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Arnold's Trivium
1991 Arnold's Trivium
1991 Arnold's Trivium
Part of
Arnold's Trivium
Subcontests
(100)
1
1
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Problem 1
Sketch the graph of the derivative and the graph of the integral of a function given by a free-hand graph.
2
1
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Problem 2
Find the limit
lim
x
→
0
sin
tan
x
−
tan
sin
x
arcsin
arctan
x
−
arctan
arcsin
x
\lim_{x\to0}\frac{\sin \tan x-\tan\sin x}{\arcsin\arctan x-\arctan\arcsin x}
x
→
0
lim
arcsin
arctan
x
−
arctan
arcsin
x
sin
tan
x
−
tan
sin
x
3
1
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Problem 3
Find the critical values and critical points of the mapping
z
↦
z
2
+
2
z
‾
z\mapsto z^2+2\overline{z}
z
↦
z
2
+
2
z
(sketch the answer).
4
1
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Problem 4
Calculate the
100
100
100
th derivative of the function
x
2
+
1
x
3
−
x
\frac{x^2+1}{x^3-x}
x
3
−
x
x
2
+
1
5
1
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Problem 5
Calculate the
100
100
100
th derivative of the function
1
x
2
+
3
x
+
2
\frac{1}{x^2+3x+2}
x
2
+
3
x
+
2
1
at
x
=
0
x=0
x
=
0
with
10
%
10\%
10%
relative error.
6
1
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Problem 6
In the
(
x
,
y
)
(x,y)
(
x
,
y
)
-plane sketch the curve given parametrically by
x
=
2
t
−
4
t
3
x=2t-4t^3
x
=
2
t
−
4
t
3
,
y
=
t
2
−
3
t
4
y=t^2-3t^4
y
=
t
2
−
3
t
4
.
7
1
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Problem 7
How many normals to an ellipse can be drawn from a given point in plane? Find the region in which the number of normals is maximal.
8
1
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Problem 8
How many maxima, minima, and saddle points does the function
x
4
+
y
4
+
z
4
+
u
4
+
v
4
x^4 + y^4 + z^4 + u^4 + v^4
x
4
+
y
4
+
z
4
+
u
4
+
v
4
have on the surface
x
+
.
.
.
+
v
=
0
x+ ... +v = 0
x
+
...
+
v
=
0
,
x
2
+
.
.
.
+
v
2
=
1
x^2+ ... + v^2 = 1
x
2
+
...
+
v
2
=
1
,
x
3
+
.
.
.
+
v
3
=
C
x^3 + ... + v^3 = C
x
3
+
...
+
v
3
=
C
?
9
1
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Problem 9
Does every positive polynomial in two real variables attain its lower bound in the plane?
10
1
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Problem 10
Investigate the asymptotic behaviour of the solutions
y
y
y
of the equation
x
5
+
x
2
y
2
=
y
6
x^5 + x^2y^2 = y^6
x
5
+
x
2
y
2
=
y
6
that tend to zero as
x
→
0
x\to0
x
→
0
.
11
1
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Problem 11
Investigate the convergence of the integral
∫
−
∞
∞
∫
−
∞
∞
d
x
d
y
1
+
x
4
y
4
\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\frac{dxdy}{1+x^4y^4}
∫
−
∞
∞
∫
−
∞
∞
1
+
x
4
y
4
d
x
d
y
12
1
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Problem 12
Find the flux of the vector field
r
→
/
r
3
\overrightarrow{r}/r^3
r
/
r
3
through the surface
(
x
−
1
)
2
+
y
2
+
z
2
=
2
(x-1)^2+y^2+z^2=2
(
x
−
1
)
2
+
y
2
+
z
2
=
2
13
1
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Problem 13
Calculate with
5
%
5\%
5%
relative error
∫
1
10
x
x
d
x
\int_1^{10}x^xdx
∫
1
10
x
x
d
x
14
1
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Problem 14
Calculate with at most
10
%
10\%
10%
relative error
∫
−
∞
∞
(
x
4
+
4
x
+
4
)
−
100
d
x
\int_{-\infty}^{\infty}(x^4+4x+4)^{-100}dx
∫
−
∞
∞
(
x
4
+
4
x
+
4
)
−
100
d
x
15
1
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Problem 15
Calculate with
10
%
10\%
10%
relative error
∫
−
∞
∞
cos
(
100
(
x
4
−
x
)
)
d
x
\int_{-\infty}^{\infty}\cos(100(x^4-x))dx
∫
−
∞
∞
cos
(
100
(
x
4
−
x
))
d
x
16
1
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Problem 16
What fraction of a
5
5
5
-dimensional cube is the volume of the inscribed sphere? What fraction is it of a
10
10
10
-dimensional cube?
17
1
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Problem 17
Find the distance of the centre of gravity of a uniform
100
100
100
-dimensional solid hemisphere of radius
1
1
1
from the centre of the sphere with
10
%
10\%
10%
relative error.
