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Vojtěch Jarník IMC
2019 VJIMC
2019 VJIMC
Part of
Vojtěch Jarník IMC
Subcontests
(4)
4
2
Hide problems
VJIMC 2019 P4
Determine the largest constant
K
≥
0
K\geq 0
K
≥
0
such that
a
a
(
b
2
+
c
2
)
(
a
a
−
1
)
2
+
b
b
(
c
2
+
a
2
)
(
b
b
−
1
)
2
+
c
c
(
a
2
+
b
2
)
(
c
c
−
1
)
2
≥
K
(
a
+
b
+
c
a
b
c
−
1
)
2
\frac{a^a(b^2+c^2)}{(a^a-1)^2}+\frac{b^b(c^2+a^2)}{(b^b-1)^2}+\frac{c^c(a^2+b^2)}{(c^c-1)^2}\geq K\left (\frac{a+b+c}{abc-1}\right)^2
(
a
a
−
1
)
2
a
a
(
b
2
+
c
2
)
+
(
b
b
−
1
)
2
b
b
(
c
2
+
a
2
)
+
(
c
c
−
1
)
2
c
c
(
a
2
+
b
2
)
≥
K
(
ab
c
−
1
a
+
b
+
c
)
2
holds for all positive real numbers
a
,
b
,
c
a,b,c
a
,
b
,
c
such that
a
b
+
b
c
+
c
a
=
a
b
c
ab+bc+ca=abc
ab
+
b
c
+
c
a
=
ab
c
.Proposed by Orif Ibrogimov (Czech Technical University of Prague).
VJIMC 2019 P4 Category II
Let
D
=
{
z
∈
C
:
Im
(
z
)
>
0
,
Re
(
z
)
>
0
}
D=\{ z \in \mathbb{C} : \operatorname{Im}(z) >0 , \operatorname{Re}(z) >0 \}
D
=
{
z
∈
C
:
Im
(
z
)
>
0
,
Re
(
z
)
>
0
}
. Let
n
≥
1
n \geq 1
n
≥
1
and let
a
1
,
a
2
,
…
a
n
∈
D
a_1,a_2,\dots a_n \in D
a
1
,
a
2
,
…
a
n
∈
D
be distinct complex numbers. Define
f
(
z
)
=
z
⋅
∏
j
=
1
n
z
−
a
j
z
−
a
j
‾
f(z)=z \cdot \prod_{j=1}^{n} \frac{z-a_j}{z-\overline{a_j}}
f
(
z
)
=
z
⋅
j
=
1
∏
n
z
−
a
j
z
−
a
j
Prove that
f
′
f'
f
′
has at least one root in
D
D
D
.Proposed by Géza Kós (Lorand Eotvos University, Budapest)
3
2
Hide problems
VJIMC 2019 P3
For an invertible
n
×
n
n\times n
n
×
n
matrix
M
M
M
with integer entries we define a sequence
S
M
=
{
M
i
}
i
=
0
∞
\mathcal{S}_M=\{M_i\}_{i=0}^{\infty}
S
M
=
{
M
i
}
i
=
0
∞
by the recurrence
M
0
=
M
M_0=M
M
0
=
M
,
M
i
+
1
=
(
M
i
T
)
−
1
M
i
M_{i+1}=(M_i^T)^{-1}M_i
M
i
+
1
=
(
M
i
T
)
−
1
M
i
for
i
≥
0
i\geq 0
i
≥
0
.Find the smallest integer
n
≥
2
n\geq 2
n
≥
2
for wich there exists a normal
n
×
n
n\times n
n
×
n
matrix with integer entries such that its sequence
S
M
\mathcal{S}_M
S
M
is not constant and has period
P
=
7
P=7
P
=
7
i.e
M
i
+
7
=
M
i
M_{i+7}=M_i
M
i
+
7
=
M
i
. (
M
T
M^T
M
T
means the transpose of a matrix
M
M
M
. A square matrix is called normal if
M
T
M
=
M
M
T
M^T M=M M^T
M
T
M
=
M
M
T
holds).Proposed by Martin Niepel (Comenius University, Bratislava)..
VJIMC 2019 P3 Category II
Let
p
p
p
be an even non-negative continous function with
∫
R
p
(
x
)
d
x
=
1
\int _{\mathbb{R}} p(x) dx =1
∫
R
p
(
x
)
d
x
=
1
and let
n
n
n
be a positive integer. Let
ξ
1
,
ξ
2
,
ξ
3
…
,
ξ
n
\xi_1,\xi_2,\xi_3 \dots ,\xi_n
ξ
1
,
ξ
2
,
ξ
3
…
,
ξ
n
be independent identically distributed random variables with density function
p
p
p
. Define\begin{align*} X_{0} & = 0 \\ X_{1} & = X_0+ \xi_1 \\ & \vdotswithin{ = }\notag \\ X_{n} & = X_{n-1} + \xi_n \end{align*}Prove that the probability that all random variables
X
1
,
X
2
…
X
n
−
1
X_1,X_2 \dots X_{n-1}
X
1
,
X
2
…
X
n
−
1
lie between
X
0
X_0
X
0
and
X
n
X_n
X
n
is
1
n
\frac{1}{n}
n
1
.Proposed by Fedor Petrov (St.Petersburg State University).
