Arnold's Trivium 1991
1
Sketch the graph of the derivative and the graph of the integral of a function given by a free-hand graph.
![\[\lim_{x\to0}\frac{\sin \tan x-\tan\sin x}{\arcsin\arctan x-\arctan\arcsin x}\]](http://latex.artofproblemsolving.com/1/9/3/19326b9559851fe57d04e3003a0983e0cde717e8.png)
3
Find the critical values and critical points of the mapping 
(sketch the answer).
(sketch the answer).![\[\frac{x^2+1}{x^3-x}\]](http://latex.artofproblemsolving.com/d/3/f/d3ff4ac44376ea8b559be3880b1f1721a2747415.png)
![\[\frac{1}{x^2+3x+2}\]](http://latex.artofproblemsolving.com/2/9/7/29710b7544d266387086e619e77e8466db3a66f5.png)
with 
-
,
.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
How many maxima, minima, and saddle points does the function 
have on the surface 
, 
, 
?
have on the surface 
, 
, 
?9
Does every positive polynomial in two real variables attain its lower bound in the plane?
10
Investigate the asymptotic behaviour of the solutions 
of the equation 
that tend to zero as 
.
of the equation 
that tend to zero as 
.![\[\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\frac{dxdy}{1+x^4y^4}\]](http://latex.artofproblemsolving.com/8/9/e/89e51f532113c7b92b105d2cb4efab41967c93e6.png)
through the surface![\[(x-1)^2+y^2+z^2=2\]](http://latex.artofproblemsolving.com/1/7/e/17e9a4eb5e1302d627c2ed6ed84903199aa67484.png)
relative error![\[\int_1^{10}x^xdx\]](http://latex.artofproblemsolving.com/d/b/2/db249a91f0e4e72d25eaaca3d198c30c73cba946.png)
![\[\int_{-\infty}^{\infty}(x^4+4x+4)^{-100}dx\]](http://latex.artofproblemsolving.com/0/7/9/079ec65fe364fbd0c8cd44185c0495e2fe0c8208.png)
![\[\int_{-\infty}^{\infty}\cos(100(x^4-x))dx\]](http://latex.artofproblemsolving.com/1/c/3/1c3c864fee522d65e61926dd4494a397dfff825f.png)
16
What fraction of a 
-dimensional cube is the volume of the inscribed sphere? What fraction is it of a 
-dimensional cube?
-dimensional cube is the volume of the inscribed sphere? What fraction is it of a 
-dimensional cube?17
Find the distance of the centre of gravity of a uniform 
-dimensional solid hemisphere of radius 
from the centre of the sphere with 
relative error.
-dimensional solid hemisphere of radius 
from the centre of the sphere with 
relative error.![\[\int\cdots\int \exp\left(-\sum_{1\le i\le j\le n}x_ix_j\right)dx_1\cdots dx_n\]](http://latex.artofproblemsolving.com/2/8/b/28b4e703b08115df295821798b9b61ce3c5f0365.png)
19
Investigate the path of a light ray in a plane medium with refractive index 
using Snell's law 
, where 
is the angle made by the ray with the -axis.
using Snell's law 
, where 
is the angle made by the ray with the -axis.20
Find the derivative of the solution of the equation 
, with initial conditions 
, 
, with respect to the parameter 
for 
.
, with initial conditions 
, 
, with respect to the parameter 
for 
.21
Find the derivative of the solution of the equation 
with initial condition 
, 
with respect to for .
with initial condition 
, 
with respect to for .22
Investigate the boundary of the domain of stability (
) in the space of coefficients of the equation 
.
) in the space of coefficients of the equation 
.![\[\frac{dy}{dx}=x+\frac{x^3}{y}\]](http://latex.artofproblemsolving.com/d/5/f/d5f77f1721d2c73f6e16b71e1a1f13b946ca8923.png)
![\[\ddot{x}=x^5+x^2\dot{x}\]](http://latex.artofproblemsolving.com/f/8/0/f8001a13e0370170f51d5f99da2013bdf41b99f5.png)
25
Can an asymptotically stable equilibrium position become unstable in the Lyapunov sense under linearization?
of solutions of the systems![\[\begin{cases}
\dot{x}=y\\
\dot{y}=2\sin y-y-x\end{cases}\]](http://latex.artofproblemsolving.com/6/6/2/662e0a566df4c8821ebe9916616f304fe7e93d07.png)
![\[\begin{cases}
\dot{x}=y\\
\dot{y}=2x-x^{3}-x^{2}-\epsilon y\end{cases}\]](http://latex.artofproblemsolving.com/4/3/2/4327598f7f65070258688afc3657593df2c8950f.png)
.27
Sketch the images of the solutions of the equation
![\[\ddot{x}=F(x)-k\dot{x},\; F=-dU/dx\]](//latex.artofproblemsolving.com/9/3/8/9386ffdcbd398992d38102681a0f970444aada57.png)
in the
-plane, where 
, near non-degenerate critical points of the potential 
.
