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Contests
Undergraduate contests
Vojtěch Jarník IMC
1998 VJIMC
1998 VJIMC
Part of
Vojtěch Jarník IMC
Subcontests
(5)
Problem 1
2
Hide problems
no primes in arithmetic seq of fixed length
Let
a
a
a
and
d
d
d
be two positive integers. Prove that there exists a constant
K
K
K
such that every set of
K
K
K
consecutive elements of the arithmetic progression
{
a
+
n
d
}
n
=
1
∞
\{a+nd\}_{n=1}^\infty
{
a
+
n
d
}
n
=
1
∞
contains at least one number which is not prime.
eigenspace complement is invariant under linear operator
Let
H
H
H
be a complex Hilbert space. Let
T
:
H
→
H
T:H\to H
T
:
H
→
H
be a bounded linear operator such that
∣
(
T
x
,
x
)
∣
≤
∥
x
∥
2
|(Tx,x)|\le\lVert x\rVert^2
∣
(
T
x
,
x
)
∣
≤
∥
x
∥
2
for each
x
∈
H
x\in H
x
∈
H
. Assume that
μ
∈
C
\mu\in\mathbb C
μ
∈
C
,
∣
μ
∣
=
1
|\mu|=1
∣
μ
∣
=
1
, is an eigenvalue with the corresponding eigenspace
E
=
{
ϕ
∈
H
:
T
ϕ
=
μ
ϕ
}
E=\{\phi\in H:T\phi=\mu\phi\}
E
=
{
ϕ
∈
H
:
Tϕ
=
μ
ϕ
}
. Prove that the orthogonal complement
E
⊥
=
{
x
∈
H
:
∀
ϕ
∈
E
:
(
x
,
ϕ
)
=
0
}
E^\perp=\{x\in H:\forall\phi\in E:(x,\phi)=0\}
E
⊥
=
{
x
∈
H
:
∀
ϕ
∈
E
:
(
x
,
ϕ
)
=
0
}
of
E
E
E
is
T
T
T
-invariant, i.e.,
T
(
E
⊥
)
⊆
E
⊥
T(E^\perp)\subseteq E^\perp
T
(
E
⊥
)
⊆
E
⊥
.
Problem 4-M
2
Hide problems
sum of sqrt(1-k^2/n^2) inequality
Prove the inequality
n
π
4
−
1
8
n
≤
1
2
+
∑
k
=
1
n
−
1
1
−
k
2
n
2
≤
n
π
4
\frac{n\pi}4-\frac1{\sqrt{8n}}\le\frac12+\sum_{k=1}^{n-1}\sqrt{1-\frac{k^2}{n^2}}\le\frac{n\pi}4
4
nπ
−
8
n
1
≤
2
1
+
k
=
1
∑
n
−
1
1
−
n
2
k
2
≤
4
nπ
for every integer
n
≥
2
n\ge2
n
≥
2
.
Jensen with weaker condition?
A function
f
:
R
→
R
f:\mathbb R\to\mathbb R
f
:
R
→
R
has the property that for every
x
,
y
∈
R
x,y\in\mathbb R
x
,
y
∈
R
there exists a real number
t
t
t
(depending on
x
x
x
and
y
y
y
) such that
0
<
t
<
1
0<t<1
0
<
t
<
1
and
f
(
t
x
+
(
1
−
t
)
y
)
=
t
f
(
x
)
+
(
1
−
t
)
f
(
y
)
.
f(tx+(1-t)y)=tf(x)+(1-t)f(y).
f
(
t
x
+
(
1
−
t
)
y
)
=
t
f
(
x
)
+
(
1
−
t
)
f
(
y
)
.
Does it imply that
f
(
x
+
y
2
)
=
f
(
x
)
+
f
(
y
)
2
f\left(\frac{x+y}2\right)=\frac{f(x)+f(y)}2
f
(
2
x
+
y
)
=
2
f
(
x
)
+
f
(
y
)
for every
x
,
y
∈
R
x,y\in\mathbb R
x
,
y
∈
R
?
Problem 3
2
Hide problems
functions converge ptwise to 0, not uniformly convergent
Give an example of a sequence of continuous functions on
R
\mathbb R
R
converging pointwise to
0
0
0
which is not uniformly convergent on any nonempty open set.
complex roots satisfy |z|>1
Show that all complex roots of the polynomial
P
(
z
)
=
a
0
z
n
+
a
1
z
n
−
1
+
…
+
a
n
−
1
z
+
a
n
P(z)=a_0z^n+a_1z^{n-1}+\ldots+a_{n-1}z+a_n
P
(
z
)
=
a
0
z
n
+
a
1
z
n
−
1
+
…
+
a
n
−
1
z
+
a
n
, where
0
<
a
0
<
…
<
a
n
0<a_0<\ldots<a_n
0
<
a
0
<
…
<
a
n
, satisfy
∣
z
∣
>
1
|z|>1
∣
z
∣
>
1
.
Problem 2
2
Hide problems
e-like limit as n-> infinity
Find the limit
lim
n
→
∞
(
(
1
+
1
n
)
n
e
)
n
.
\lim_{n\to\infty}\left(\frac{\left(1+\frac1n\right)^n}e\right)^n.
n
→
∞
lim
(
e
(
1
+
n
1
)
n
)
n
.
palindrome in arithmetic sequence exists?
Decide whether there is a member in the arithmetic sequence
{
a
n
}
n
=
1
∞
\{a_n\}_{n=1}^\infty
{
a
n
}
n
=
1
∞
whose first member is
a
1
=
1998
a_1=1998
a
1
=
1998
and the common difference
d
=
131
d=131
d
=
131
which is a palindrome (palindrome is a number such that its decimal expansion is symmetric, e.g.,
7
7
7
,
33
33
33
,
433334
433334
433334
,
2135312
2135312
2135312
and so on).
Problem 4-I
2
Hide problems
quine in Pascal (VJIMC 1998 1.4-I)
Prove that there exists a program in standard Pascal which prints out its own ASCII code. No disk operations are permitted.
algorithm determining truth of statement (VJIMC 1998 2.4-I)
Let us consider a first-order language
L
L
L
with a ternary predicate
Plus
\operatorname{Plus}
Plus
. Hence (well-formed) formulas of
L
L
L
are built of symbols for variables, logical connectives, quantifiers, brackets, and the predicate symbol
Plus
\operatorname{Plus}
Plus
.
(
∃
x
1
)
(
∀
x
2
)
:
Plus
(
x
2
,
x
1
,
x
2
)
∧
(
∀
x
3
)
:
¬
Plus
(
x
1
,
x
3
,
x
3
)
(\exists x_1)(\forall x_2):\operatorname{Plus}(x_2,x_1,x_2)\wedge(\forall x_3):\neg\operatorname{Plus}(x_1,x_3,x_3)
(
∃
x
1
)
(
∀
x
2
)
:
Plus
(
x
2
,
x
1
,
x
2
)
∧
(
∀
x
3
)
:
¬
Plus
(
x
1
,
x
3
,
x
3
)
is an example of such a formula. Recall that a formula is closed iff each variable symbol occurs within the scope of a quantifier. Show that there exists an algorithm which decides whether or not a given closed formula of
L
L
L
is true for the set
N
\mathbb N
N
of natural numbers (
{
0
,
1
,
2
,
…
}
\{0,1,2,\ldots\}
{
0
,
1
,
2
,
…
}
) where
Plus
(
x
,
y
,
z
)
\operatorname{Plus}(x,y,z)
Plus
(
x
,
y
,
z
)
is interpreted as
x
+
y
=
z
x+y=z
x
+
y
=
z
.