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Problems
Contests
National and Regional Contests
Austria Contests
Austrian MO Beginners' Competition
2000 Austria Beginners' Competition
2000 Austria Beginners' Competition
Part of
Austrian MO Beginners' Competition
Subcontests
(4)
4
1
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AC \cap GE =circumcenter of AGP in half regular dodecagon ABCDEFG
Let
A
B
C
D
E
F
G
ABCDEFG
A
BC
D
EFG
be half of a regular dodecagon . Let
P
P
P
be the intersection of the lines
A
B
AB
A
B
and
G
F
GF
GF
, and let
Q
Q
Q
be the intersection of the lines
A
C
AC
A
C
and
G
E
GE
GE
. Prove that
Q
Q
Q
is the circumcenter of the triangle
A
G
P
AGP
A
GP
.
3
1
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multiple of product of its digits = multiple of sum of its digits
A two-digit number is nice if it is both a multiple of the product of its digits and a multiple of the sum of its digits. How many numbers satisfy this property? What is the ratio of the number to the sum of digits for each of the nice numbers?
2
1
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(a+b)^3 / a^2b >= 27/4
Let
a
,
b
a,b
a
,
b
positive real numbers. Prove that
(
a
+
b
)
3
a
2
b
≥
27
4
.
\frac{(a+b)^3}{a^2b}\ge \frac{27}{4}.
a
2
b
(
a
+
b
)
3
≥
4
27
.
When does equality occur?
1
1
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(x-y^2)(y-x^2)+x^3+y^3=a
Let
a
a
a
be a real number. Determine, for all
a
a
a
, all pairs
(
x
,
y
)
(x,y)
(
x
,
y
)
of real numbers such that
(
x
−
y
2
)
(
y
−
x
2
)
+
x
3
+
y
3
=
a
(x-y^2)(y-x^2)+x^3+y^3=a
(
x
−
y
2
)
(
y
−
x
2
)
+
x
3
+
y
3
=
a
.