![\[\ddot{x}=F(x)-k\dot{x},\; F=-dU/dx\]](http://latex.artofproblemsolving.com/9/3/8/9386ffdcbd398992d38102681a0f970444aada57.png)
in the
-plane, where 
, near non-degenerate critical points of the potential 
.28
Sketch the phase portrait and investigate its variation under variation of the small complex parameter 
:
![\[\dot{z}=\epsilon z-(1+i)z|z|^2+\overline{z}^4\]](//latex.artofproblemsolving.com/c/b/9/cb9fd94f983d20e23ed30fe874f5196623b27ac4.png)
:![\[\dot{z}=\epsilon z-(1+i)z|z|^2+\overline{z}^4\]](http://latex.artofproblemsolving.com/c/b/9/cb9fd94f983d20e23ed30fe874f5196623b27ac4.png)
29
A charge moves with velocity in a plane under the action of a strong magnetic field 
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 
as 
.]
in a plane under the action of a strong magnetic field 
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 
as 
.]30
Find the sum of the indexes of the singular points other than zero of the vector field
![\[z\overline{z}^2+z^4+2\overline{z}^4\]](//latex.artofproblemsolving.com/7/4/c/74c046cec0e7d7e19dfe13eca5a693b2efef477d.png)
![\[z\overline{z}^2+z^4+2\overline{z}^4\]](http://latex.artofproblemsolving.com/7/4/c/74c046cec0e7d7e19dfe13eca5a693b2efef477d.png)
![\[(x^4+y^4+z^4,x^3y-xy^3,xyz^2)\]](http://latex.artofproblemsolving.com/6/4/2/642907fd365012734ebd50a13f970cfa9242b20f.png)
![\[(xy+yz+xz)\]](http://latex.artofproblemsolving.com/4/5/6/45613860f9c752f3999742f99257e34c8b1123c2.png)
33
Find the linking coefficient of the phase trajectories of the equation of small oscillations 
, 
on a level surface of the total energy.
, 
on a level surface of the total energy.![\[(x^2+y^2-2)^2+z^2=1\]](http://latex.artofproblemsolving.com/8/f/9/8f92a5ec96a2e56707267278313cd28badc6b6d5.png)
36
Sketch the evolvent of the cubic parabola 
(the evolvent is the locus of the points 
, where 
is the arc-length of the curve 
and 
is a constant).
(the evolvent is the locus of the points 
, where 
is the arc-length of the curve 
and 
is a constant).37
Prove that in Euclidean space the surfaces
![\[((A-\lambda E)^{-1}x,x)=1\]](//latex.artofproblemsolving.com/8/5/0/85073a841307c9abd5fea6068412caf0b315232f.png)
passing through the point
and corresponding to different values of 
are pairwise orthogonal ( is a symmetric operator without multiple eigenvalues).
![\[((A-\lambda E)^{-1}x,x)=1\]](http://latex.artofproblemsolving.com/8/5/0/85073a841307c9abd5fea6068412caf0b315232f.png)
passing through the point
and corresponding to different values of 
are pairwise orthogonal ( is a symmetric operator without multiple eigenvalues).![\[z^4+(x^2+y^2-1)(2x^2+3y^2-1)=0\]](http://latex.artofproblemsolving.com/b/6/d/b6d82d92290ade5be102db2e397ba39c9b5af330.png)
39
Calculate the Gauss integral
![\[\oint\frac{(d\overrightarrow{A},d\overrightarrow{B},\overrightarrow{A}-\overrightarrow{B})}{|\overrightarrow{A}-\overrightarrow{B}|^3}\]](//latex.artofproblemsolving.com/3/7/5/375061b97d23eb9459c9c8e80bb121557d45914c.png)
where
runs along the curve 
, 
, 
, and 
along the curve 
, 
, 
.
Note: that
was supposed to be oiint (i.e. 
with a circle) but the command does not work on AoPS.
![\[\oint\frac{(d\overrightarrow{A},d\overrightarrow{B},\overrightarrow{A}-\overrightarrow{B})}{|\overrightarrow{A}-\overrightarrow{B}|^3}\]](http://latex.artofproblemsolving.com/3/7/5/375061b97d23eb9459c9c8e80bb121557d45914c.png)
where
runs along the curve 
, 
, 
, and 
along the curve 
, 
, 
.Note: that
was supposed to be oiint (i.e. 
with a circle) but the command does not work on AoPS.40
Find the parallel displacement of a vector pointing north at Leningrad (latitude 
) from west to east along a closed parallel.
) from west to east along a closed parallel.41
Find the geodesic curvature of the line 
in the upper half-plane with the Lobachevskii—Poincare metric
![\[ds^2=(dx^2+dy^2)/y^2\]](//latex.artofproblemsolving.com/a/d/c/adcbb66e2015ce55c214062f28978e00a2511733.png)
in the upper half-plane with the Lobachevskii—Poincare metric![\[ds^2=(dx^2+dy^2)/y^2\]](http://latex.artofproblemsolving.com/a/d/c/adcbb66e2015ce55c214062f28978e00a2511733.png)
42
Do the medians of a triangle meet in a single point in the Lobachevskii plane? What about the altitudes?
