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Problems
Contests
National and Regional Contests
Austria Contests
Austrian MO National Competition
2000 Federal Competition For Advanced Students, Part 2
2000 Federal Competition For Advanced Students, Part 2
Part of
Austrian MO National Competition
Subcontests
(3)
3
2
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All functions so that f(x + f(y + z)) + f(f(x + y) + z) = 2y
Find all functions
f
:
R
→
R
f : \mathbb R \to \mathbb R
f
:
R
→
R
such that for all reals
x
,
y
,
z
x, y, z
x
,
y
,
z
it holds that
f
(
x
+
f
(
y
+
z
)
)
+
f
(
f
(
x
+
y
)
+
z
)
=
2
y
.
f(x + f(y + z)) + f(f(x + y) + z) = 2y.
f
(
x
+
f
(
y
+
z
))
+
f
(
f
(
x
+
y
)
+
z
)
=
2
y
.
All real solutions of the absolue value problem
Find all real solutions to the equation
∣
∣
∣
∣
∣
∣
∣
x
2
−
x
−
1
∣
−
3
∣
−
5
∣
−
7
∣
−
9
∣
−
11
∣
−
13
∣
=
x
2
−
2
x
−
48.
| | | | | | |x^2 -x - 1| - 3| - 5| - 7| - 9| - 11| - 13| = x^2 - 2x - 48.
∣∣∣∣∣∣∣
x
2
−
x
−
1∣
−
3∣
−
5∣
−
7∣
−
9∣
−
11∣
−
13∣
=
x
2
−
2
x
−
48.
2
2
Hide problems
Show that points P,Q,X, Y lie on a circle
A trapezoid
A
B
C
D
ABCD
A
BC
D
with
A
B
∥
C
D
AB \parallel CD
A
B
∥
C
D
is inscribed in a circle
k
k
k
. Points
P
P
P
and
Q
Q
Q
are chose on the arc
A
D
C
B
ADCB
A
D
CB
in the order
A
−
P
−
Q
−
B
A-P -Q-B
A
−
P
−
Q
−
B
. Lines
C
P
CP
CP
and
A
Q
AQ
A
Q
meet at
X
X
X
, and lines
B
P
BP
BP
and
D
Q
DQ
D
Q
meet at
Y
Y
Y
. Show that points
P
,
Q
,
X
,
Y
P,Q,X, Y
P
,
Q
,
X
,
Y
lie on a circle.
On the equation |(m^2+2000m+999999) - (3n^3+9n^2+27n) |=1
Find all pairs of integers
(
m
,
n
)
(m, n)
(
m
,
n
)
such that \left| (m^2 + 2000m+ 999999)- (3n^3 + 9n^2 + 27n) \right|= 1.
1
2
Hide problems
Prove that the Euler line HU bisects the angle BHD
In a non-equilateral acute-angled triangle
A
B
C
ABC
A
BC
with
∠
C
=
6
0
∘
\angle C = 60^\circ
∠
C
=
6
0
∘
,
U
U
U
is the circumcenter,
H
H
H
the orthocenter and
D
D
D
the intersection of
A
H
AH
A
H
and
B
C
BC
BC
. Prove that the Euler line
H
U
HU
H
U
bisects the angle
B
H
D
BHD
B
HD
.
Find the explicit form of the sequence b_n
The sequence an is defined by
a
0
=
4
,
a
1
=
1
a_0 = 4, a_1 = 1
a
0
=
4
,
a
1
=
1
and the recurrence formula
a
n
+
1
=
a
n
+
6
a
n
−
1
a_{n+1} = a_n + 6a_{n-1}
a
n
+
1
=
a
n
+
6
a
n
−
1
. The sequence
b
n
b_n
b
n
is given by
b
n
=
∑
k
=
0
n
(
n
k
)
a
k
.
b_n=\sum_{k=0}^n \binom nk a_k.
b
n
=
k
=
0
∑
n
(
k
n
)
a
k
.
Find the coefficients
α
,
β
\alpha,\beta
α
,
β
so that
b
n
b_n
b
n
satisfies the recurrence formula b_{n+1} = \alpha b_n + \beta b_{n-1}. Find the explicit form of
b
n
b_n
b
n
.