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Problems
Contests
National and Regional Contests
Belgium Contests
Flanders Math Olympiad
1994 Flanders Math Olympiad
1994 Flanders Math Olympiad
Part of
Flanders Math Olympiad
Subcontests
(4)
3
1
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tetrahedron volume
Two regular tetrahedrons
A
A
A
and
B
B
B
are made with the 8 vertices of a unit cube. (this way is unique) What's the volume of
A
∪
B
A\cup B
A
∪
B
?
4
1
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functional sequence
Let
(
f
i
)
(f_i)
(
f
i
)
be a sequence of functions defined by:
f
1
(
x
)
=
x
,
f
n
(
x
)
=
f
n
−
1
(
x
)
−
1
4
f_1(x)=x, f_n(x) = \sqrt{f_{n-1}(x)}-\dfrac14
f
1
(
x
)
=
x
,
f
n
(
x
)
=
f
n
−
1
(
x
)
−
4
1
. (
n
∈
N
,
n
≥
2
n\in \mathbb{N}, n\ge2
n
∈
N
,
n
≥
2
) (a) Prove that
f
n
(
x
)
≤
f
n
−
1
(
x
)
f_n(x) \le f_{n-1}(x)
f
n
(
x
)
≤
f
n
−
1
(
x
)
for all x where both functions are defined. (b) Find for each
n
n
n
the points of
x
x
x
inside the domain for which
f
n
(
x
)
=
x
f_n(x)=x
f
n
(
x
)
=
x
.
1
1
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Triangle inequality
Let
a
,
b
,
c
>
0
a,b,c>0
a
,
b
,
c
>
0
the sides of a right triangle. Find all real
x
x
x
for which
a
x
>
b
x
+
c
x
a^x>b^x+c^x
a
x
>
b
x
+
c
x
, with
a
a
a
is the longest side.
2
1
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not my favorite (algebra + number theory)
Determine all integer solutions (a,b,c) with
c
≤
94
c\leq 94
c
≤
94
for which:
(
a
+
c
)
2
+
(
b
+
c
)
2
=
60
+
20
c
(a+\sqrt c)^2+(b+\sqrt c)^2 = 60 + 20\sqrt c
(
a
+
c
)
2
+
(
b
+
c
)
2
=
60
+
20
c