18
1
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Problem 18
Calculate
∫
⋯
∫
exp
(
−
∑
1
≤
i
≤
j
≤
n
x
i
x
j
)
d
x
1
⋯
d
x
n
\int\cdots\int \exp\left(-\sum_{1\le i\le j\le n}x_ix_j\right)dx_1\cdots dx_n
∫
⋯
∫
exp
(
−
1
≤
i
≤
j
≤
n
∑
x
i
x
j
)
d
x
1
⋯
d
x
n
19
1
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Problem 19
Investigate the path of a light ray in a plane medium with refractive index
n
(
y
)
=
y
4
−
y
2
+
1
n(y)=y^4-y^2+1
n
(
y
)
=
y
4
−
y
2
+
1
using Snell's law
n
(
y
)
sin
α
=
const
n(y)\sin\alpha = \text{const}
n
(
y
)
sin
α
=
const
, where
α
\alpha
α
is the angle made by the ray with the
y
y
y
-axis.
20
1
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Problem 20
Find the derivative of the solution of the equation
x
¨
=
x
+
A
x
˙
2
\ddot{x} =x + A\dot{x}^2
x
¨
=
x
+
A
x
˙
2
, with initial conditions
x
(
0
)
=
1
x(0) = 1
x
(
0
)
=
1
,
x
˙
(
0
)
=
0
\dot{x}(0) = 0
x
˙
(
0
)
=
0
, with respect to the parameter
A
A
A
for
A
=
0
A = 0
A
=
0
.
21
1
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Problem 21
Find the derivative of the solution of the equation
x
¨
=
x
˙
2
+
x
3
\ddot{x} = \dot{x}^2 + x^3
x
¨
=
x
˙
2
+
x
3
with initial condition
x
(
0
)
=
0
x(0) = 0
x
(
0
)
=
0
,
x
˙
(
0
)
=
A
\dot{x}(0) = A
x
˙
(
0
)
=
A
with respect to
A
A
A
for
A
=
0
A = 0
A
=
0
.
22
1
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Problem 22
Investigate the boundary of the domain of stability (
max
Re
λ
j
<
0
\max \text{Re }\lambda_j < 0
max
Re
λ
j
<
0
) in the space of coefficients of the equation \dddot{x} + a\ddot{x} + b\dot{x} + cx = 0.
23
1
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Problem 23
Solve the quasi-homogeneous equation
d
y
d
x
=
x
+
x
3
y
\frac{dy}{dx}=x+\frac{x^3}{y}
d
x
d
y
=
x
+
y
x
3
24
1
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Problem 24
Solve the quasi-homogeneous equation
x
¨
=
x
5
+
x
2
x
˙
\ddot{x}=x^5+x^2\dot{x}
x
¨
=
x
5
+
x
2
x
˙
25
1
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Problem 25
Can an asymptotically stable equilibrium position become unstable in the Lyapunov sense under linearization?
26
1
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Problem 26
Investigate the behaviour as
t
→
+
∞
t\to+\infty
t
→
+
∞
of solutions of the systems
{
x
˙
=
y
y
˙
=
2
sin
y
−
y
−
x
\begin{cases} \dot{x}=y\\ \dot{y}=2\sin y-y-x\end{cases}
{
x
˙
=
y
y
˙
=
2
sin
y
−
y
−
x
{
x
˙
=
y
y
˙
=
2
x
−
x
3
−
x
2
−
ϵ
y
\begin{cases} \dot{x}=y\\ \dot{y}=2x-x^{3}-x^{2}-\epsilon y\end{cases}
{
x
˙
=
y
y
˙
=
2
x
−
x
3
−
x
2
−
ϵy
where
ϵ
≪
1
\epsilon\ll 1
ϵ
≪
1
.
27
1
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Problem 27
Sketch the images of the solutions of the equation
x
¨
=
F
(
x
)
−
k
x
˙
,
F
=
−
d
U
/
d
x
\ddot{x}=F(x)-k\dot{x},\; F=-dU/dx
x
¨
=
F
(
x
)
−
k
x
˙
,
F
=
−
d
U
/
d
x
in the
(
x
,
E
)
(x,E)
(
x
,
E
)
-plane, where
E
=
x
˙
2
/
2
+
U
(
x
)
E=\dot{x}^2/2+U(x)
E
=
x
˙
2
/2
+
U
(
x
)
, near non-degenerate critical points of the potential
U
U
U
.
28
1
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Problem 28
Sketch the phase portrait and investigate its variation under variation of the small complex parameter
ϵ
\epsilon
ϵ
:
z
˙
=
ϵ
z
−
(
1
+
i
)
z
∣
z
∣
2
+
z
‾
4
\dot{z}=\epsilon z-(1+i)z|z|^2+\overline{z}^4
z
˙
=
ϵz
−
(
1
+
i
)
z
∣
z
∣
2
+
z
4
29
1
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Problem 29
A charge moves with velocity
1
1
1
in a plane under the action of a strong magnetic field
B
(
x
,
y
)
B(x, y)
B
(
x
,
y
)
perpendicular to the plane. To which side will the centre of the Larmor neighbourhood drift? Calculate the velocity of this drift (to a first approximation). [Mathematically, this concerns the curves of curvature
N
B
NB
NB
as
N
→
+
∞
N\to + \infty
N
→
+
∞
.]
