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Problems
Contests
National and Regional Contests
Austria Contests
Austrian MO National Competition
2003 Federal Competition For Advanced Students, Part 2
2003 Federal Competition For Advanced Students, Part 2
Part of
Austrian MO National Competition
Subcontests
(3)
3
2
Hide problems
Compute the area of the union of all squares
For every lattice point
(
x
,
y
)
(x, y)
(
x
,
y
)
with
x
,
y
x, y
x
,
y
non-negative integers, a square of side
0.9
2
x
5
y
\frac{0.9}{2^x5^y}
2
x
5
y
0.9
with center at the point
(
x
,
y
)
(x, y)
(
x
,
y
)
is constructed. Compute the area of the union of all these squares.
Show that C lies on the line RS.
Let
A
B
C
ABC
A
BC
be an acute-angled triangle. The circle
k
k
k
with diameter
A
B
AB
A
B
intersects
A
C
AC
A
C
and
B
C
BC
BC
again at
P
P
P
and
Q
Q
Q
, respectively. The tangents to
k
k
k
at
A
A
A
and
Q
Q
Q
meet at
R
R
R
, and the tangents at
B
B
B
and
P
P
P
meet at
S
S
S
. Show that
C
C
C
lies on the line
R
S
RS
RS
.
2
2
Hide problems
the minimum value of [(a^3+b^3+c^3)/abc]^2
Let
a
,
b
,
c
a, b, c
a
,
b
,
c
be nonzero real numbers for which there exist
α
,
β
,
γ
∈
{
−
1
,
1
}
\alpha, \beta, \gamma \in\{-1, 1\}
α
,
β
,
γ
∈
{
−
1
,
1
}
with \alpha a + \beta b + \gamma c = 0. What is the smallest possible value of
(
a
3
+
b
3
+
c
3
a
b
c
)
2
?
\left( \frac{a^3+b^3+c^3}{abc}\right)^2 ?
(
ab
c
a
3
+
b
3
+
c
3
)
2
?
In how many ways can one tile the rectangle?
We are given sufficiently many stones of the forms of a rectangle
2
×
1
2\times 1
2
×
1
and square
1
×
1
1\times 1
1
×
1
. Let
n
>
3
n > 3
n
>
3
be a natural number. In how many ways can one tile a rectangle
3
×
n
3 \times n
3
×
n
using these stones, so that no two
2
×
1
2 \times 1
2
×
1
rectangles have a common point, and each of them has the longer side parallel to the shorter side of the big rectangle?
1
2
Hide problems
P(n) = n^3 -n^2 -5n+ 2 is a prime
Consider the polynomial
P
(
n
)
=
n
3
−
n
2
−
5
n
+
2
P(n) = n^3 -n^2 -5n+ 2
P
(
n
)
=
n
3
−
n
2
−
5
n
+
2
. Determine all integers
n
n
n
for which
P
(
n
)
2
P(n)^2
P
(
n
)
2
is a square of a prime.[hide="Remark."]I'm not sure if the statement of this problem is correct, because if
P
(
n
)
2
P(n)^2
P
(
n
)
2
be a square of a prime, then
P
(
n
)
P(n)
P
(
n
)
should be that prime, and I don't think the problem means that.
Number abc in base g will be cba in base h = g ± 1
Prove that, for any integer
g
>
2
g > 2
g
>
2
, there is a unique three-digit number
a
b
c
‾
g
\overline{abc}_g
ab
c
g
in base
g
g
g
whose representation in some base
h
=
g
±
1
h = g \pm 1
h
=
g
±
1
is
c
b
a
‾
h
\overline{cba}_h
c
ba
h
.