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Undergraduate contests
Vojtěch Jarník IMC
2003 VJIMC
2003 VJIMC
Part of
Vojtěch Jarník IMC
Subcontests
(4)
Problem 1
2
Hide problems
d(n^2+1) is not monotone for n>n_0, any n_0
Let
d
(
k
)
d(k)
d
(
k
)
denote the number of natural divisors of a natural number
k
k
k
. Prove that for any natural number
n
0
n_0
n
0
the sequence
{
d
(
n
2
+
1
)
}
n
=
n
0
∞
\left\{d(n^2+1)\right\}^\infty_{n=n_0}
{
d
(
n
2
+
1
)
}
n
=
n
0
∞
is not strictly monotone.
A^2002=B^2003=I and AB=BA implies A+B+I invertible
Two real square matrices
A
A
A
and
B
B
B
satisfy the conditions
A
2002
=
B
2003
=
I
A^{2002}=B^{2003}=I
A
2002
=
B
2003
=
I
and
A
B
=
B
A
AB=BA
A
B
=
B
A
. Prove that
A
+
B
+
I
A+B+I
A
+
B
+
I
is invertible. (The symbol
I
I
I
denotes the identity matrix.)
Problem 4
2
Hide problems
eigenvalues of sum of hermitian matrices (2x2)
Let
A
A
A
and
B
B
B
be complex Hermitian
2
×
2
2\times2
2
×
2
matrices having the pairs of eigenvalues
(
α
1
,
α
2
)
(\alpha_1,\alpha_2)
(
α
1
,
α
2
)
and
(
β
1
,
β
2
)
(\beta_1,\beta_2)
(
β
1
,
β
2
)
, respectively. Determine all possible pairs of eigenvalues
(
γ
1
,
γ
2
)
(\gamma_1,\gamma_2)
(
γ
1
,
γ
2
)
of the matrix
C
=
A
+
B
C=A+B
C
=
A
+
B
. (We recall that a matrix
A
=
(
a
i
j
)
A=(a_{ij})
A
=
(
a
ij
)
is Hermitian if and only if
a
i
j
=
a
j
i
‾
a_{ij}=\overline{a_{ji}}
a
ij
=
a
ji
for all
i
i
i
and
j
j
j
.)
double integral inequality in two functions
Let
f
,
g
:
[
0
,
1
]
→
(
0
,
+
∞
)
f,g:[0,1]\to(0,+\infty)
f
,
g
:
[
0
,
1
]
→
(
0
,
+
∞
)
be two continuous functions such that
f
f
f
and
g
f
\frac gf
f
g
are increasing. Prove that
∫
0
1
∫
0
x
f
(
t
)
d
t
∫
0
x
g
(
t
)
d
t
d
x
≤
2
∫
0
1
f
(
t
)
g
(
t
)
d
t
.
\int^1_0\frac{\int^x_0f(t)\text dt}{\int^x_0g(t)\text dt}\text dx\le2\int^1_0\frac{f(t)}{g(t)}\text dt.
∫
0
1
∫
0
x
g
(
t
)
d
t
∫
0
x
f
(
t
)
d
t
d
x
≤
2
∫
0
1
g
(
t
)
f
(
t
)
d
t
.
Problem 3
2
Hide problems
complicated limit of sequence, a_(n+2)=a_(n+1)+a_n/2^n
Let
{
a
n
}
n
=
0
∞
\{a_n\}^\infty_{n=0}
{
a
n
}
n
=
0
∞
be the sequence of real numbers satisfying
a
0
=
0
a_0=0
a
0
=
0
,
a
1
=
1
a_1=1
a
1
=
1
and
a
n
+
2
=
a
n
+
1
+
a
n
2
n
a_{n+2}=a_{n+1}+\frac{a_n}{2^n}
a
n
+
2
=
a
n
+
1
+
2
n
a
n
for every
n
≥
0
n\ge0
n
≥
0
. Prove that
lim
n
→
∞
a
n
=
1
+
∑
n
=
1
∞
1
2
n
(
n
−
1
)
2
∏
k
=
1
n
(
2
k
−
1
)
.
