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Contests
Undergraduate contests
Vojtěch Jarník IMC
1999 VJIMC
1999 VJIMC
Part of
Vojtěch Jarník IMC
Subcontests
(4)
Problem 4
2
Hide problems
x^n+y^n in Z for n in [4], then n=5 holds for complex x,y
Show that the following implication holds for any two complex numbers
x
x
x
and
y
y
y
: if
x
+
y
x+y
x
+
y
,
x
2
+
y
2
x^2+y^2
x
2
+
y
2
,
x
3
+
y
3
x^3+y^3
x
3
+
y
3
,
x
4
+
y
4
∈
Z
x^4+y^4\in\mathbb Z
x
4
+
y
4
∈
Z
, then
x
n
+
y
n
∈
Z
x^n+y^n\in\mathbb Z
x
n
+
y
n
∈
Z
for all natural n.
integral inequality in [0,1]^n
Let
u
1
,
u
2
,
…
,
u
n
∈
C
(
[
0
,
1
]
n
)
u_1,u_2,\ldots,u_n\in C([0,1]^n)
u
1
,
u
2
,
…
,
u
n
∈
C
([
0
,
1
]
n
)
be nonnegative and continuous functions, and let
u
j
u_j
u
j
do not depend on the
j
j
j
-th variable for
j
=
1
,
…
,
n
j=1,\ldots,n
j
=
1
,
…
,
n
. Show that
(
∫
[
0
,
1
]
n
∏
j
=
1
n
u
j
)
n
−
1
≤
∏
j
=
1
n
∫
[
0
,
1
]
n
u
j
n
−
1
.
\left(\int_{[0,1]^n}\prod_{j=1}^nu_j\right)^{n-1}\le\prod_{j=1}^n\int_{[0,1]^n}u_j^{n-1}.
(
∫
[
0
,
1
]
n
j
=
1
∏
n
u
j
)
n
−
1
≤
j
=
1
∏
n
∫
[
0
,
1
]
n
u
j
n
−
1
.
Problem 2
2
Hide problems
11|a^n+b^n -> 11|a,11|b for all a,b, find n
Find all natural numbers
n
≥
1
n\ge1
n
≥
1
such that the implication
(
11
∣
a
n
+
b
n
)
⟹
(
11
∣
a
∧
11
∣
b
)
(11\mid a^n+b^n)\implies(11\mid a\wedge11\mid b)
(
11
∣
a
n
+
b
n
)
⟹
(
11
∣
a
∧
11
∣
b
)
holds for any two natural numbers
a
a
a
and
b
b
b
.
x_(n+1)=avg(x_n,f(x_n)) converges to fixed point of f
Let
a
,
b
∈
R
a,b\in\mathbb R
a
,
b
∈
R
,
a
≤
b
a\le b
a
≤
b
. Assume that
f
:
[
a
,
b
]
→
[
a
,
b
]
f:[a,b]\to[a,b]
f
:
[
a
,
b
]
→
[
a
,
b
]
satisfies
f
(
x
)
−
f
(
y
)
≤
∣
x
−
y
∣
f(x)-f(y)\le|x-y|
f
(
x
)
−
f
(
y
)
≤
∣
x
−
y
∣
for every
x
,
y
∈
[
a
,
b
]
x,y\in[a,b]
x
,
y
∈
[
a
,
b
]
. Choose an
x
1
∈
[
a
,
b
]
x_1\in[a,b]
x
1
∈
[
a
,
b
]
and define
x
n
+
1
=
x
n
+
f
(
x
n
)
2
,
n
=
1
,
2
,
3
,
…
.
x_{n+1}=\frac{x_n+f(x_n)}2,\qquad n=1,2,3,\ldots.
x
n
+
1
=
2
x
n
+
f
(
x
n
)
,
n
=
1
,
2
,
3
,
…
.
Show that
{
x
n
}
n
=
1
∞
\{x_n\}^\infty_{n=1}
{
x
n
}
n
=
1
∞
converges to some fixed point of
f
f
f
.
Problem 3
2
Hide problems
distance independent of n points on ellipsoid
Let
A
1
,
…
,
A
n
A_1,\ldots,A_n
A
1
,
…
,
A
n
be points of an ellipsoid with center
O
O
O
in
R
n
\mathbb R^n
R
n
such that
O
A
i
OA_i
O
A
i
, for
i
=
1
,
…
,
n
i=1,\ldots,n
i
=
1
,
…
,
n
, are mutually orthogonal. Prove that the distance of the point
O
O
O
from the hyperplane
A
1
A
2
…
A
n
A_1A_2\ldots A_n
A
1
A
2
…
A
n
does not depend on the choice of the points
A
1
,
…
,
A
n
A_1,\ldots,A_n
A
1
,
…
,
A
n
.
countable set of balls, finite subsets have disjoint interiors
Suppose that we have a countable set
A
A
A
of balls and a unit cube in
R
3
\mathbb R^3
R
3
. Assume that for every finite subset
B
B
B
of
A
A
A
it is possible to put all balls of
B
B
B
into the cube in such a way that they have disjoint interiors. Show that it is possible to arrange all the balls in the cube so that all of them have pairwise disjoint interiors.
Problem 1
2
Hide problems
infinite product of k/(k+∞)
Find the limit
lim
n
→
∞
(
∏
k
=
1
n
k
k
+
n
)
e
1999
n
−
1
.
\lim_{n\to\infty}\left(\prod_{k=1}^n\frac k{k+n}\right)^{e^{\frac{1999}n}-1}.
n
→
∞
lim
(
k
=
1
∏
n
k
+
n
k
)
e
n
1999
−
1
.
set of k lines has 3 skew or 3 concurrent
Find the minimal
k
k
k
such that every set of
k
k
k
different lines in
R
3
\mathbb R^3
R
3
contains either
3
3
3
mutually parallel lines or
3
3
3
mutually intersecting lines or
3
3
3
mutually skew lines.