30
1
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Problem 30
Find the sum of the indexes of the singular points other than zero of the vector field
z
z
‾
2
+
z
4
+
2
z
‾
4
z\overline{z}^2+z^4+2\overline{z}^4
z
z
2
+
z
4
+
2
z
4
31
1
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Problem 31
Find the index of the singular point
0
0
0
of the vector field with components
(
x
4
+
y
4
+
z
4
,
x
3
y
−
x
y
3
,
x
y
z
2
)
(x^4+y^4+z^4,x^3y-xy^3,xyz^2)
(
x
4
+
y
4
+
z
4
,
x
3
y
−
x
y
3
,
x
y
z
2
)
32
1
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Problem 32
Find the index of the singular point
0
0
0
of the vector field
(
x
y
+
y
z
+
x
z
)
(xy+yz+xz)
(
x
y
+
yz
+
x
z
)
33
1
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Problem 33
Find the linking coefficient of the phase trajectories of the equation of small oscillations
x
¨
=
−
4
x
\ddot{x}=-4x
x
¨
=
−
4
x
,
y
¨
=
−
9
y
\ddot{y}=-9y
y
¨
=
−
9
y
on a level surface of the total energy.
34
1
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Problem 34
Investigate the singular points on the curve
y
=
x
3
y=x^3
y
=
x
3
in the projective plane.
35
1
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Problem 35
Sketch the geodesics on the surface
(
x
2
+
y
2
−
2
)
2
+
z
2
=
1
(x^2+y^2-2)^2+z^2=1
(
x
2
+
y
2
−
2
)
2
+
z
2
=
1
36
1
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Problem 36
Sketch the evolvent of the cubic parabola
y
=
x
3
y=x^3
y
=
x
3
(the evolvent is the locus of the points
r
→
(
s
)
+
(
c
−
s
)
r
→
˙
(
s
)
\overrightarrow{r}(s)+(c-s)\dot{\overrightarrow{r}}(s)
r
(
s
)
+
(
c
−
s
)
r
˙
(
s
)
, where
s
s
s
is the arc-length of the curve
r
→
(
s
)
\overrightarrow{r}(s)
r
(
s
)
and
c
c
c
is a constant).
37
1
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Problem 37
Prove that in Euclidean space the surfaces
(
(
A
−
λ
E
)
−
1
x
,
x
)
=
1
((A-\lambda E)^{-1}x,x)=1
((
A
−
λ
E
)
−
1
x
,
x
)
=
1
passing through the point
x
x
x
and corresponding to different values of
λ
\lambda
λ
are pairwise orthogonal (
A
A
A
is a symmetric operator without multiple eigenvalues).
38
1
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Problem 38
Calculate the integral of the Gaussian curvature of the surface
z
4
+
(
x
2
+
y
2
−
1
)
(
2
x
2
+
3
y
2
−
1
)
=
0
z^4+(x^2+y^2-1)(2x^2+3y^2-1)=0
z
4
+
(
x
2
+
y
2
−
1
)
(
2
x
2
+
3
y
2
−
1
)
=
0
39
1
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Problem 39
Calculate the Gauss integral
∮
(
d
A
→
,
d
B
→
,
A
→
−
B
→
)
∣
A
→
−
B
→
∣
3
\oint\frac{(d\overrightarrow{A},d\overrightarrow{B},\overrightarrow{A}-\overrightarrow{B})}{|\overrightarrow{A}-\overrightarrow{B}|^3}
∮
∣
A
−
B
∣
3
(
d
A
,
d
B
,
A
−
B
)
where
A
→
\overrightarrow{A}
A
runs along the curve
x
=
cos
α
x=\cos\alpha
x
=
cos
α
,
y
=
sin
α
y=\sin\alpha
y
=
sin
α
,
z
=
0
z=0
z
=
0
, and
B
→
\overrightarrow{B}
B
along the curve
x
=
2
cos
2
β
x=2\cos^2\beta
x
=
2
cos
2
β
,
y
=
1
2
sin
β
y=\frac12\sin\beta
y
=
2
1
sin
β
,
z
=
sin
2
β
z=\sin2\beta
z
=
sin
2
β
.Note: that
∮
\oint
∮
was supposed to be oiint (i.e.
∬
\iint
∬
with a circle) but the command does not work on AoPS.
40
1
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Problem 40
Find the parallel displacement of a vector pointing north at Leningrad (latitude
6
0
∘
60^{\circ}
6
0
∘
) from west to east along a closed parallel.
