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Problems
Contests
National and Regional Contests
Austria Contests
Austrian MO National Competition
2009 Federal Competition For Advanced Students, P1
2009 Federal Competition For Advanced Students, P1
Part of
Austrian MO National Competition
Subcontests
(4)
4
1
Hide problems
midpoints, feet of altitudes and other midpoints, lead to concurrency
Let
D
,
E
D, E
D
,
E
, and
F
F
F
be respectively the midpoints of the sides
B
C
,
C
A
BC, CA
BC
,
C
A
, and
A
B
AB
A
B
of
△
A
B
C
\vartriangle ABC
△
A
BC
. Let
H
a
,
H
b
,
H
c
H_a, H_b, H_c
H
a
,
H
b
,
H
c
be the feet of perpendiculars from
A
,
B
,
C
A, B, C
A
,
B
,
C
to the opposite sides, respectively. Let
P
,
Q
,
R
P, Q, R
P
,
Q
,
R
be the midpoints of the
H
b
H
c
,
H
c
H
a
H_bH_c, H_cH_a
H
b
H
c
,
H
c
H
a
, and
H
a
H
b
H_aH_b
H
a
H
b
respectively. Prove that
P
D
,
Q
E
PD, QE
P
D
,
QE
, and
R
F
RF
RF
are concurrent.
3
1
Hide problems
n bus stops placed around a circular lake, no of bus routes
There are
n
n
n
bus stops placed around the circular lake. Each bus stop is connected by a road to the two adjacent stops (we call a segment the entire road between two stops). Determine the number of bus routes that start and end in the fixed bus stop A, pass through each bus stop at least once and travel through exactly
n
+
1
n+1
n
+
1
segments.
1
1
Hide problems
3^{n^2} > (n!)^4
Show that for all positive integer
n
n
n
the following inequality holds
3
n
2
>
(
n
!
)
4
3^{n^2} > (n!)^4
3
n
2
>
(
n
!
)
4
.
2
1
Hide problems
Old Austrian
For a positive integers
n
,
k
n,k
n
,
k
we define k-multifactorial of n as
F
k
(
n
)
Fk(n)
F
k
(
n
)
=
(
n
)
(n)
(
n
)
.
(
n
−
k
)
(n-k)
(
n
−
k
)
(
n
−
2
k
)
(n-2k)
(
n
−
2
k
)
...
(
r
)
(r)
(
r
)
, where
r
r
r
is the reminder when
n
n
n
is divided by
k
k
k
that satisfy
1
<
=
r
<
=
k
1<=r<=k
1
<=
r
<=
k
Determine all non-negative integers
n
n
n
such that
F
20
(
n
)
+
2009
F20(n)+2009
F
20
(
n
)
+
2009
is a perfect square.