MathDB
Problems
Contests
National and Regional Contests
Brazil Contests
Brazil National Olympiad
1990 Brazil National Olympiad
1990 Brazil National Olympiad
Part of
Brazil National Olympiad
Subcontests
(5)
2
1
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$a^3 + 1990b^3 = c^4$
There exists infinitely many positive integers such that
a
3
+
1990
b
3
=
c
4
a^3 + 1990b^3 = c^4
a
3
+
1990
b
3
=
c
4
.
1
1
Hide problems
Polyhedron faces and their edges
Show that a convex polyhedron with an odd number of faces has at least one face with an even number of edges.
4
1
Hide problems
Area phae = area pebf = area pfcg = area pgdh.
A
B
C
D
ABCD
A
BC
D
is a quadrilateral,
E
,
F
,
G
,
H
E,F,G,H
E
,
F
,
G
,
H
are midpoints of
A
B
,
B
C
,
C
D
,
D
A
AB,BC,CD,DA
A
B
,
BC
,
C
D
,
D
A
. Find the point P such that
a
r
e
a
(
P
H
A
E
)
=
a
r
e
a
(
P
E
B
F
)
=
a
r
e
a
(
P
F
C
G
)
=
a
r
e
a
(
P
G
D
H
)
area (PHAE) = area (PEBF) = area (PFCG) = area (PGDH)
a
re
a
(
P
H
A
E
)
=
a
re
a
(
PEBF
)
=
a
re
a
(
PFCG
)
=
a
re
a
(
PG
DH
)
.
3
1
Hide problems
Tetrahedron with all faces equal and unit curcumsphere
Each face of a tetrahedron is a triangle with sides
a
,
b
,
a, b,
a
,
b
,
c and the tetrahedon has circumradius 1. Find
a
2
+
b
2
+
c
2
a^2 + b^2 + c^2
a
2
+
b
2
+
c
2
.
5
1
Hide problems
F^n(0)=0 iff f^n(t)=t
Let
f
(
x
)
=
a
x
+
b
c
x
+
d
f(x)=\frac{ax+b}{cx+d}
f
(
x
)
=
c
x
+
d
a
x
+
b
F
n
(
x
)
=
f
(
f
(
f
.
.
.
f
(
x
)
.
.
.
)
)
F_n(x)=f(f(f...f(x)...))
F
n
(
x
)
=
f
(
f
(
f
...
f
(
x
)
...
))
(with
n
f
′
s
n\ f's
n
f
′
s
)Suppose that
f
(
0
)
≠
0
f(0) \not =0
f
(
0
)
=
0
,
f
(
f
(
0
)
)
≠
0
f(f(0)) \not = 0
f
(
f
(
0
))
=
0
, and for some
n
n
n
we have
F
n
(
0
)
=
0
F_n(0)=0
F
n
(
0
)
=
0
, show that
F
n
(
x
)
=
x
F_n(x)=x
F
n
(
x
)
=
x
(for any valid x).