41
1
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Problem 41
Find the geodesic curvature of the line
y
=
1
y=1
y
=
1
in the upper half-plane with the Lobachevskii—Poincare metric
d
s
2
=
(
d
x
2
+
d
y
2
)
/
y
2
ds^2=(dx^2+dy^2)/y^2
d
s
2
=
(
d
x
2
+
d
y
2
)
/
y
2
42
1
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Problem 42
Do the medians of a triangle meet in a single point in the Lobachevskii plane? What about the altitudes?
43
1
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Problem 43
Find the Betti numbers of the surface
x
1
2
+
⋯
+
x
k
2
−
y
1
2
−
⋯
−
y
l
2
=
1
x_1^2+\cdots+x_k^2-y_1^2-\cdots-y_l^2=1
x
1
2
+
⋯
+
x
k
2
−
y
1
2
−
⋯
−
y
l
2
=
1
and the set
x
1
2
+
⋯
+
x
k
2
≤
1
+
y
1
2
+
⋯
+
y
l
2
x_1^2+\cdots+x_k^2\le1+y_1^2+\cdots+y_l^2
x
1
2
+
⋯
+
x
k
2
≤
1
+
y
1
2
+
⋯
+
y
l
2
in a
(
k
+
l
)
(k+l)
(
k
+
l
)
-dimensional linear space.
44
1
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Problem 44
Find the Betti numbers of the surface
x
2
+
y
2
=
1
+
z
2
x^2+y^2 = 1 + z^2
x
2
+
y
2
=
1
+
z
2
in three-dimensional projective space. The same for the surfaces
z
=
x
y
z = xy
z
=
x
y
,
z
=
x
2
z=x^2
z
=
x
2
,
z
2
=
x
2
+
y
2
z^2 = x^2 + y^2
z
2
=
x
2
+
y
2
.
45
1
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Problem 45
Find the self-intersection index of the surface
x
4
+
y
4
=
1
x^4+y^4=1
x
4
+
y
4
=
1
in the projective plane
CP
2
\text{CP}^2
CP
2
.
46
1
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Problem 46
Map the interior of the unit disc conformally onto the first quadrant.
47
1
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Problem 47
Map the exterior of the disc conformally onto the exterior of a given ellipse.
48
1
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Problem 48
Map the half-plane without a segment perpendicular to its boundary conformally onto the half-plane.
49
1
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Problem 49
Calculate
∮
∣
z
∣
=
2
d
z
1
+
z
10
\oint_{|z|=2}\frac{dz}{\sqrt{1+z^{10}}}
∮
∣
z
∣
=
2
1
+
z
10
d
z
50
1
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Problem 50
Calculate
∫
−
∞
+
∞
e
i
k
x
1
+
x
2
d
x
\int_{-\infty}^{+\infty}\frac{e^{ikx}}{1+x^2}dx
∫
−
∞
+
∞
1
+
x
2
e
ik
x
d
x
51
1
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Problem 51
Calculate the integral
∫
−
∞
+
∞
e
i
k
x
1
−
e
x
1
+
e
x
d
x
\int_{-\infty}^{+\infty}e^{ikx}\frac{1-e^x}{1+e^x}dx
∫
−
∞
+
∞
e
ik
x
1
+
e
x
1
−
e
x
d
x
52
1
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Problem 52
Calculate the first term of the asymptotic expression as
k
→
∞
k\to\infty
k
→
∞
of the integral
∫
−
∞
+
∞
e
i
k
x
1
+
x
2
n
d
x
\int_{-\infty}^{+\infty}\frac{e^{ikx}}{\sqrt{1+x^{2n}}}dx
∫
−
∞
+
∞
1
+
x
2
n
e
ik
x
d
x
53
1
Hide problems
Problem 53
Investigate the singular points of the differential form
d
t
=
d
x
/
y
dt = dx/y
d
t
=
d
x
/
y
on the compact Riemann surface
y
2
/
2
+
U
(
x
)
=
E
y^2/2 + U(x) = E
y
2
/2
+
U
(
x
)
=
E
, where
U
U
U
is a polynomial and
E
E
E
is not a critical value.
54
1
Hide problems
Problem 54
x
¨
=
3
x
−
x
3
−
1
\ddot{x}=3x-x^3-1
x
¨
=
3
x
−
x
3
−
1
. In which of the potential wells is the period of oscillation greater (in the shallower or the deeper) with equal values of the total energy?
55
1
Hide problems
Problem 55
Investigate topologically the Riemann surface of the function
w
=
arctan
z
w=\arctan z
w
=
arctan
z
56
1
Hide problems
Problem 56
How many handles has the Riemann surface of the function
w
=
1
+
z
n
w=\sqrt{1+z^n}
w
=
1
+
z
n
57
1
Hide problems
Problem 57
Find the dimension of the solution space of the problem
∂
u
/
∂
z
‾
=
δ
(
z
−
i
)
\partial u/\partial \overline{z} = \delta(z - i)
∂
u
/
∂
z
=
δ
(
z
−
i
)
for
Im
z
≥
0
\text{Im } z \ge 0
Im
z
≥
0
,
Im
u
(
z
)
=
0
\text{Im } u(z) = 0
Im
u
(
z
)
=
0
for
Im
z
=
0
\text{Im } z = 0
Im
z
=
0
,
u
→
0
u\to 0
u
→
0
as
z
→
∞
z\to\infty
z
→
∞
.
