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Brazil Undergrad MO
2005 Brazil Undergrad MO
2005 Brazil Undergrad MO
Part of
Brazil Undergrad MO
Subcontests
(6)
3
1
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More like combinatorics, i think, but it concerns vectors
Let
v
1
,
v
2
,
…
,
v
n
v_1,v_2,\ldots,v_n
v
1
,
v
2
,
…
,
v
n
vectors in
R
2
\mathbb{R}^2
R
2
such that
∣
v
i
∣
≤
1
|v_i|\leq 1
∣
v
i
∣
≤
1
for
1
≤
i
≤
n
1 \leq i \leq n
1
≤
i
≤
n
and
∑
i
=
1
n
v
i
=
0
\sum_{i=1}^n v_i=0
∑
i
=
1
n
v
i
=
0
. Prove that there exists a permutation
σ
\sigma
σ
of
(
1
,
2
,
…
,
n
)
(1,2,\ldots,n)
(
1
,
2
,
…
,
n
)
such that
∣
∑
j
=
1
k
v
σ
(
j
)
∣
≤
5
\left|\sum_{j=1}^k v_{\sigma(j)}\right| \leq\sqrt 5
∑
j
=
1
k
v
σ
(
j
)
≤
5
for every
k
k
k
,
1
≤
k
≤
n
1\leq k \leq n
1
≤
k
≤
n
. Remark: If
v
=
(
x
,
y
)
∈
R
2
v = (x,y)\in \mathbb{R}^2
v
=
(
x
,
y
)
∈
R
2
,
∣
v
∣
=
x
2
+
y
2
|v| = \sqrt{x^2 + y^2}
∣
v
∣
=
x
2
+
y
2
.
5
1
Hide problems
A nice identity concerning an integral and a serie
Prove that
∑
n
=
1
∞
1
n
n
=
∫
0
1
x
−
x
d
x
.
\sum_{n=1}^\infty {1\over n^n} = \int_0^1 x^{-x}\,dx.
n
=
1
∑
∞
n
n
1
=
∫
0
1
x
−
x
d
x
.
1
1
Hide problems
A imc-like matrix problem
Determine the number of possible values for the determinant of
A
A
A
, given that
A
A
A
is a
n
×
n
n\times n
n
×
n
matrix with real entries such that
A
3
−
A
2
−
3
A
+
2
I
=
0
A^3 - A^2 - 3A + 2I = 0
A
3
−
A
2
−
3
A
+
2
I
=
0
, where
I
I
I
is the identity and
0
0
0
is the all-zero matrix.
6
1
Hide problems
Lots of nonsingular matrices
Prove that for any natural numbers
0
≤
i
1
<
i
2
<
⋯
<
i
k
0 \leq i_1 < i_2 < \cdots < i_k
0
≤
i
1
<
i
2
<
⋯
<
i
k
and
0
≤
j
1
<
j
2
<
⋯
<
j
k
0 \leq j_1 < j_2 < \cdots < j_k
0
≤
j
1
<
j
2
<
⋯
<
j
k
, the matrix
A
=
(
a
r
s
)
1
≤
r
,
s
≤
k
A = (a_{rs})_{1\leq r,s\leq k}
A
=
(
a
rs
)
1
≤
r
,
s
≤
k
,
a
r
s
=
(
i
r
+
j
s
i
r
)
=
(
i
r
+
j
s
)
!
i
r
!
j
s
!
a_{rs} = {i_r + j_s\choose i_r} = {(i_r + j_s)!\over i_r!\, j_s!}
a
rs
=
(
i
r
i
r
+
j
s
)
=
i
r
!
j
s
!
(
i
r
+
j
s
)!
(
1
≤
r
,
s
≤
k
1\leq r,s\leq k
1
≤
r
,
s
≤
k
) is nonsingular.
2
1
Hide problems
Sequence
Let
f
f
f
and
g
g
g
be two continuous, distinct functions from
[
0
,
1
]
→
(
0
,
+
∞
)
[0,1] \rightarrow (0,+\infty)
[
0
,
1
]
→
(
0
,
+
∞
)
such that
∫
0
1
f
(
x
)
d
x
=
∫
0
1
g
(
x
)
d
x
\int_{0}^{1}f(x)dx = \int_{0}^{1}g(x)dx
∫
0
1
f
(
x
)
d
x
=
∫
0
1
g
(
x
)
d
x
Let
y
n
=
∫
0
1
f
n
+
1
(
x
)
g
n
(
x
)
d
x
y_n=\int_{0}^{1}{\frac{f^{n+1}(x)}{g^{n}(x)}dx}
y
n
=
∫
0
1
g
n
(
x
)
f
n
+
1
(
x
)
d
x
, for
n
≥
0
n\geq 0
n
≥
0
, natural. Prove that
(
y
n
)
(y_n)
(
y
n
)
is an increasing and divergent sequence.
4
1
Hide problems
Convergence - beautifull
Let
a
n
+
1
=
a
n
+
1
a
n
2005
a_{n+1} = a_n + \frac{1}{{a_n}^{2005}}
a
n
+
1
=
a
n
+
a
n
2005
1
and
a
1
=
1
a_1=1
a
1
=
1
. Show that
∑
n
=
1
∞
1
n
a
n
\sum^{\infty}_{n=1}{\frac{1}{n a_n}}
∑
n
=
1
∞
n
a
n
1
converge.