\lim_{n\to\infty}a_n=1+\sum_{n=1}^\infty\frac1{2^{\frac{n(n-1)}2}\displaystyle\prod_{k=1}^n(2^k-1)}.
n
→
∞
lim
a
n
=
1
+
n
=
1
∑
∞
2
2
n
(
n
−
1
)
k
=
1
∏
n
(
2
k
−
1
)
1
.
formalization of Ramanujan's result
Find the limit
lim
n
→
∞
1
+
2
1
+
3
…
+
(
n
−
1
)
1
+
n
.
\lim_{n\to\infty}\sqrt{1+2\sqrt{1+3\sqrt{\ldots+(n-1)\sqrt{1+n}}}}.
n
→
∞
lim
1
+
2
1
+
3
…
+
(
n
−
1
)
1
+
n
.
Problem 2
2
Hide problems
sum of rows and columns of matrix, sign
Let
A
=
(
a
i
j
)
A=(a_{ij})
A
=
(
a
ij
)
be an
m
×
n
m\times n
m
×
n
real matrix with at least one non-zero element. For each
i
∈
{
1
,
…
,
m
}
i\in\{1,\ldots,m\}
i
∈
{
1
,
…
,
m
}
, let
R
i
=
∑
j
=
1
n
a
i
j
R_i=\sum_{j=1}^na_{ij}
R
i
=
∑
j
=
1
n
a
ij
be the sum of the
i
i
i
-th row of the matrix
A
A
A
, and for each
j
∈
{
1
,
…
,
n
}
j\in\{1,\ldots,n\}
j
∈
{
1
,
…
,
n
}
, let
C
j
=
∑
i
=
1
m
a
i
j
C_j =\sum_{i=1}^ma_{ij}
C
j
=
∑
i
=
1
m
a
ij
be the sum of the
j
j
j
-th column of the matrix
A
A
A
. Prove that there exist indices
k
∈
{
1
,
…
,
m
}
k\in\{1,\ldots,m\}
k
∈
{
1
,
…
,
m
}
and
l
∈
{
1
,
…
,
n
}
l\in\{1,\ldots,n\}
l
∈
{
1
,
…
,
n
}
such that
a
k
l
>
0
,
R
k
≥
0
,
C
l
≥
0
,
a_{kl}>0,\qquad R_k\ge0,\qquad C_l\ge0,
a
k
l
>
0
,
R
k
≥
0
,
C
l
≥
0
,
or
a
k
l
<
0
,
R
k
≤
0
,
C
l
≤
0.
a_{kl}<0,\qquad R_k\le0,\qquad C_l\le0.
a
k
l
<
0
,
R
k
≤
0
,
C
l
≤
0.
Disks in the Euclidean plane
Let
{
D
1
,
D
2
,
.
.
.
,
D
n
}
\{D_1, D_2, ..., D_n \}
{
D
1
,
D
2
,
...
,
D
n
}
be a set of disks in the Euclidean plane. Let
a
i
,
j
=
S
(
D
i
∩
D
j
)
a_ {i, j} = S (D_i \cap D_j)
a
i
,
j
=
S
(
D
i
∩
D
j
)
be the area of
D
i
∩
D
j
D_i \cap D_j
D
i
∩
D
j
. Prove that
∑
i
=
1
n
∑
j
=
1
n
a
i
,
j
x
i
x
j
≥
0
\sum_ {i = 1} ^ n \sum_ {j = 1} ^ n a_ {i, j} x_ix_j \geq 0
i
=
1
∑
n
j
=
1
∑
n
a
i
,
j
x
i
x
j
≥
0
for any real numbers
x
1
,
x
2
,
.
.
.
,
x
n
x_1, x_2, ..., x_n
x
1
,
x
2
,
...
,
x
n
.