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Contests
National and Regional Contests
Austria Contests
Austrian MO National Competition
2014 Federal Competition For Advanced Students
2014 Federal Competition For Advanced Students
Part of
Austrian MO National Competition
Subcontests
(4)
4
1
Hide problems
right triangle, 2 circles given, angle bisector wanted
We are given a right-angled triangle
M
N
P
MNP
MNP
with right angle in
P
P
P
. Let
k
M
k_M
k
M
be the circle with center
M
M
M
and radius
M
P
MP
MP
, and let
k
N
k_N
k
N
be the circle with center
N
N
N
and radius
N
P
NP
NP
. Let
A
A
A
and
B
B
B
be the common points of
k
M
k_M
k
M
and the line
M
N
MN
MN
, and let
C
C
C
and
D
D
D
be the common points of
k
N
k_N
k
N
and the line
M
N
MN
MN
with with
C
C
C
between
A
A
A
and
B
B
B
. Prove that the line
P
C
PC
PC
bisects the angle
∠
A
P
B
\angle APB
∠
A
PB
.
3
1
Hide problems
a_{n+1} = a_n + 2 \cdot 3^n , find rational a_o such a^j_k / a^k_j
Let
a
n
a_n
a
n
be a sequence de fined by some
a
0
a_0
a
0
and the recursion
a
n
+
1
=
a
n
+
2
⋅
3
n
a_{n+1} = a_n + 2 \cdot 3^n
a
n
+
1
=
a
n
+
2
⋅
3
n
for
n
≥
0
n \ge 0
n
≥
0
. Determine all rational values of
a
0
a_0
a
0
such that
a
k
j
/
a
j
k
a^j_k / a^k_j
a
k
j
/
a
j
k
is an integer for all integers
j
j
j
and
k
k
k
with
0
<
j
<
k
0 < j < k
0
<
j
<
k
.
2
1
Hide problems
lattice set of squares, friendly sets
We call a set of squares with sides parallel to the coordinate axes and vertices with integer coordinates friendly if any two of them have exactly two points in common. We consider friendly sets in which each of the squares has sides of length
n
n
n
. Determine the largest possible number of squares in such a friendly set.
1
1
Hide problems
system x^2 + x = y^3 - y, y^2 + y = x^3 - x
Determine all real numbers
x
x
x
and
y
y
y
such that
x
2
+
x
=
y
3
−
y
x^2 + x = y^3 - y
x
2
+
x
=
y
3
−
y
,
y
2
+
y
=
x
3
−
x
y^2 + y = x^3 - x
y
2
+
y
=
x
3
−
x