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Problems
Contests
National and Regional Contests
Austria Contests
Austrian MO National Competition
2004 Federal Competition For Advanced Students, P2
2004 Federal Competition For Advanced Students, P2
Part of
Austrian MO National Competition
Subcontests
(6)
5
1
Hide problems
4x4 system, symmetry, a^2 = (\sqrt{bc}\sqrt[3]{bcd})/(b+c)(b+c+d)
Solve the following system of equations in real numbers:
{
a
2
=
b
c
b
c
d
3
(
b
+
c
)
(
b
+
c
+
d
)
b
2
=
c
d
c
d
a
3
(
c
+
d
)
(
c
+
d
+
a
)
c
2
=
d
a
d
a
b
3
(
d
+
a
)
(
d
+
a
+
b
)
d
2
=
a
b
a
b
c
3
(
a
+
b
)
(
a
+
b
+
c
)
\begin{cases} a^2 = \cfrac{\sqrt{bc}\sqrt[3]{bcd}}{(b+c)(b+c+d)} \\ b^2 =\cfrac{\sqrt{cd}\sqrt[3]{cda}}{(c+d)(c+d+a)} \\ c^2 =\cfrac{\sqrt{da}\sqrt[3]{dab}}{(d+a)(d+a+b)} \\ d^2 =\cfrac{\sqrt{ab}\sqrt[3]{abc}}{(a+b)(a+b+c)} \end{cases}
⎩
⎨
⎧
a
2
=
(
b
+
c
)
(
b
+
c
+
d
)
b
c
3
b
c
d
b
2
=
(
c
+
d
)
(
c
+
d
+
a
)
c
d
3
c
d
a
c
2
=
(
d
+
a
)
(
d
+
a
+
b
)
d
a
3
d
ab
d
2
=
(
a
+
b
)
(
a
+
b
+
c
)
ab
3
ab
c
4
1
Hide problems
a^2_1+a^2_2+ ...+a^2_N is a perfect square, find recurrence formula
Show that there is an infinite sequence
a
1
,
a
2
,
.
.
.
a_1,a_2,...
a
1
,
a
2
,
...
of natural numbers such that
a
1
2
+
a
2
2
+
.
.
.
+
a
N
2
a^2_1+a^2_2+ ...+a^2_N
a
1
2
+
a
2
2
+
...
+
a
N
2
is a perfect square for all
N
N
N
. Give a recurrent formula for one such sequence.
3
1
Hide problems
cyclic trapezoid with perpendicular diagonals, area between circles wanted
A trapezoid
A
B
C
D
ABCD
A
BC
D
with perpendicular diagonals
A
C
AC
A
C
and
B
D
BD
B
D
is inscribed in a circle
k
k
k
. Let
k
a
k_a
k
a
and
k
c
k_c
k
c
respectively be the circles with diameters
A
B
AB
A
B
and
C
D
CD
C
D
. Compute the area of the region which is inside the circle
k
k
k
, but outside the circles
k
a
k_a
k
a
and
k
c
k_c
k
c
.
1
1
Hide problems
(i)a^6+b^6+c^6+d^6−6abcd >= −2 (ii) a^k +b^k +c^k +d^k −kabcd >= M_k
Prove without using advanced (differential) calculus: (a) For any real numbers a,b,c,d it holds that
a
6
+
b
6
+
c
6
+
d
6
−
6
a
b
c
d
≥
−
2
a^6+b^6+c^6+d^6-6abcd \ge -2
a
6
+
b
6
+
c
6
+
d
6
−
6
ab
c
d
≥
−
2
. When does equality hold? (b) For which natural numbers
k
k
k
does some inequality of the form
a
k
+
b
k
+
c
k
+
d
k
−
k
a
b
c
d
≥
M
k
a^k +b^k +c^k +d^k -kabcd \ge M_k
a
k
+
b
k
+
c
k
+
d
k
−
kab
c
d
≥
M
k
hold for all real
a
,
b
,
c
,
d
a,b,c,d
a
,
b
,
c
,
d
? For each such
k
k
k
,
6
1
Hide problems
35th Austrian Mathematical Olympiad 2004
Over the sides of an equilateral triangle with area
1
1
1
are triangles with the opposite angle
6
0
∘
60^{\circ}
6
0
∘
to each side drawn outside of the triangle. The new corners are
P
P
P
,
Q
Q
Q
and
R
R
R
. (and the new triangles
A
P
B
APB
A
PB
,
B
Q
C
BQC
BQC
and
A
R
C
ARC
A
RC
) 1)What is the highest possible area of the triangle
P
Q
R
PQR
PQR
? 2)What is the highest possible area of the triangle whose vertexes are the midpoints of the inscribed circles of the triangles
A
P
B
APB
A
PB
,
B
Q
C
BQC
BQC
and
A
R
C
ARC
A
RC
?
2
1
Hide problems
35th Austrian Mathematical Olympiad 2004
Show that every set
{
p
1
,
p
2
,
…
,
p
k
}
\{p_1,p_2,\dots,p_k\}
{
p
1
,
p
2
,
…
,
p
k
}
of prime numbers fulfils the following: The sum of all unit fractions (that are fractions of the type
1
n
\frac{1}{n}
n
1
), whose denominators are exactly the
k
k
k
given prime factors (but in arbitrary powers with exponents unequal zero), is an unit fraction again. How big is this sum if
1
2004
\frac{1}{2004}
2004
1
is among this summands? Show that for every set
{
p
1
,
p
2
,
…
,
p
k
}
\{p_1,p_2,\dots,p_k\}
{
p
1
,
p
2
,
…
,
p
k
}
containing
k
k
k
prime numbers (
k
>
2
k>2
k
>
2
) is the sum smaller than
1
N
\frac{1}{N}
N
1
with
N
=
2
⋅
3
k
−
2
(
k
−
2
)
!
N=2\cdot 3^{k-2}(k-2)!
N
=
2
⋅
3
k
−
2
(
k
−
2
)!