58
1
Hide problems
Problem 58
Find the dimension of the solution space of the problem
∂
u
/
∂
z
‾
=
a
δ
(
z
—
−
i
)
+
b
δ
(
z
+
i
)
\partial u/\partial\overline{z} = a\delta(z —-i) + b\delta(z + i)
∂
u
/
∂
z
=
a
δ
(
z
—
−
i
)
+
b
δ
(
z
+
i
)
for
∣
z
∣
≤
2
|z|\le 2
∣
z
∣
≤
2
,
Im
u
=
0
\text{Im } u = 0
Im
u
=
0
for
∣
z
∣
=
2
|z| = 2
∣
z
∣
=
2
.
59
1
Hide problems
Problem 59
Investigate the existence and uniqueness of the solution of the problem
y
u
x
=
x
u
y
,
u
∣
x
=
1
=
cos
y
yu_x = xu_y, u|_{x=1} =\cos y
y
u
x
=
x
u
y
,
u
∣
x
=
1
=
cos
y
in a neighbourhood of the point
(
1
,
y
0
)
(1, y_0)
(
1
,
y
0
)
.
60
1
Hide problems
Problem 60
Is there a solution of the Cauchy problem
x
(
x
2
+
y
2
)
∂
u
∂
x
+
y
3
∂
u
∂
y
=
0
,
u
∣
y
=
0
=
1
x(x^2+y^2)\frac{\partial u}{\partial x}+y^3\frac{\partial u}{\partial y}=0,\;u|_{y=0}=1
x
(
x
2
+
y
2
)
∂
x
∂
u
+
y
3
∂
y
∂
u
=
0
,
u
∣
y
=
0
=
1
in a neighbourhood of the point
(
x
0
,
0
)
(x_0,0)
(
x
0
,
0
)
of the
x
x
x
-axis? Is it unique?
61
1
Hide problems
Problem 61
What is the largest value of
t
t
t
for which the solution of the problem
∂
u
∂
t
+
u
∂
u
∂
x
=
sin
x
,
u
∣
t
=
0
=
0
\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}=\sin x,\; u|_{t=0}=0
∂
t
∂
u
+
u
∂
x
∂
u
=
sin
x
,
u
∣
t
=
0
=
0
can be extended to the interval
[
0
,
t
)
[0,t)
[
0
,
t
)
.
62
1
Hide problems
Problem 62
Find all solutions of the equation
y
∂
u
/
∂
x
−
sin
x
∂
u
/
∂
y
=
u
2
y\partial u/\partial x-\sin x\partial u/\partial y=u^2
y
∂
u
/
∂
x
−
sin
x
∂
u
/
∂
y
=
u
2
in a neighbourhood of the point
0
,
0
0,0
0
,
0
.
63
1
Hide problems
Problem 63
Is there a solution of the Cauchy problem
y
∂
u
/
∂
x
+
sin
x
∂
u
/
∂
y
=
y
y\partial u/\partial x+\sin x\partial u/\partial y=y
y
∂
u
/
∂
x
+
sin
x
∂
u
/
∂
y
=
y
,
u
∣
x
=
0
=
y
4
u|_{x=0}=y^4
u
∣
x
=
0
=
y
4
on the whole
(
x
,
y
)
(x,y)
(
x
,
y
)
plane? Is it unique?
64
1
Hide problems
Problem 64
Does the Cauchy problem
u
∣
y
=
x
2
=
1
u|_{y=x^2}=1
u
∣
y
=
x
2
=
1
,
(
∇
u
)
2
=
1
(\nabla u)^2=1
(
∇
u
)
2
=
1
have a smooth solution in the domain
y
≥
x
2
y\ge x^2
y
≥
x
2
? In the domain
y
≤
x
2
y\le x^2
y
≤
x
2
?
65
1
Hide problems
Problem 65
Find the mean value of the function
ln
r
\ln r
ln
r
on the circle
(
x
−
a
)
2
+
(
y
−
b
)
2
=
R
2
(x - a)^2 + (y-b)^2 = R^2
(
x
−
a
)
2
+
(
y
−
b
)
2
=
R
2
(of the function
1
/
r
1/r
1/
r
on the sphere).
