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Problems
Contests
National and Regional Contests
Austria Contests
Austrian MO National Competition
2011 Federal Competition For Advanced Students, Part 2
2011 Federal Competition For Advanced Students, Part 2
Part of
Austrian MO National Competition
Subcontests
(3)
3
2
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Secant lines and a trapezoid
We are given a non-isosceles triangle
A
B
C
ABC
A
BC
with incenter
I
I
I
. Show that the circumcircle
k
k
k
of the triangle
A
I
B
AIB
A
I
B
does not touch the lines
C
A
CA
C
A
and
C
B
CB
CB
. Let
P
P
P
be the second point of intersection of
k
k
k
with
C
A
CA
C
A
and let
Q
Q
Q
be the second point of intersection of
k
k
k
with
C
B
CB
CB
. Show that the four points
A
A
A
,
B
B
B
,
P
P
P
and
Q
Q
Q
(not necessarily in this order) are the vertices of a trapezoid.
All lines UV pass through a fixed point
Two circles
k
1
k_1
k
1
and
k
2
k_2
k
2
with radii
r
1
r_1
r
1
and
r
2
r_2
r
2
touch each outside at point
Q
Q
Q
. The other endpoints of the diameters through
Q
Q
Q
are
P
P
P
on
k
1
k_1
k
1
and
R
R
R
on
k
2
k_2
k
2
. We choose two points
A
A
A
and
B
B
B
, one on each of the arcs
P
Q
PQ
PQ
of
k
1
k_1
k
1
. (
P
B
Q
A
PBQA
PBQ
A
is a convex quadrangle.) Further, let
C
C
C
be the second point of intersection of the line
A
Q
AQ
A
Q
with
k
2
k_2
k
2
and let
D
D
D
be the second point of intersection of the line
B
Q
BQ
BQ
with
k
2
k_2
k
2
. The lines
P
B
PB
PB
and
R
C
RC
RC
intersect in
U
U
U
and the lines
P
A
PA
P
A
and
R
D
RD
R
D
intersect in
V
V
V
. Show that there is a point
Z
Z
Z
that lies on all of these lines
U
V
UV
U
V
.
2
2
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Permutations with special properties
We consider permutations
f
f
f
of the set
N
\mathbb{N}
N
of non-negative integers, i.e. bijective maps
f
f
f
from
N
\mathbb{N}
N
to
N
\mathbb{N}
N
, with the following additional properties: f(f(x)) = x \mbox{and} \left| f(x)-x\right| \leqslant 3 \mbox{for all } x \in\mathbb{N}\mbox{.} Further, for all integers
n
>
42
n > 42
n
>
42
, \left.M(n)=\frac{1}{n+1}\sum_{j=0}^n \left|f(j)-j\right|<2,011\mbox{.}\right. Show that there are infinitely many natural numbers
K
K
K
such that
f
f
f
maps the set
{
n
∣
0
⩽
n
⩽
K
}
\left\{ n\mid 0\leqslant n\leqslant K\right\}
{
n
∣
0
⩽
n
⩽
K
}
onto itself.
Inequality for n real numbers
Let
k
k
k
and
n
n
n
be positive integers. Show that if
x
j
x_j
x
j
(
1
⩽
j
⩽
n
1\leqslant j\leqslant n
1
⩽
j
⩽
n
) are real numbers with
∑
j
=
1
n
1
x
j
2
k
+
k
=
1
k
\sum_{j=1}^n\frac{1}{x_j^{2^k}+k}=\frac{1}{k}
∑
j
=
1
n
x
j
2
k
+
k
1
=
k
1
, then \sum_{j=1}^n\frac{1}{x_j^{2^{k+1}}+k+2}\leqslant\frac{1}{k+1}\mbox{.}
1
2
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Bricks with bolts and braces
Every brick has
5
5
5
holes in a line. The holes can be filled with bolts (fi tting in one hole) and braces (fi tting into two neighboring holes). No hole may remain free. One puts
n
n
n
of these bricks in a line to form a pattern from left to right. In this line no two braces and no three bolts may be adjacent. How many diff erent such patterns can be produced with
n
n
n
bricks?
Divisibility relation: a^b+b | a^(2b)+2b
Determine all pairs
(
a
,
b
)
(a,b)
(
a
,
b
)
of non-negative integers, such that
a
b
+
b
a^b+b
a
b
+
b
divides
a
2
b
+
2
b
a^{2b}+2b
a
2
b
+
2
b
. (Remark:
0
0
=
1
0^0=1
0
0
=
1
.)