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Vojtěch Jarník IMC
2011 VJIMC
2011 VJIMC
Part of
Vojtěch Jarník IMC
Subcontests
(4)
Problem 3
2
Hide problems
infinite sums equal
Prove that
∑
k
=
0
∞
x
k
1
+
x
2
k
+
2
(
1
−
x
2
k
+
2
)
2
=
∑
k
=
0
∞
(
−
1
)
k
x
k
(
1
−
x
k
+
1
)
2
\sum_{k=0}^\infty x^k\frac{1+x^{2k+2}}{(1-x^{2k+2})^2}=\sum_{k=0}^\infty(-1)^k\frac{x^k}{(1-x^{k+1})^2}
k
=
0
∑
∞
x
k
(
1
−
x
2
k
+
2
)
2
1
+
x
2
k
+
2
=
k
=
0
∑
∞
(
−
1
)
k
(
1
−
x
k
+
1
)
2
x
k
for all
x
∈
(
−
1
,
1
)
x\in(-1,1)
x
∈
(
−
1
,
1
)
.
complex rational function
Let
p
p
p
and
q
q
q
be complex polynomials with
deg
p
>
deg
q
\deg p>\deg q
de
g
p
>
de
g
q
and let
f
(
z
)
=
p
(
z
)
q
(
z
)
f(z)=\frac{p(z)}{q(z)}
f
(
z
)
=
q
(
z
)
p
(
z
)
. Suppose that all roots of
p
p
p
lie inside the unit circle
∣
z
∣
=
1
|z|=1
∣
z
∣
=
1
and that all roots of
q
q
q
lie outside the unit circle. Prove that
max
∣
z
∣
=
1
∣
f
′
(
z
)
∣
>
deg
p
−
deg
q
2
max
∣
z
∣
=
1
∣
f
(
z
)
∣
.
\max_{|z|=1}|f'(z)|>\frac{\deg p-\deg q}2\max_{|z|=1}|f(z)|.
∣
z
∣
=
1
max
∣
f
′
(
z
)
∣
>
2
de
g
p
−
de
g
q
∣
z
∣
=
1
max
∣
f
(
z
)
∣.
Problem 2
2
Hide problems
convergence of a_(n+1)/a_n
Let
(
a
n
)
n
=
1
∞
(a_n)^\infty_{n=1}
(
a
n
)
n
=
1
∞
be an unbounded and strictly increasing sequence of positive reals such that the arithmetic mean of any four consecutive terms
a
n
,
a
n
+
1
,
a
n
+
2
,
a
n
+
3
a_n,a_{n+1},a_{n+2},a_{n+3}
a
n
,
a
n
+
1
,
a
n
+
2
,
a
n
+
3
belongs to the same sequence. Prove that the sequence
a
n
+
1
a
n
\frac{a_{n+1}}{a_n}
a
n
a
n
+
1
converges and find all possible values of its limit.
k-repeated sum
Let
k
k
k
be a positive integer. Compute
∑
n
1
=
1
∞
∑
n
2
=
1
∞
⋯
∑
n
k
=
1
∞
1
n
1
n
2
⋯
n
k
(
n
1
+
n
2
+
…
+
n
k
+
1
)
.
\sum_{n_1=1}^\infty\sum_{n_2=1}^\infty\cdots\sum_{n_k=1}^\infty\frac1{n_1n_2\cdots n_k(n_1+n_2+\ldots+n_k+1)}.
n
1
=
1
∑
∞
n
2
=
1
∑
∞
⋯
n
k
=
1
∑
∞
n
1
n
2
⋯
n
k
(
n
1
+
n
2
+
…
+
n
k
+
1
)
1
.
Problem 1
2
Hide problems
existence given P(1/n)=f(n)
(a) Is there a polynomial
P
(
x
)
P(x)
P
(
x
)
with real coefficients such that
P
(
1
k
)
=
k
+
2
k
P\left(\frac1k\right)=\frac{k+2}k
P
(
k
1
)
=
k
k
+
2
for all positive integers
k
k
k
? (b) Is there a polynomial
P
(
x
)
P(x)
P
(
x
)
with real coefficients such that
P
(
1
k
)
=
1
2
k
+
1
P\left(\frac1k\right)=\frac1{2k+1}
P
(
k
1
)
=
2
k
+
1
1
for all positive integers
k
k
k
?
matrix product is nonzero
Let
n
>
k
n>k
n
>
k
and let
A
1
,
…
,
A
k
A_1,\ldots,A_k
A
1
,
…
,
A
k
be real
n
×
n
n\times n
n
×
n
matrices of rank
n
−
1
n-1
n
−
1
. Prove that
A
1
⋯
A
k
≠
0.
A_1\cdots A_k\ne0.
A
1
⋯
A
k
=
0.
Problem 4
2
Hide problems
syseq in group, 3 eqns/variables
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be elements of finite order in some group. Prove that if
a
−
1
b
a
=
b
2
a^{-1}ba=b^2
a
−
1
ba
=
b
2
,
b
−
2
c
b
2
=
c
2
b^{-2}cb^2=c^2
b
−
2
c
b
2
=
c
2
, and
c
−
3
a
c
3
=
a
2
c^{-3}ac^3=a^2
c
−
3
a
c
3
=
a
2
then
a
=
b
=
c
=
e
a=b=c=e
a
=
b
=
c
=
e
, where
e
e
e
is the unit element.
nice FE, irreducibility map
Find all
Q
\mathbb Q
Q
-linear maps
Φ
:
Q
[
x
]
→
Q
[
x
]
\Phi:\mathbb Q[x]\to\mathbb Q[x]
Φ
:
Q
[
x
]
→
Q
[
x
]
such that for any irreducible polynomial
p
∈
Q
[
x
]
p\in\mathbb Q[x]
p
∈
Q
[
x
]
the polynomial
Φ
(
p
)
\Phi(p)
Φ
(
p
)
is also irreducible.