66
1
Hide problems
Problem 66
Solve the Dirichlet problem
Δ
u
=
0
for
x
2
+
y
2
<
1
\Delta u=0\text{ for }x^2+y^2<1
Δ
u
=
0
for
x
2
+
y
2
<
1
u
=
1
for
x
2
+
y
2
=
1
,
y
>
0
u=1\text{ for }x^2+y^2=1,\;y>0
u
=
1
for
x
2
+
y
2
=
1
,
y
>
0
u
=
−
1
for
x
2
+
y
2
=
1
,
y
<
0
u=-1\text{ for }x^2+y^2=1,\;y<0
u
=
−
1
for
x
2
+
y
2
=
1
,
y
<
0
67
1
Hide problems
Problem 67
What is the dimension of the space of solutions continuous on
x
2
+
y
2
≥
1
x^2+y^2\ge1
x
2
+
y
2
≥
1
of the problem
Δ
u
=
0
for
x
2
+
y
2
>
1
\Delta u=0\text{ for }x^2+y^2>1
Δ
u
=
0
for
x
2
+
y
2
>
1
∂
u
/
∂
n
=
0
for
x
2
+
y
2
=
1
\partial u/\partial n=0\text{ for }x^2+y^2=1
∂
u
/
∂
n
=
0
for
x
2
+
y
2
=
1
68
1
Hide problems
Problem 68
Find
inf
∬
x
2
+
y
2
≤
1
(
∂
u
∂
x
)
2
+
(
∂
u
∂
y
)
2
d
x
d
y
\inf\iint_{x^2+y^2\le1}\left(\frac{\partial u}{\partial x}\right)^2+\left(\frac{\partial u}{\partial y}\right)^2dxdy
in
f
∬
x
2
+
y
2
≤
1
(
∂
x
∂
u
)
2
+
(
∂
y
∂
u
)
2
d
x
d
y
69
1
Hide problems
Problem 69
Prove that the solid angle based on a given closed contour is a function of the vertex of the angle that is harmonic outside the contour.
70
1
Hide problems
Problem 70
Calculate the mean value of the solid angle by which the disc
x
2
+
y
2
≤
1
x^2 + y^2 \le 1
x
2
+
y
2
≤
1
lying in the plane
z
=
0
z = 0
z
=
0
is seen from points of the sphere
x
2
+
y
2
+
(
z
−
2
)
2
=
1
x^2 + y^2 + (z-2)^2 = 1
x
2
+
y
2
+
(
z
−
2
)
2
=
1
.
71
1
Hide problems
Problem 71
Calculate the charge density on the conducting boundary
x
2
+
y
2
+
z
2
=
1
x^2 + y^2 + z^2 = 1
x
2
+
y
2
+
z
2
=
1
of a cavity in which a charge
q
=
1
q = 1
q
=
1
is placed at distance
r
r
r
from the centre.
72
1
Hide problems
Problem 72
Calculate to the first order in
ϵ
\epsilon
ϵ
the effect that the influence of the flattening of the earth (
ϵ
≈
1
/
300
\epsilon\approx 1/300
ϵ
≈
1/300
) on the gravitational field of the earth has on the distance of the moon (assuming the earth to be homogeneous).
73
1
Hide problems
Problem 73
Find (to the first order in
ϵ
\epsilon
ϵ
) the influence of the imperfection of an almost spherical capacitor
R
=
1
+
ϵ
f
(
φ
,
θ
)
R = 1 + \epsilon f(\varphi, \theta)
R
=
1
+
ϵ
f
(
φ
,
θ
)
on its capacity.
74
1
Hide problems
Problem 74
Sketch the graph of
u
(
x
,
1
)
u(x, 1)
u
(
x
,
1
)
, if
0
≤
x
≤
1
0 \le x\le1
0
≤
x
≤
1
,
∂
u
∂
t
=
∂
2
u
∂
x
2
,
u
∣
t
=
0
=
x
2
,
u
∣
x
2
=
x
=
x
2
\frac{\partial u}{\partial t}=\frac{\partial^2 u}{\partial x^2},\;u|_{t=0}=x^2,\;u|_{x^2=x}=x^2
∂
t
∂
u
=
∂
x
2
∂
2
u
,
u
∣
t
=
0
=
x
2
,
u
∣
x
2
=
x
=
x
2
75
1
Hide problems
Problem 75
On account of the annual fluctuation of temperature the ground at the town of Ν freezes to a depth of 2 metres. To what depth would it freeze on account of a daily fluctuation of the same amplitude?
76
1
Hide problems
Problem 76
Investigate the behaviour at
t
→
∞
t\to\infty
t
→
∞
of the solution of the problem
u
t
+
(
u
sin
x
)
x
=
ϵ
u
x
x
,
u
∣
t
=
0
=
1
,
ϵ
≪
1
u_t+(u\sin x)_x=\epsilon u_{xx},\;u|_{t=0}=1,\;\epsilon\ll1
u
t
+
(
u
sin
x
)
x
=
ϵ
u
xx
,
u
∣
t
=
0
=
1
,
ϵ
≪
1
77
1
Hide problems
Problem 77
Find the eigenvalues and their multiplicities of the Laplace operator
Δ
=
div grad
\Delta = \text{div grad}
Δ
=
div grad
on a sphere of radius
R
R
R
in Euclidean space of dimension
n
n
n
.
