MathDB
Problems
Contests
National and Regional Contests
Netherlands Contests
Dutch Mathematical Olympiad
1991 Dutch Mathematical Olympiad
1991 Dutch Mathematical Olympiad
Part of
Dutch Mathematical Olympiad
Subcontests
(5)
5
1
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triangle
Let
H
H
H
be the orthocenter,
O
O
O
the circumcenter, and
R
R
R
the circumradius of an acute-angled triangle
A
B
C
ABC
A
BC
. Consider the circles
k
a
,
k
b
,
k
c
,
k
h
,
k
k_a,k_b,k_c,k_h,k
k
a
,
k
b
,
k
c
,
k
h
,
k
, all with radius
R
R
R
, centered at
A
,
B
,
C
,
H
,
M
,
A,B,C,H,M,
A
,
B
,
C
,
H
,
M
,
respectively. Circles
k
a
k_a
k
a
and
k
b
k_b
k
b
meet at
M
M
M
and
F
F
F
;
k
a
k_a
k
a
and
k
c
k_c
k
c
meet at
M
M
M
and
E
E
E
; and
k
b
k_b
k
b
and
k
c
k_c
k
c
meet at
M
M
M
and
D
D
D
.
(
a
)
(a)
(
a
)
Prove that the points
D
,
E
,
F
D,E,F
D
,
E
,
F
lie on the circle
k
h
k_h
k
h
.
(
b
)
(b)
(
b
)
Prove that the set of the points inside
k
h
k_h
k
h
that are inside exactly one of the circles
k
a
,
k
b
,
k
c
k_a,k_b,k_c
k
a
,
k
b
,
k
c
has the area twice the area of
△
A
B
C
\triangle ABC
△
A
BC
.
4
1
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nice problem
Three real numbers
a
,
b
,
c
a,b,c
a
,
b
,
c
satisfy the equations a\plus{}b\plus{}c\equal{}3, a^2\plus{}b^2\plus{}c^2\equal{}9, a^3\plus{}b^3\plus{}c^3\equal{}24. Find a^4\plus{}b^4\plus{}c^4.
3
1
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prove that f(0)=0
A real function
f
f
f
satisfies 4f(f(x))\minus{}2f(x)\minus{}3x\equal{}0 for all real numbers
x
x
x
. Prove that f(0)\equal{}0.
2
1
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compute the length
An angle with vertex
A
A
A
and measure
α
\alpha
α
and a point
P
0
P_0
P
0
on one of its rays are given so that AP_0\equal{}2. Point
P
1
P_1
P
1
is chose on the other ray. The sequence of points
P
1
,
P
2
,
P
3
,
.
.
.
P_1,P_2,P_3,...
P
1
,
P
2
,
P
3
,
...
is defined so that
P
n
P_n
P
n
lies on the segment AP_{n\minus{}2} and the triangle P_n P_{n\minus{}1} P_{n\minus{}2} is isosceles with P_n P_{n\minus{}1}\equal{}P_n P_{n\minus{}2} for all
n
≥
2
n \ge 2
n
≥
2
.
(
a
)
(a)
(
a
)
Prove that for each value of
α
\alpha
α
there is a unique point
P
1
P_1
P
1
for which the sequence
P
1
,
P
2
,
.
.
.
,
P
n
,
.
.
.
P_1,P_2,...,P_n,...
P
1
,
P
2
,
...
,
P
n
,
...
does not terminate.
(
b
)
(b)
(
b
)
Suppose that the sequence
P
1
,
P
2
,
.
.
.
P_1,P_2,...
P
1
,
P
2
,
...
does not terminate and that the length of the polygonal line
P
0
P
1
P
2
.
.
.
P
k
P_0 P_1 P_2 ... P_k
P
0
P
1
P
2
...
P
k
tends to
5
5
5
when
k
→
∞
k \rightarrow \infty
k
→
∞
. Compute the length of
P
0
P
1
P_0 P_1
P
0
P
1
.
1
1
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easy inequality
Prove that for any three positive real numbers a,b,c, \frac{1}{a\plus{}b}\plus{}\frac{1}{b\plus{}c}\plus{}\frac{1}{c\plus{}a} \ge \frac{9}{2} \cdot \frac{1}{a\plus{}b\plus{}c}.