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Austria Contests
Austrian MO National Competition
2006 Federal Competition For Advanced Students, Part 1
2006 Federal Competition For Advanced Students, Part 1
Part of
Austrian MO National Competition
Subcontests
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1
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37th Austrian Mathematical Olympiad 2006
Given is the function f\equal{} \lfloor x^2 \rfloor \plus{} \{ x \} for all positive reals
x
x
x
. (
⌊
x
⌋
\lfloor x \rfloor
⌊
x
⌋
denotes the largest integer less than or equal
x
x
x
and \{ x \} \equal{} x \minus{} \lfloor x \rfloor.) Show that there exists an arithmetic sequence of different positive rational numbers, which all have the denominator
3
3
3
, if they are a reduced fraction, and don’t lie in the range of the function
f
f
f
.
1
1
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37th Austrian Mathematical Olympiad 2006
Let
n
n
n
be a non-negative integer, which ends written in decimal notation on exactly
k
k
k
zeros, but which is bigger than
1
0
k
10^k
1
0
k
. For a
n
n
n
is only k\equal{}k(n)\geq2 known. In how many different ways (as a function of k\equal{}k(n)\geq2) can
n
n
n
be written as difference of two squares of non-negative integers at least?
3
1
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37th Austrian Mathematical Olympiad 2006
In the triangle
A
B
C
ABC
A
BC
let
D
D
D
and
E
E
E
be the boundary points of the incircle with the sides
B
C
BC
BC
and
A
C
AC
A
C
. Show that if AD\equal{}BE holds, then the triangle is isoceles.
2
1
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37th Austrian Mathematical Olympiad 2006
Show that the sequence a_n \equal{} \frac {(n \plus{} 1)^nn^{2 \minus{} n}}{7n^2 \plus{} 1} is strictly monotonically increasing, where n \equal{} 0,1,2, \dots.