78
1
Hide problems
Problem 78
Solve the Cauchy problem
∂
2
A
∂
t
2
=
9
∂
2
A
∂
x
2
−
2
B
,
∂
2
B
∂
t
2
=
6
∂
2
B
∂
x
2
−
2
A
\frac{\partial ^2A}{\partial t^2}=9\frac{\partial^2 A}{\partial x^2}-2B,\;\frac{\partial^2 B}{\partial t^2}=6\frac{\partial^2 B}{\partial x^2}-2A
∂
t
2
∂
2
A
=
9
∂
x
2
∂
2
A
−
2
B
,
∂
t
2
∂
2
B
=
6
∂
x
2
∂
2
B
−
2
A
A
∣
t
=
0
=
cos
x
,
B
∣
t
=
0
=
0
,
∂
A
∂
t
∣
t
=
0
=
∂
B
∂
t
∣
t
=
0
=
0
A|_{t=0}=\cos x,\; B|_{t=0}=0,\; \left.\frac{\partial A}{\partial t}\right|_{t=0}=\left.\frac{\partial B}{\partial t}\right|_{t=0}=0
A
∣
t
=
0
=
cos
x
,
B
∣
t
=
0
=
0
,
∂
t
∂
A
t
=
0
=
∂
t
∂
B
t
=
0
=
0
79
1
Hide problems
Problem 79
How many solutions has the boundary-value problem
u
x
x
+
λ
u
=
sin
x
,
u
(
0
)
=
u
(
π
)
=
0
u_{xx}+\lambda u=\sin x,\;u(0)=u(\pi)=0
u
xx
+
λ
u
=
sin
x
,
u
(
0
)
=
u
(
π
)
=
0
80
1
Hide problems
Problem 80
Solve the equation
∫
0
1
(
x
+
y
)
2
u
(
x
)
d
x
=
λ
u
(
y
)
+
1
\int_0^1(x+y)^2u(x)dx=\lambda u(y)+1
∫
0
1
(
x
+
y
)
2
u
(
x
)
d
x
=
λ
u
(
y
)
+
1
81
1
Hide problems
Problem 81
Find the Green's function of the operator
d
2
/
d
x
2
−
1
d^2/dx^2-1
d
2
/
d
x
2
−
1
and solve the equation
∫
−
∞
+
∞
e
−
∣
x
−
y
∣
u
(
y
)
d
y
=
e
−
x
2
\int_{-\infty}^{+\infty}e^{-|x-y|}u(y)dy=e^{-x^2}
∫
−
∞
+
∞
e
−
∣
x
−
y
∣
u
(
y
)
d
y
=
e
−
x
2
82
1
Hide problems
Problem 82
For what values of the velocity
c
c
c
does the equation
u
t
=
u
−
u
2
+
u
x
x
u_t = u -u^2 + u_{xx}
u
t
=
u
−
u
2
+
u
xx
have a solution in the form of a traveling wave
u
=
φ
(
x
−
c
t
)
u = \varphi(x-ct)
u
=
φ
(
x
−
c
t
)
,
φ
(
−
∞
)
=
1
\varphi(-\infty) = 1
φ
(
−
∞
)
=
1
,
φ
(
∞
)
=
0
\varphi(\infty) = 0
φ
(
∞
)
=
0
,
0
≤
u
≤
1
0 \le u \le 1
0
≤
u
≤
1
?
83
1
Hide problems
Problem 83
Find solutions of the equation
u
t
=
u
x
x
x
+
u
u
x
u_t=u_{xxx}+uu_x
u
t
=
u
xxx
+
u
u
x
in the form of a traveling wave
u
=
φ
(
x
−
c
t
)
u=\varphi(x-ct)
u
=
φ
(
x
−
c
t
)
,
φ
(
±
∞
)
=
0
\varphi(\pm\infty)=0
φ
(
±
∞
)
=
0
.
84
1
Hide problems
Problem 84
Find the number of positive and negative squares in the canonical form of the quadratic form
∑
i
<
j
(
x
i
−
x
j
)
2
\sum_{i<j}(x_i-x_j)^2
∑
i
<
j
(
x
i
−
x
j
)
2
in
n
n
n
variables. The same for the form
∑
i
<
j
x
i
x
j
\sum_{i<j}x_i x_j
∑
i
<
j
x
i
x
j
.
85
1
Hide problems
Problem 85
Find the lengths of the principal axes of the ellipsoid
∑
i
≤
j
x
i
x
j
=
1
\sum_{i\le j}x_i x_j=1
i
≤
j
∑
x
i
x
j
=
1
86
1
Hide problems
Problem 86
Through the centre of a cube (tetrahedron, icosahedron) draw a straight line in such a way that the sum of the squares of its distances from the vertices is a) minimal, b) maximal.
