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Problems
Contests
National and Regional Contests
Austria Contests
Austrian MO National Competition
2005 Federal Competition For Advanced Students, Part 1
2005 Federal Competition For Advanced Students, Part 1
Part of
Austrian MO National Competition
Subcontests
(4)
4
1
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3 concurrent diagonals
We're given two congruent, equilateral triangles
A
B
C
ABC
A
BC
and
P
Q
R
PQR
PQR
with parallel sides, but one has one vertex pointing up and the other one has the vertex pointing down. One is placed above the other so that the area of intersection is a hexagon
A
1
A
2
A
3
A
4
A
5
A
6
A_1A_2A_3A_4A_5A_6
A
1
A
2
A
3
A
4
A
5
A
6
(labelled counterclockwise). Prove that
A
1
A
4
A_1A_4
A
1
A
4
,
A
2
A
5
A_2A_5
A
2
A
5
and
A
3
A
6
A_3A_6
A
3
A
6
are concurrent.
3
1
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$a^n+b^n+c^n$
For 3 real numbers
a
,
b
,
c
a,b,c
a
,
b
,
c
let
s
n
=
a
n
+
b
n
+
c
n
s_n=a^{n}+b^{n}+c^{n}
s
n
=
a
n
+
b
n
+
c
n
. It is known that
s
1
=
2
s_1=2
s
1
=
2
,
s
2
=
6
s_2=6
s
2
=
6
and
s
3
=
14
s_3=14
s
3
=
14
. Prove that for all natural numbers
n
>
1
n>1
n
>
1
, we have
∣
s
n
2
−
s
n
−
1
s
n
+
1
∣
=
8
|s^2_n-s_{n-1}s_{n+1}|=8
∣
s
n
2
−
s
n
−
1
s
n
+
1
∣
=
8
2
1
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a < 2005
For how many integers
a
a
a
with
∣
a
∣
≤
2005
|a| \leq 2005
∣
a
∣
≤
2005
, does the system
x
2
=
y
+
a
x^2=y+a
x
2
=
y
+
a
y
2
=
x
+
a
y^2=x+a
y
2
=
x
+
a
have integer solutions?
1
1
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All 10 Digits
Prove that there are infinitely many multiples of 2005 that contain all the digits 0, 1, 2,...,9 an equal number of times.