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Problems
Contests
National and Regional Contests
Austria Contests
Austrian MO National Competition
2008 Federal Competition For Advanced Students, P1
2008 Federal Competition For Advanced Students, P1
Part of
Austrian MO National Competition
Subcontests
(4)
2
1
Hide problems
nxn system x_1x_2(3a-2x_3) = a^3 ... x_nx_1(3a-2x_2) = a^3
Given
a
∈
R
+
a \in R^{+}
a
∈
R
+
and an integer
n
>
4
n > 4
n
>
4
determine all n-tuples (
x
1
,
.
.
.
,
x
n
x_1, ...,x_n
x
1
,
...
,
x
n
) of positive real numbers that satisfy the following system of equations:
{
x
1
x
2
(
3
a
−
2
x
3
)
=
a
3
x
2
x
3
(
3
a
−
2
x
4
)
=
a
3
.
.
.
x
n
−
2
x
n
−
1
(
3
a
−
2
x
n
)
=
a
3
x
n
−
1
x
n
(
3
a
−
2
x
1
)
=
a
3
x
n
x
1
(
3
a
−
2
x
2
)
=
a
3
\begin {cases} x_1x_2(3a-2x_3) = a^3\\ x_2x_3(3a-2x_4) = a^3\\ ...\\ x_{n-2}x_{n-1}(3a-2x_n) = a^3\\ x_{n-1}x_n(3a-2x_1) = a^3 \\ x_nx_1(3a-2x_2) = a^3 \end {cases}
⎩
⎨
⎧
x
1
x
2
(
3
a
−
2
x
3
)
=
a
3
x
2
x
3
(
3
a
−
2
x
4
)
=
a
3
...
x
n
−
2
x
n
−
1
(
3
a
−
2
x
n
)
=
a
3
x
n
−
1
x
n
(
3
a
−
2
x
1
)
=
a
3
x
n
x
1
(
3
a
−
2
x
2
)
=
a
3
.
3
1
Hide problems
a_{n+1} = (p+1)a_n - pa_{n-1} , a_n \le b_n
Let
p
>
1
p > 1
p
>
1
be a natural number. Consider the set
F
p
F_p
F
p
of all non-constant sequences of non-negative integers that satisfy the recursive relation
a
n
+
1
=
(
p
+
1
)
a
n
−
p
a
n
−
1
a_{n+1} = (p+1)a_n - pa_{n-1}
a
n
+
1
=
(
p
+
1
)
a
n
−
p
a
n
−
1
for all
n
>
0
n > 0
n
>
0
. Show that there exists a sequence (
a
n
a_n
a
n
) in
F
p
F_p
F
p
with the property that for every other sequence (
b
n
b_n
b
n
) in
F
p
F_p
F
p
, the inequality
a
n
≤
b
n
a_n \le b_n
a
n
≤
b
n
holds for all
n
n
n
.
1
1
Hide problems
1 \binom{2008}{0 }+2\binom{2008}{1}+ ...+2009\binom{2008}{2008} mod2008
What is the remainder of the number
1
(
2008
0
)
+
2
(
2008
1
)
+
.
.
.
+
2009
(
2008
2008
)
1 \binom{2008}{0 }+2\binom{2008}{1}+ ...+2009\binom{2008}{2008}
1
(
0
2008
)
+
2
(
1
2008
)
+
...
+
2009
(
2008
2008
)
when divided by
2008
2008
2008
?
4
1
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triangle EFG is isosceles iff triangle ABC is isosceles
In a triangle
A
B
C
ABC
A
BC
let
E
E
E
be the midpoint of the side
A
C
AC
A
C
and
F
F
F
the midpoint of the side
B
C
BC
BC
. Let
G
G
G
be the foot of the perpendicular from
C
C
C
to
A
B
AB
A
B
. Show that
△
E
F
G
\vartriangle EFG
△
EFG
is isosceles if and only if
△
A
B
C
\vartriangle ABC
△
A
BC
is isosceles.