87
1
Hide problems
Problem 87
Find the derivatives of the lengths of the semiaxes of the ellipsoid
x
2
+
y
2
+
z
2
+
x
y
+
y
z
+
z
x
=
1
+
ϵ
x
y
x^2 + y^2 + z^2 + xy + yz + zx = 1 + \epsilon xy
x
2
+
y
2
+
z
2
+
x
y
+
yz
+
z
x
=
1
+
ϵ
x
y
with respect to
ϵ
\epsilon
ϵ
at
ϵ
=
0
\epsilon = 0
ϵ
=
0
.
88
1
Hide problems
Problem 88
How many figures can be obtained by intersecting the infinite-dimensional cube
∣
x
k
∣
≤
1
|x_k| \le 1
∣
x
k
∣
≤
1
,
k
=
1
,
2
,
…
k = 1,2,\ldots
k
=
1
,
2
,
…
with a two-dimensional plane?
89
1
Hide problems
Problem 89
Calculate the sum of vector products
[
[
x
,
y
]
,
z
]
+
[
[
y
,
z
]
,
x
]
+
[
[
z
,
x
]
,
y
]
[[x, y], z] + [[y, z], x] + [[z, x], y]
[[
x
,
y
]
,
z
]
+
[[
y
,
z
]
,
x
]
+
[[
z
,
x
]
,
y
]
90
1
Hide problems
Problem 90
Calculate the sum of matrix commutators
[
A
,
[
B
,
C
]
]
+
[
B
,
[
C
,
A
]
]
+
[
C
,
[
A
,
B
]
]
[A, [B, C]] + [B, [C, A]] + [C, [A, B]]
[
A
,
[
B
,
C
]]
+
[
B
,
[
C
,
A
]]
+
[
C
,
[
A
,
B
]]
, where
[
A
,
B
]
=
A
B
−
B
A
[A, B] = AB-BA
[
A
,
B
]
=
A
B
−
B
A
91
1
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Problem 91
Find the Jordan normal form of the operator
e
d
/
d
t
e^{d/dt}
e
d
/
d
t
in the space of quasi-polynomials
{
e
λ
t
p
(
t
)
}
\{e^{\lambda t}p(t)\}
{
e
λ
t
p
(
t
)}
where the degree of the polynomial
p
p
p
is less than
5
5
5
, and of the operator
ad
A
\text{ad}_A
ad
A
,
B
↦
[
A
,
B
]
B\mapsto [A, B]
B
↦
[
A
,
B
]
, in the space of
n
×
n
n\times n
n
×
n
matrices
B
B
B
, where
A
A
A
is a diagonal matrix.
92
1
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Problem 92
Find the orders of the subgroups of the group of rotations of the cube, and find its normal subgroups.
93
1
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Problem 93
Decompose the space of functions defined on the vertices of a cube into invariant subspaces irreducible with respect to the group of a) its symmetries, b) its rotations.
94
1
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Problem 94
Decompose a
5
5
5
-dimensional real linear space into the irreducible invariant subspaces of the group generated by cyclic permutations of the basis vectors.
95
1
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Problem 95
Decompose the space of homogeneous polynomials of degree
5
5
5
in
(
x
,
y
,
z
)
(x, y, z)
(
x
,
y
,
z
)
into irreducible subspaces invariant with respect to the rotation group
S
O
(
3
)
SO(3)
SO
(
3
)
.
96
1
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Problem 96
Each of
3600
3600
3600
subscribers of a telephone exchange calls it once an hour on average. What is the probability that in a given second
5
5
5
or more calls are received? Estimate the mean interval of time between such seconds
(
i
,
i
+
1
)
(i, i + 1)
(
i
,
i
+
1
)
.
97
1
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Problem 97
A particle performing a random walk on the integer points of the semi-axis
x
≥
0
x \ge 0
x
≥
0
moves a distance
1
1
1
to the right with probability
a
a
a
, and to the left with probability
b
b
b
, and stands still in the remaining cases (if
x
=
0
x = 0
x
=
0
, it stands still instead of moving to the left). Determine the steady-state probability distribution, and also the expectation of
x
x
x
and
x
2
x^2
x
2
over a long time, if the particle starts at the point
0
0
0
.
98
1
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Problem 98
In the game of "Fingers",
N
N
N
players stand in a circle and simultaneously thrust out their right hands, each with a certain number of fingers showing. The total number of fingers shown is counted out round the circle from the leader, and the player on whom the count stops is the winner. How large must
N
N
N
be for a suitably chosen group of
N
/
10
N/10
N
/10
players to contain a winner with probability at least
0.9
0.9
0.9
? How does the probability that the leader wins behave as
N
→
∞
N\to\infty
N
→
∞
?
99
1
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Problem 99
One player conceals a
10
10
10
or
20
20
20
copeck coin, and the other guesses its value. If he is right he gets the coin, if wrong he pays
15
15
15
copecks. Is this a fair game? What are the optimal mixed strategies for both players?
100
1
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Problem 100
Find the mathematical expectation of the area of the projection of a cube with edge of length
1
1
1
onto a plane with an isotropically distributed random direction of projection.