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Problems
Contests
National and Regional Contests
Austria Contests
Austrian MO Beginners' Competition
2006 Austria Beginners' Competition
2006 Austria Beginners' Competition
Part of
Austrian MO Beginners' Competition
Subcontests
(4)
4
1
Hide problems
isosceles if it has 2 equal excircles
Show that if a triangle has two excircles of the same size, then the triangle is isosceles.(Note: The excircle
A
B
C
ABC
A
BC
to the side
a
a
a
touches the extensions of the sides
A
B
AB
A
B
and
A
C
AC
A
C
and the side
B
C
BC
BC
.)
3
1
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sum of areas of rectangles equals n(n + 2)(7n + 4)/24
Let
n
n
n
be an even positive integer. We consider rectangles with integer side lengths
k
k
k
and
k
+
1
k +1
k
+
1
, where
k
k
k
is greater than
n
2
\frac{n}{2}
2
n
and at most equal to
n
n
n
. Show that for all even positive integers
n
n
n
the sum of the areas of these rectangles equals
n
(
n
+
2
)
(
7
n
+
4
)
24
.
\frac{n(n + 2)(7n + 4)}{24}.
24
n
(
n
+
2
)
(
7
n
+
4
)
.
2
1
Hide problems
(x^2 + ax + 4)(x^2 - 5x + 6) < 0, set of solutions is interval
For which real numbers
a
a
a
is the set of all solutions of the inequality
(
x
2
+
a
x
+
4
)
(
x
2
−
5
x
+
6
)
<
0
(x^2 + ax + 4)(x^2 - 5x + 6) < 0
(
x
2
+
a
x
+
4
)
(
x
2
−
5
x
+
6
)
<
0
an interval?
1
1
Hide problems
a^{2006} + b^{2006} + 1 is divisible by 2006^2
Do integers
a
,
b
a, b
a
,
b
exist such that
a
2006
+
b
2006
+
1
a^{2006} + b^{2006} + 1
a
2006
+
b
2006
+
1
is divisible by
200
6
2
2006^2
200
6
2
?