Prove that independent of the choice of
g
1
g_1
g
1
: (a)
g
n
+
1
≤
g
n
g_{n+1}\le g_n
g
n
+
1
≤
g
n
(b)
g
10
=
000
g_{10}= 000
g
10
=
000
.
1
1
Hide problems
point construction, trapezoid with diagonals of 60^o
Given a triangle
A
B
C
ABC
A
BC
,
∠
C
=
6
0
o
\angle C= 60^o
∠
C
=
6
0
o
. Construct a point
P
P
P
on the side
A
C
AC
A
C
and a point
Q
Q
Q
on side
B
C
BC
BC
such that
A
B
Q
P
ABQP
A
BQP
is a trapezoid whose diagonals make an angle of
6
0
o
60^o
6
0
o
with each other.
3
1
Hide problems
(n + 1)(n +10) = q^2 for maxinum n
Solve
(
n
+
1
)
(
n
+
10
)
=
q
2
(n + 1)(n +10) = q^2
(
n
+
1
)
(
n
+
10
)
=
q
2
, for certain
q
q
q
and maximum
n
n
n
.