2
2
Hide problems
VJIMC 2019 P2
A triplet of polynomials
u
,
v
,
w
∈
R
[
x
,
y
,
z
]
u,v,w \in \mathbb{R}[x,y,z]
u
,
v
,
w
∈
R
[
x
,
y
,
z
]
is called smart if there exists polynomials
P
,
Q
,
R
∈
R
[
x
,
y
,
z
]
P,Q,R\in \mathbb{R}[x,y,z]
P
,
Q
,
R
∈
R
[
x
,
y
,
z
]
such that the following polynomial identity holds :
u
2019
P
+
v
2019
Q
+
w
2019
R
=
2019
u^{2019}P +v^{2019 }Q+w^{2019} R=2019
u
2019
P
+
v
2019
Q
+
w
2019
R
=
2019
a) Is the triplet of polynomials
u
=
x
+
2
y
+
3
,
v
=
y
+
z
+
2
,
w
=
x
+
y
+
z
u=x+2y+3 , \;\;\;\; v=y+z+2, \;\;\;\;\;w=x+y+z
u
=
x
+
2
y
+
3
,
v
=
y
+
z
+
2
,
w
=
x
+
y
+
z
smart? b) Is the triplet of polynomials
u
=
x
+
2
y
+
3
,
v
=
y
+
z
+
2
,
w
=
x
+
y
−
z
u=x+2y+3 , \;\;\;\; v=y+z+2, \;\;\;\;\;w=x+y-z
u
=
x
+
2
y
+
3
,
v
=
y
+
z
+
2
,
w
=
x
+
y
−
z
smart?Proposed by Arturas Dubickas (Vilnius University).
VJIMC 2019 P2 Category II
Find all twice differentiable functions
f
:
R
→
R
f : \mathbb{R} \to \mathbb{R}
f
:
R
→
R
such that
f
′
′
(
x
)
cos
(
f
(
x
)
)
≥
(
f
′
(
x
)
)
2
sin
(
f
(
x
)
)
f''(x) \cos(f(x))\geq(f'(x))^2 \sin(f(x))
f
′′
(
x
)
cos
(
f
(
x
))
≥
(
f
′
(
x
)
)
2
sin
(
f
(
x
))
for every
x
∈
R
x\in \mathbb{R}
x
∈
R
.Proposed by Orif Ibrogimov (Czech Technical University of Prague), Karim Rakhimov (University of Pisa)
1
2
Hide problems
VJIMC 2019 P1
Let
{
a
n
}
n
=
0
∞
\{a_n \}_{n=0}^{\infty}
{
a
n
}
n
=
0
∞
be a sequence given recrusively such that
a
0
=
1
a_0=1
a
0
=
1
and
a
n
+
1
=
7
a
n
+
45
a
n
2
−
36
2
a_{n+1}=\frac{7a_n+\sqrt{45a_n^2-36}}{2}
a
n
+
1
=
2
7
a
n
+
45
a
n
2
−
36
for
n
≥
0
n\geq 0
n
≥
0
Show that : a)
a
n
a_n
a
n
is a positive integer. b)
a
n
a
n
+
1
−
1
a_n a_{n+1}-1
a
n
a
n
+
1
−
1
is a square of an integer.Proposed by Stefan Gyurki (Matej Bel University, Banska Bystrica).
VJIMC 2019 P1 Category II
a)Is it true that for every non-empty set
A
A
A
and every associative operation
∗
:
A
×
A
→
A
*:A \times A \to A
∗
:
A
×
A
→
A
the conditions
x
∗
x
∗
y
=
y
and
y
∗
x
∗
x
=
y
x*x*y=y \;\;\; \text{and}\; \;\; y*x*x=y
x
∗
x
∗
y
=
y
and
y
∗
x
∗
x
=
y
for every
x
,
y
∈
A
x,y\in A
x
,
y
∈
A
imply commutativity of
∗
*
∗
?b)a)Is it true that for every non-empty set
A
A
A
and every associative operation
∗
:
A
×
A
→
A
*:A \times A \to A
∗
:
A
×
A
→
A
the condition
x
∗
x
∗
y
=
y
x*x*y=y
x
∗
x
∗
y
=
y
for every
x
,
y
∈
A
x,y\in A
x
,
y
∈
A
implies commutativity of
∗
*
∗
?Proposed by Paulius Drungilas, Arturas Dubickas (Vilnius University).