MathDB
Problems
Contests
Undergraduate contests
Vojtěch Jarník IMC
2013 VJIMC
2013 VJIMC
Part of
Vojtěch Jarník IMC
Subcontests
(4)
Problem 3
2
Hide problems
Z[x] on set of integers, bounding closure measure of set
Let
S
S
S
be a finite set of integers. Prove that there exists a number
c
c
c
depending on
S
S
S
such that for each non-constant polynomial
f
f
f
with integer coefficients the number of integers
k
k
k
satisfying
f
(
k
)
∈
S
f(k)\in S
f
(
k
)
∈
S
does not exceed
max
(
deg
f
,
c
)
\max(\deg f,c)
max
(
de
g
f
,
c
)
.
Z[x], P(\sqrt[3]5+\sqrt[3]25)=5+\sqrt[3]5
Prove that there is no polynomial
P
P
P
with integer coefficients such that
P
(
5
3
+
25
3
)
=
5
+
5
3
P\left(\sqrt[3]5+\sqrt[3]{25}\right)=5+\sqrt[3]5
P
(
3
5
+
3
25
)
=
5
+
3
5
.
Problem 4
2
Hide problems
a binomial sum in two variables
Let
n
n
n
and
k
k
k
be positive integers. Evaluate the following sum
∑
j
=
0
k
(
k
j
)
2
(
n
+
2
k
−
j
2
k
)
\sum_{j=0}^k\binom kj^2\binom{n+2k-j}{2k}
j
=
0
∑
k
(
j
k
)
2
(
2
k
n
+
2
k
−
j
)
where
(
n
k
)
=
n
!
k
!
(
n
−
k
)
!
\binom nk=\frac{n!}{k!(n-k)!}
(
k
n
)
=
k
!
(
n
−
k
)!
n
!
.
integral of f(t)/(sqrt(x-t))
Let
F
\mathcal F
F
be the set of all continuous functions
f
:
[
0
,
1
]
→
R
f:[0,1]\to\mathbb R
f
:
[
0
,
1
]
→
R
with the property
∣
∫
0
x
f
(
t
)
x
−
t
d
t
∣
≤
1
for all
x
∈
(
0
,
1
]
.
\left|\int^x_0\frac{f(t)}{\sqrt{x-t}}\text dt\right|\le1\enspace\text{for all }x\in(0,1].
∫
0
x
x
−
t
f
(
t
)
d
t
≤
1
for all
x
∈
(
0
,
1
]
.
Compute
sup
f
∈
F
∣
∫
0
1
f
(
x
)
d
x
∣
\sup_{f\in\mathcal F}\left|\int^1_0f(x)\text dx\right|
sup
f
∈
F
∫
0
1
f
(
x
)
d
x
.
Problem 2
2
Hide problems
matrix, a_(ij)=b_(ij)+1
Let
A
=
(
a
i
j
)
A=(a_{ij})
A
=
(
a
ij
)
and
B
=
(
b
i
j
)
B=(b_{ij})
B
=
(
b
ij
)
be two real
10
×
10
10\times10
10
×
10
matrices such that
a
i
j
=
b
i
j
+
1
a_{ij}=b_{ij}+1
a
ij
=
b
ij
+
1
for all
i
,
j
i,j
i
,
j
and
A
3
=
0
A^3=0
A
3
=
0
. Prove that
det
B
=
0
\det B=0
det
B
=
0
.
diagonals of n-hypercube, # of intersection points
An
n
n
n
-dimensional cube is given. Consider all the segments connecting any two different vertices of the cube. How many distinct intersection points do these segments have (excluding the vertices)?
Problem 1
2
Hide problems
f(x)f'(x)≥cos, f(∞)=undef. if f is bounded
Let
f
:
[
0
,
∞
)
→
R
f:[0,\infty)\to\mathbb R
f
:
[
0
,
∞
)
→
R
be a differentiable function with
∣
f
(
x
)
∣
≤
M
|f(x)|\le M
∣
f
(
x
)
∣
≤
M
and
f
(
x
)
f
′
(
x
)
≥
cos
x
f(x)f'(x)\ge\cos x
f
(
x
)
f
′
(
x
)
≥
cos
x
for
x
∈
[
0
,
∞
)
x\in[0,\infty)
x
∈
[
0
,
∞
)
, where
M
>
0
M>0
M
>
0
. Prove that
f
(
x
)
f(x)
f
(
x
)
does not have a limit as
x
→
∞
x\to\infty
x
→
∞
.
Σ first n primes, gaps
Let
S
n
S_n
S
n
denote the sum of the first
n
n
n
prime numbers. Prove that for any
n
n
n
there exists the square of an integer between
S
n
S_n
S
n
and
S
n
+
1
S_{n+1}
S
n
+
1
.