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Problems
Contests
National and Regional Contests
Austria Contests
Austrian MO National Competition
2004 Federal Competition For Advanced Students, Part 1
2004 Federal Competition For Advanced Students, Part 1
Part of
Austrian MO National Competition
Subcontests
(4)
4
1
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Which integral values can the sum take?
Each of the
2
N
=
2004
2N = 2004
2
N
=
2004
real numbers
x
1
,
x
2
,
…
,
x
2004
x_1, x_2, \ldots , x_{2004}
x
1
,
x
2
,
…
,
x
2004
equals either
2
−
1
\sqrt 2 -1
2
−
1
or
2
+
1
\sqrt 2 +1
2
+
1
. Can the sum
∑
k
=
1
N
x
2
k
−
1
x
2
k
\sum_{k=1}^N x_{2k-1}x_2k
∑
k
=
1
N
x
2
k
−
1
x
2
k
take the value
2004
2004
2004
? Which integral values can this sum take?
3
1
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Z(a,b) is an integer for a \leq b (Austria 2004 -Part1)
For natural numbers
a
,
b
a, b
a
,
b
, define
Z
(
a
,
b
)
=
(
3
a
)
!
⋅
(
4
b
)
!
a
!
4
⋅
b
!
3
Z(a,b)=\frac{(3a)!\cdot (4b)!}{a!^4 \cdot b!^3}
Z
(
a
,
b
)
=
a
!
4
⋅
b
!
3
(
3
a
)!
⋅
(
4
b
)!
.(a) Prove that
Z
(
a
,
b
)
Z(a, b)
Z
(
a
,
b
)
is an integer for
a
≤
b
a \leq b
a
≤
b
.(b) Prove that for each natural number
b
b
b
there are infinitely many natural numbers a such that
Z
(
a
,
b
)
Z(a, b)
Z
(
a
,
b
)
is not an integer.
2
1
Hide problems
Three pairs of perpendicular diagonals (Austria 2004 -Part1)
A convex hexagon
A
B
C
D
E
F
ABCDEF
A
BC
D
EF
with
A
B
=
B
C
=
a
,
C
D
=
D
E
=
b
,
E
F
=
F
A
=
c
AB = BC = a, CD = DE = b, EF = FA = c
A
B
=
BC
=
a
,
C
D
=
D
E
=
b
,
EF
=
F
A
=
c
is inscribed in a circle. Show that this hexagon has three (pairwise disjoint) pairs of mutually perpendicular diagonals.
1
1
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All reals a,b,c,d s.t. a + bcd = b + cda = c + dab = d + abc
Find all quadruples
(
a
,
b
,
c
,
d
)
(a, b, c, d)
(
a
,
b
,
c
,
d
)
of real numbers such that
a
+
b
c
d
=
b
+
c
d
a
=
c
+
d
a
b
=
d
+
a
b
c
.
a + bcd = b + cda = c + dab = d + abc.
a
+
b
c
d
=
b
+
c
d
a
=
c
+
d
ab
=
d
+
ab
c
.