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Problems
Contests
National and Regional Contests
Brazil Contests
Brazil National Olympiad
1995 Brazil National Olympiad
1995 Brazil National Olympiad
Part of
Brazil National Olympiad
Subcontests
(6)
6
1
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About ternary families of an n-set
X
X
X
has
n
n
n
elements.
F
F
F
is a family of subsets of
X
X
X
each with three elements, such that any two of the subsets have at most one element in common. Show that there is a subset of
X
X
X
with at least
2
n
\sqrt{2n}
2
n
members which does not contain any members of
F
F
F
.
4
1
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Knot in a tetrahedron
A regular tetrahedron has side
L
L
L
. What is the smallest
x
x
x
such that the tetrahedron can be passed through a loop of twine of length
x
x
x
?
1
1
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About bicentric quadrilaterals
A
B
C
D
ABCD
A
BC
D
is a quadrilateral with a circumcircle centre
O
O
O
and an inscribed circle centre
I
I
I
. The diagonals intersect at
S
S
S
. Show that if two of
O
,
I
,
S
O,I,S
O
,
I
,
S
coincide, then it must be a square.
5
1
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No nth- roots of rationals
Show that no one
n
n
n
-th root of a rational (for
n
n
n
a positive integer) can be a root of the polynomial
x
5
−
x
4
−
4
x
3
+
4
x
2
+
2
x^5 - x^4 - 4x^3 + 4x^2 + 2
x
5
−
x
4
−
4
x
3
+
4
x
2
+
2
.
2
1
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Functions and number theory
Find all real-valued functions on the positive integers such that
f
(
x
+
1019
)
=
f
(
x
)
f(x + 1019) = f(x)
f
(
x
+
1019
)
=
f
(
x
)
for all
x
x
x
, and
f
(
x
y
)
=
f
(
x
)
f
(
y
)
f(xy) = f(x) f(y)
f
(
x
y
)
=
f
(
x
)
f
(
y
)
for all
x
,
y
x,y
x
,
y
.
3
1
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Brazil 1995
For any positive integer
n
>
1
n>1
n
>
1
, let
P
(
n
)
P\left(n\right)
P
(
n
)
denote the largest prime divisor of
n
n
n
. Prove that there exist infinitely many positive integers
n
n
n
for which P\left(n\right)