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Undergraduate contests
Vojtěch Jarník IMC
2017 VJIMC
2017 VJIMC
Part of
Vojtěch Jarník IMC
Subcontests
(4)
4
2
Hide problems
Divergent improper integral
Let
f
:
(
1
,
∞
)
→
R
f:(1,\infty) \to \mathbb{R}
f
:
(
1
,
∞
)
→
R
be a continuously differentiable function satisfying
f
(
x
)
≤
x
2
log
(
x
)
f(x) \le x^2 \log(x)
f
(
x
)
≤
x
2
lo
g
(
x
)
and
f
′
(
x
)
>
0
f'(x)>0
f
′
(
x
)
>
0
for every
x
∈
(
1
,
∞
)
x \in (1,\infty)
x
∈
(
1
,
∞
)
. Prove that
∫
1
∞
1
f
′
(
x
)
d
x
=
∞
.
\int_1^{\infty} \frac{1}{f'(x)} dx=\infty.
∫
1
∞
f
′
(
x
)
1
d
x
=
∞.
Certain amount of numbers t=x^3+y^2 in a given set
A positive integer
t
t
t
is called a Jane's integer if
t
=
x
3
+
y
2
t = x^3+y^2
t
=
x
3
+
y
2
for some positive integers
x
x
x
and
y
y
y
. Prove that for every integer
n
≥
2
n \ge 2
n
≥
2
there exist infinitely many positive integers
m
m
m
such that the set of
n
2
n^2
n
2
consecutive integers
{
m
+
1
,
m
+
2
,
…
,
m
+
n
2
}
\{m+1,m+2,\dotsc,m+n^2\}
{
m
+
1
,
m
+
2
,
…
,
m
+
n
2
}
contains exactly
n
+
1
n + 1
n
+
1
Jane's integers.
3
2
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Convex polyhedron with numbers at its vertices
Let
P
P
P
be a convex polyhedron. Jaroslav writes a non-negative real number to every vertex of
P
P
P
in such a way that the sum of these numbers is
1
1
1
. Afterwards, to every edge he writes the product of the numbers at the two endpoints of that edge. Prove that the sum of the numbers at the edges is at most
3
8
\frac{3}{8}
8
3
.
Cyclic system of equations with many solutions
Let
n
≥
2
n \ge 2
n
≥
2
be an integer. Consider the system of equations \begin{align} x_1+\frac{2}{x_2}=x_2+\frac{2}{x_3}=\dots=x_n+\frac{2}{x_1} \end{align} 1. Prove that
(
1
)
(1)
(
1
)
has infinitely many real solutions
(
x
1
,
…
,
x
n
)
(x_1,\dotsc,x_n)
(
x
1
,
…
,
x
n
)
such that the numbers
x
1
,
…
,
x
n
x_1,\dotsc,x_n
x
1
,
…
,
x
n
are distinct. 2. Prove that every solution of
(
1
)
(1)
(
1
)
, such that the numbers
x
1
,
…
,
x
n
x_1,\dotsc,x_n
x
1
,
…
,
x
n
are not all equal, satisfies
∣
x
1
x
2
⋯
x
n
∣
=
2
n
/
2
\vert x_1x_2\cdots x_n\vert=2^{n/2}
∣
x
1
x
2
⋯
x
n
∣
=
2
n
/2
.
2
2
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Integer sequence with iterated operation
We say that we extend a finite sequence of positive integers
(
a
1
,
…
,
a
n
)
(a_1,\dotsc,a_n)
(
a
1
,
…
,
a
n
)
if we replace it by
(
1
,
2
,
…
,
a
1
−
1
,
a
1
,
1
,
2
,
…
,
a
2
−
1
,
a
2
,
1
,
2
,
…
,
a
3
−
1
,
a
3
,
…
,
1
,
2
,
…
,
a
n
−
1
,
a
n
)
(1,2,\dotsc,a_1-1,a_1,1,2,\dotsc,a_2-1,a_2,1,2,\dotsc,a_3-1,a_3,\dotsc,1,2,\dotsc,a_n-1,a_n)
(
1
,
2
,
…
,
a
1
−
1
,
a
1
,
1
,
2
,
…
,
a
2
−
1
,
a
2
,
1
,
2
,
…
,
a
3
−
1
,
a
3
,
…
,
1
,
2
,
…
,
a
n
−
1
,
a
n
)
i.e., each element
k
k
k
of the original sequence is replaced by
1
,
2
,
…
,
k
1,2,\dotsc,k
1
,
2
,
…
,
k
. Géza takes the sequence
(
1
,
2
,
…
,
9
)
(1,2,\dotsc,9)
(
1
,
2
,
…
,
9
)
and he extends it
2017
2017
2017
times. Then he chooses randomly one element of the resulting sequence. What is the probability that the chosen element is
1
1
1
?
Increasing function implies existence of non-increasing one
Prove or disprove the following statement. If
g
:
(
0
,
1
)
→
(
0
,
1
)
g:(0,1) \to (0,1)
g
:
(
0
,
1
)
→
(
0
,
1
)
is an increasing function and satisfies
g
(
x
)
>
x
g(x) > x
g
(
x
)
>
x
for all
x
∈
(
0
,
1
)
x \in (0,1)
x
∈
(
0
,
1
)
, then there exists a continuous function
f
:
(
0
,
1
)
→
R
f:(0,1) \to \mathbb{R}
f
:
(
0
,
1
)
→
R
satisfying
f
(
x
)
<
f
(
g
(
x
)
)
f(x) < f(g(x))
f
(
x
)
<
f
(
g
(
x
))
for all
x
∈
(
0
,
1
)
x \in (0,1)
x
∈
(
0
,
1
)
, but
f
f
f
is not an increasing function.
1
2
Hide problems
Power Series with special coefficients is rational function
Let
(
a
n
)
n
=
1
∞
(a_n)_{n=1}^{\infty}
(
a
n
)
n
=
1
∞
be a sequence with
a
n
∈
{
0
,
1
}
a_n \in \{0,1\}
a
n
∈
{
0
,
1
}
for every
n
n
n
. Let
F
:
(
−
1
,
1
)
→
R
F:(-1,1) \to \mathbb{R}
F
:
(
−
1
,
1
)
→
R
be defined by
F
(
x
)
=
∑
n
=
1
∞
a
n
x
n
F(x)=\sum_{n=1}^{\infty} a_nx^n
F
(
x
)
=
n
=
1
∑
∞
a
n
x
n
and assume that
F
(
1
2
)
F\left(\frac{1}{2}\right)
F
(
2
1
)
is rational. Show that
F
F
F
is the quotient of two polynomials with integer coefficients.
Functional equation with continuity implies constant
Let
f
:
R
→
R
f:\mathbb{R} \to \mathbb{R}
f
:
R
→
R
be a continuous function satisfying
f
(
x
+
2
y
)
=
2
f
(
x
)
f
(
y
)
f(x+2y)=2f(x)f(y)
f
(
x
+
2
y
)
=
2
f
(
x
)
f
(
y
)
for every
x
,
y
∈
R
x,y \in \mathbb{R}
x
,
y
∈
R
. Prove that
f
f
f
is constant.