MathDB
Problems
Contests
National and Regional Contests
Austria Contests
Austrian MO Regional Competition
2013 Regional Competition For Advanced Students
2013 Regional Competition For Advanced Students
Part of
Austrian MO Regional Competition
Subcontests
(4)
4
1
Hide problems
every very distinguished pentagon has an axis of symmetry
We call a pentagon distinguished if either all side lengths or all angles are equal. We call it very distinguished if in addition two of the other parts are equal. i.e.
5
5
5
sides and
2
2
2
angles or
2
2
2
sides and
5
5
5
angles.Show that every very distinguished pentagon has an axis of symmetry.
3
1
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inequality with sequence by arithmetic and geometric means
For non-negative real numbers
a
,
a,
a
,
b
b
b
let
A
(
a
,
b
)
A(a, b)
A
(
a
,
b
)
be their arithmetic mean and
G
(
a
,
b
)
G(a, b)
G
(
a
,
b
)
their geometric mean. We consider the sequence
⟨
a
n
⟩
\langle a_n \rangle
⟨
a
n
⟩
with
a
0
=
0
,
a_0 = 0,
a
0
=
0
,
a
1
=
1
a_1 = 1
a
1
=
1
and
a
n
+
1
=
A
(
A
(
a
n
−
1
,
a
n
)
,
G
(
a
n
−
1
,
a
n
)
)
a_{n+1} = A(A(a_{n-1}, a_n), G(a_{n-1}, a_n))
a
n
+
1
=
A
(
A
(
a
n
−
1
,
a
n
)
,
G
(
a
n
−
1
,
a
n
))
for
n
>
0.
n > 0.
n
>
0.
(a) Show that each
a
n
=
b
n
2
a_n = b^2_n
a
n
=
b
n
2
is the square of a rational number (with
b
n
≥
0
b_n \geq 0
b
n
≥
0
). (b) Show that the inequality
∣
b
n
−
2
3
∣
<
1
2
n
\left|b_n - \frac{2}{3}\right| < \frac{1}{2^n}
b
n
−
3
2
<
2
n
1
holds for all
n
>
0.
n > 0.
n
>
0.
2
1
Hide problems
[x/2] [x/3][x/4]=x^2
Determine all integers
x
x
x
satisfying
[
x
2
]
[
x
3
]
[
x
4
]
=
x
2
.
\left[\frac{x}{2}\right] \left[\frac{x}{3}\right] \left[\frac{x}{4}\right] = x^2.
[
2
x
]
[
3
x
]
[
4
x
]
=
x
2
.
(
[
y
]
[y]
[
y
]
is the largest integer which is not larger than
y
.
y.
y
.
)
1
1
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between 2000 and 2010, probability that a random divisor is <=45 largest?
For which integers between
2000
2000
2000
and
2010
2010
2010
(including) is the probability that a random divisor is smaller or equal
45
45
45
the largest?