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Problems
Contests
National and Regional Contests
Austria Contests
Austrian MO Regional Competition
2012 Regional Competition For Advanced Students
2012 Regional Competition For Advanced Students
Part of
Austrian MO Regional Competition
Subcontests
(4)
4
1
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Orthic triangle is isosceles
In a triangle
A
B
C
ABC
A
BC
, let
H
a
H_a
H
a
,
H
b
H_b
H
b
and
H
c
H_c
H
c
denote the base points of the altitudes on the sides
B
C
BC
BC
,
C
A
CA
C
A
and
A
B
AB
A
B
, respectively. Determine for which triangles
A
B
C
ABC
A
BC
two of the lengths
H
a
H
b
H_aH_b
H
a
H
b
,
H
b
H
c
H_bH_c
H
b
H
c
and
H
a
H
c
H_aH_c
H
a
H
c
are equal.
3
1
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Integer sequence containing square numbers
In an arithmetic sequence, the di fference of consecutive terms in constant. We consider sequences of integers in which the di fference of consecutive terms equals the sum of the differences of all preceding consecutive terms. Which of these sequences with
a
0
=
2012
a_0 = 2012
a
0
=
2012
and 1\leqslant d = a_1-a_0 \leqslant 43 contain square numbers?
2
1
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Integer equation: (x-1)x(x+1) + (y-1)y(y+1) = 24 - 9xy
Determine all integer solutions
(
x
,
y
)
(x, y)
(
x
,
y
)
of the equation (x - 1)x(x + 1) + (y - 1)y(y + 1) = 24 - 9xy\mbox{.}
1
1
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Inequality for real a: a + a^3 - a^4 - a^6 < 1
Prove that the inequality
a
+
a
3
−
a
4
−
a
6
<
1
a + a^3 - a^4 - a^6 < 1
a
+
a
3
−
a
4
−
a
6
<
1
holds for all real numbers
a
a
a
.