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National and Regional Contests
Austria Contests
Austrian MO Beginners' Competition
2003 Austria Beginners' Competition
2003 Austria Beginners' Competition
Part of
Austrian MO Beginners' Competition
Subcontests
(4)
2
1
Hide problems
(x -4) (x^2 - 8x + 14)^2 = (x - 4)^3
Find all real solutions of the equation
(
x
−
4
)
(
x
2
−
8
x
+
14
)
2
=
(
x
−
4
)
3
(x -4) (x^2 - 8x + 14)^2 = (x - 4)^3
(
x
−
4
)
(
x
2
−
8
x
+
14
)
2
=
(
x
−
4
)
3
.
1
1
Hide problems
max [\sqrt{[\sqrt{x+y}]}] if [\sqrt{x}] = 10 and [\sqrt{x}] =14
For the real numbers
x
x
x
and
y
y
y
,
[
x
]
=
10
[\sqrt{x}] = 10
[
x
]
=
10
and
[
y
]
=
14
[\sqrt{y}] =14
[
y
]
=
14
. How large is
[
[
x
+
y
]
]
\left[\sqrt{[ \sqrt{x+y} ]}\right]
[
[
x
+
y
]
]
?(Note: the square roots are the positive values and
[
x
]
[x]
[
x
]
is the largest integer less than or equal to x.)
3
1
Hide problems
max D, product of 5 consecutive even integers is always divisible by D
a) Show that the product of
5
5
5
consecutive even integers is divisible by
15
15
15
. b) Determine the largest integer
D
D
D
such that the product of
5
5
5
consecutive even integers is always divisible by
D
D
D
.
4
1
Hide problems
every rectangle circumscribed by a square is itself a square
Prove that every rectangle circumscribed by a square is itself a square.(A rectangle is circumscribed by a square if there is exactly one corner point of the square on each side of the rectangle.)