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Contests
National and Regional Contests
Austria Contests
Austrian MO Regional Competition
2000 Regional Competition For Advanced Students
2000 Regional Competition For Advanced Students
Part of
Austrian MO Regional Competition
Subcontests
(4)
2
1
Hide problems
(2x + 1)^4 + ax(x + 1) - x/2= 0
For any real number
a
a
a
, find all real numbers
x
x
x
that satisfy the following equation.
(
2
x
+
1
)
4
+
a
x
(
x
+
1
)
−
x
2
=
0
(2x + 1)^4 + ax(x + 1) - \frac{x}{2}= 0
(
2
x
+
1
)
4
+
a
x
(
x
+
1
)
−
2
x
=
0
1
1
Hide problems
2^n > 10n^2 - 60n + 80
For which natural numbers
n
n
n
does
2
n
>
10
n
2
−
60
n
+
80
2^n > 10n^2 -60n + 80
2
n
>
10
n
2
−
60
n
+
80
hold?
4
1
Hide problems
u_{n+1} =u_n(u_n + 1)/ n, rational numbers
We consider the sequence
{
u
n
}
\{u_n\}
{
u
n
}
defined by recursion
u
n
+
1
=
u
n
(
u
n
+
1
)
n
u_{n+1} =\frac{u_n(u_n + 1)}{n}
u
n
+
1
=
n
u
n
(
u
n
+
1
)
for
n
≥
1
n \ge 1
n
≥
1
. (a) Determine the terms of the sequence for
u
1
=
1
u_1 = 1
u
1
=
1
. (b) Show that if a member of the sequence is rational, then all subsequent members are also rational numbers. (c) Show that for every natural number
K
K
K
there is a
u
1
>
1
u_1 > 1
u
1
>
1
such that the first
K
K
K
terms of the sequence are natural numbers.
3
1
Hide problems
locus of midpoint of tangent segment
We consider two circles
k
1
(
M
1
,
r
1
)
k_1(M_1, r_1)
k
1
(
M
1
,
r
1
)
and
k
2
(
M
2
,
r
2
)
k_2(M_2, r_2)
k
2
(
M
2
,
r
2
)
with
z
=
M
1
M
2
>
r
1
+
r
2
z = M_1M_2 > r_1+r_2
z
=
M
1
M
2
>
r
1
+
r
2
and a common outer tangent with the tangent points
P
1
P_1
P
1
and
P
2
P2
P
2
(that is, they lie on the same side of the connecting line
M
1
M
2
M_1M_2
M
1
M
2
). We now change the radii so that their sum is
r
1
+
r
2
=
c
r_1+r_2 = c
r
1
+
r
2
=
c
remains constant. What set of points does the midpoint of the tangent segment
P
1
P
2
P_1P_2
P
1
P
2
run through, when
r
1
r_1
r
1
varies from
0
0
0
to
c
c
c
?