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Problems
Contests
National and Regional Contests
Austria Contests
Austrian MO National Competition
2003 Federal Competition For Advanced Students, Part 1
2003 Federal Competition For Advanced Students, Part 1
Part of
Austrian MO National Competition
Subcontests
(4)
4
1
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The segments divide the parallelogram into four triangles
In a parallelogram
A
B
C
D
ABCD
A
BC
D
, points
E
E
E
and
F
F
F
are the midpoints of
A
B
AB
A
B
and
B
C
BC
BC
, respectively, and
P
P
P
is the intersection of
E
C
EC
EC
and
F
D
FD
F
D
. Prove that the segments
A
P
,
B
P
,
C
P
AP,BP,CP
A
P
,
BP
,
CP
and
D
P
DP
D
P
divide the parallelogram into four triangles whose areas are in the ratio
1
:
2
:
3
:
4
1 : 2 : 3 : 4
1
:
2
:
3
:
4
.
3
1
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Solve a(1 - b^2) = b(1 - c^2) = c(1 - d^2) = d(1 - a^2) = t
Given a positive real number
t
t
t
, find the number of real solutions
a
,
b
,
c
,
d
a, b, c, d
a
,
b
,
c
,
d
of the system
a
(
1
−
b
2
)
=
b
(
1
−
c
2
)
=
c
(
1
−
d
2
)
=
d
(
1
−
a
2
)
=
t
.
a(1 - b^2) = b(1 -c^2) = c(1 -d^2) = d(1 - a^2) = t.
a
(
1
−
b
2
)
=
b
(
1
−
c
2
)
=
c
(
1
−
d
2
)
=
d
(
1
−
a
2
)
=
t
.
2
1
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Find max and min of f(x, y) = y-2x if (x^2+y^2)/(x+y) \leq 4
Find the greatest and smallest value of
f
(
x
,
y
)
=
y
−
2
x
f(x, y) = y-2x
f
(
x
,
y
)
=
y
−
2
x
, if x, y are distinct non-negative real numbers with
x
2
+
y
2
x
+
y
≤
4
\frac{x^2+y^2}{x+y}\leq 4
x
+
y
x
2
+
y
2
≤
4
.
1
1
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All primes p,q,r such that p^q + p^r is a perfect square
Find all triples of prime numbers
(
p
,
q
,
r
)
(p, q, r)
(
p
,
q
,
r
)
such that
p
q
+
p
r
p^q + p^r
p
q
+
p
r
is a perfect square.