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Contests
Undergraduate contests
Vojtěch Jarník IMC
2022 VJIMC
2022 VJIMC
Part of
Vojtěch Jarník IMC
Subcontests
(4)
3
2
Hide problems
limit sum integral function
Let
f
:
[
0
,
1
]
→
R
f:[0,1]\to\mathbb R
f
:
[
0
,
1
]
→
R
be a given continuous function. Find the limit
lim
n
→
∞
(
n
+
1
)
∑
k
=
0
n
∫
0
1
x
k
(
1
−
x
)
n
−
k
f
(
x
)
d
x
.
\lim_{n\to\infty}(n+1)\sum_{k=0}^n\int^1_0x^k(1-x)^{n-k}f(x)dx.
n
→
∞
lim
(
n
+
1
)
k
=
0
∑
n
∫
0
1
x
k
(
1
−
x
)
n
−
k
f
(
x
)
d
x
.
variance inequality
Let
x
1
,
…
,
x
n
x_1,\ldots,x_n
x
1
,
…
,
x
n
be given real numbers with
0
<
m
≤
x
i
≤
M
0<m\le x_i\le M
0
<
m
≤
x
i
≤
M
for each
i
∈
{
1
,
…
,
n
}
i\in\{1,\ldots,n\}
i
∈
{
1
,
…
,
n
}
. Let
X
X
X
be the discrete random variable uniformly distributed on
{
x
1
,
…
,
x
n
}
\{x_1,\ldots,x_n\}
{
x
1
,
…
,
x
n
}
. The mean
μ
\mu
μ
and the variance
σ
2
\sigma^2
σ
2
of
X
X
X
are defined as
μ
(
X
)
=
x
1
+
…
+
x
n
n
and
σ
2
(
X
)
=
(
x
1
−
μ
(
X
)
)
2
+
…
+
(
x
n
−
μ
(
X
)
)
2
n
.
\mu(X)=\frac{x_1+\ldots+x_n}n\text{ and }\sigma^2(X)=\frac{(x_1-\mu(X))^2+\ldots+(x_n-\mu(X))^2}n.
μ
(
X
)
=
n
x
1
+
…
+
x
n
and
σ
2
(
X
)
=
n
(
x
1
−
μ
(
X
)
)
2
+
…
+
(
x
n
−
μ
(
X
)
)
2
.
By
X
2
X^2
X
2
denote the discrete random variable uniformly distributed on
{
x
1
2
,
…
,
x
n
2
}
\{x_1^2,\ldots,x_n^2\}
{
x
1
2
,
…
,
x
n
2
}
. Prove that
σ
2
(
X
)
≥
(
m
2
M
2
)
2
σ
2
(
X
2
)
.
\sigma^2(X)\ge\left(\frac m{2M^2}\right)^2\sigma^2(X^2).
σ
2
(
X
)
≥
(
2
M
2
m
)
2
σ
2
(
X
2
)
.
2
2
Hide problems
A^7+A^5+A^3+A=I implies det(A)>0
Let
n
≥
1
n\ge1
n
≥
1
. Assume that
A
A
A
is a real
n
×
n
n\times n
n
×
n
matrix which satisfies the equality
A
7
+
A
5
+
A
3
+
A
−
I
=
0.
A^7+A^5+A^3+A-I=0.
A
7
+
A
5
+
A
3
+
A
−
I
=
0.
Show that
det
(
A
)
>
0
\det(A)>0
det
(
A
)
>
0
.
irreducibility of a polynomial
For any given pair of positive integers
m
>
n
m>n
m
>
n
find all
a
∈
R
a\in\mathbb R
a
∈
R
for which the polynomial
x
m
−
a
x
n
+
1
x^m-ax^n+1
x
m
−
a
x
n
+
1
can be expressed as a quotient of two nonzero polynomials with real nonnegative coefficients.
1
2
Hide problems
polynomial has no real roots
Assume that a real polynomial
P
(
x
)
P(x)
P
(
x
)
has no real roots. Prove that the polynomial
Q
(
x
)
=
P
(
x
)
+
P
′
′
(
x
)
2
!
+
P
(
4
)
(
x
)
4
!
+
…
Q(x)=P(x)+\frac{P''(x)}{2!}+\frac{P^{(4)}(x)}{4!}+\ldots
Q
(
x
)
=
P
(
x
)
+
2
!
P
′′
(
x
)
+
4
!
P
(
4
)
(
x
)
+
…
also has no real roots.
integral inequality, existence of function
Determine whether there exists a differentiable function
f
:
[
0
,
1
]
→
R
f:[0,1]\to\mathbb R
f
:
[
0
,
1
]
→
R
such that
f
(
0
)
=
f
(
1
)
=
1
,
∣
f
′
(
x
)
∣
≤
2
for all
x
∈
[
0
,
1
]
and
∣
∫
0
1
f
(
x
)
d
x
∣
≤
1
2
.
f(0)=f(1)=1,\qquad|f'(x)|\le2\text{ for all }x\in[0,1]\qquad\text{and}\qquad\left|\int^1_0f(x)dx\right|\le\frac12.
f
(
0
)
=
f
(
1
)
=
1
,
∣
f
′
(
x
)
∣
≤
2
for all
x
∈
[
0
,
1
]
and
∫
0
1
f
(
x
)
d
x
≤
2
1
.
4
2
Hide problems
nim-like game
In a box there are
31
31
31
,
41
41
41
and
59
59
59
stones coloured, respectively, red, green and blue. Three players, having t-shirts of these three colours, play the following game. They sequentially make one of two moves: (I) either remove three stones of one colour from the box, (II) or replace two stones of different colours by two stones of the third colour. The game ends when all the stones in the box have the same colour and the winner is the player whose t-shirt has this colour. Assuming that the players play optimally, is it possible to decide whether the game ends and who will win, depending on who the starting player is?
Additive property of a multiplicative function
Let
g
g
g
be the multiplicative function given by
g
(
p
α
)
=
α
p
α
−
1
,
g(p^{\alpha}) = \alpha p^{\alpha-1},
g
(
p
α
)
=
α
p
α
−
1
,
for all
α
∈
Z
+
\alpha\in\mathbb Z^+
α
∈
Z
+
and primes
p
p
p
. Prove that there exist infinitely many integers
n
n
n
such that
g
(
n
+
1
)
=
g
(
n
)
+
g
(
1
)
.
g(n+1) = g(n) + g(1).
g
(
n
+
1
)
=
g
(
n
)
+
g
(
1
)
.