math prob1

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2005 Brazil Undergrad MO
Brazil Undergrad MO 2005
Day 1
October 22nd
1
Determine the number of possible values for the determinant of
a
matrix with real entries such that
identity and is the all-zero matrix.
, given that
, where
is
is the
cyshine
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2
Let
and
be two continuous, distinct functions from
such that
Let
, for
Prove that
, natural.
is an increasing and divergent sequence.
aodeath
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Cezar Lupu
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3
Let
vectors in
such that
. Prove that there exists a permutation
every ,
.
Remark: If
,
for
of
and
such that
for
.
cyshine
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Day 2
October 23rd
4
Let
and
. Show that
converge.
IgorCastro
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alekk
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5
Prove that
cyshine
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Kent Merryfield
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6
Prove that for any natural
numbers
matrix
nonsingular.
and
,
,the
(
cyshine
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) is
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