43
Find the Betti numbers of the surface 
and the set 
in a 
-dimensional linear space.
and the set 
in a 
-dimensional linear space.44
Find the Betti numbers of the surface 
in three-dimensional projective space. The same for the surfaces 
, 
, 
.
in three-dimensional projective space. The same for the surfaces 
, 
, 
.
in the projective plane 
.47
Map the exterior of the disc conformally onto the exterior of a given ellipse.
48
Map the half-plane without a segment perpendicular to its boundary conformally onto the half-plane.
![\[\oint_{|z|=2}\frac{dz}{\sqrt{1+z^{10}}}\]](http://latex.artofproblemsolving.com/d/4/6/d464ee949252432fe34538361851b41e6a8d28c8.png)
![\[\int_{-\infty}^{+\infty}\frac{e^{ikx}}{1+x^2}dx\]](http://latex.artofproblemsolving.com/a/b/f/abf7d2b77c49f8338b5aaaa7da97a2b2781c83b5.png)
![\[\int_{-\infty}^{+\infty}e^{ikx}\frac{1-e^x}{1+e^x}dx\]](http://latex.artofproblemsolving.com/2/6/d/26d4743f7c3f9f83a2aa5a1f314d134b120aad14.png)
of the integral![\[\int_{-\infty}^{+\infty}\frac{e^{ikx}}{\sqrt{1+x^{2n}}}dx\]](http://latex.artofproblemsolving.com/9/9/8/9980f3c44b77f957dbaaeb8dbb069764fb014e05.png)
53
Investigate the singular points of the differential form 
on the compact Riemann surface 
, where is a polynomial and 
is not a critical value.
on the compact Riemann surface 
, where is a polynomial and 
is not a critical value.54
. 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?![\[w=\arctan z\]](http://latex.artofproblemsolving.com/c/9/6/c964562ac780172fa20f9f57ffbb8ffbf2038a62.png)
![\[w=\sqrt{1+z^n}\]](http://latex.artofproblemsolving.com/7/c/e/7cec180e5bb8449fa75d956f47d88a724bd7c4c5.png)
for 
,
for 
,
as 
.58
Find the dimension of the solution space of the problem $\partial u/\partial\overline{z} = a\delta(z —-i) + b\delta(z + i)$ for
, 
for 
.
, 
for 
.59
Investigate the existence and uniqueness of the solution of the problem 
in a neighbourhood of the point 
.
in a neighbourhood of the point 
.60
Is there a solution of the Cauchy problem
![\[x(x^2+y^2)\frac{\partial u}{\partial x}+y^3\frac{\partial u}{\partial y}=0,\;u|_{y=0}=1\]](//latex.artofproblemsolving.com/6/0/3/603156ef57533ea88961763262c64298fa99ee53.png)
in a neighbourhood of the point
of the -axis? Is it unique?
![\[x(x^2+y^2)\frac{\partial u}{\partial x}+y^3\frac{\partial u}{\partial y}=0,\;u|_{y=0}=1\]](http://latex.artofproblemsolving.com/6/0/3/603156ef57533ea88961763262c64298fa99ee53.png)
in a neighbourhood of the point
of the -axis? Is it unique?61
What is the largest value of 
for which the solution of the problem
![\[\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}=\sin x,\; u|_{t=0}=0\]](//latex.artofproblemsolving.com/7/a/d/7adeb2774fab8b4c3ccc74bda74991ed505d7b86.png)
can be extended to the interval
.
for which the solution of the problem![\[\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}=\sin x,\; u|_{t=0}=0\]](http://latex.artofproblemsolving.com/7/a/d/7adeb2774fab8b4c3ccc74bda74991ed505d7b86.png)
can be extended to the interval
.
in a neighbourhood of the point 
.
,
on the whole 64
Does the Cauchy problem 
, 
have a smooth solution in the domain 
? In the domain 
?
, 
have a smooth solution in the domain 
? In the domain 
?65
Find the mean value of the function 
on the circle 
(of the function 
on the sphere).
on the circle 
(of the function 
on the sphere).![\[\Delta u=0\text{ for }x^2+y^2<1\]](http://latex.artofproblemsolving.com/7/9/1/7918ea3b0e81750cbd2667da089f282e20690d20.png)
![\[u=1\text{ for }x^2+y^2=1,\;y>0\]](http://latex.artofproblemsolving.com/3/e/3/3e32c6620d88eb037f9404df8e3853cf505afddb.png)
![\[u=-1\text{ for }x^2+y^2=1,\;y<0\]](http://latex.artofproblemsolving.com/8/b/7/8b7f394c58a6dcbebb043ab04713becf27b03ef3.png)
of the problem![\[\Delta u=0\text{ for }x^2+y^2>1\]](http://latex.artofproblemsolving.com/9/e/7/9e712ac32647262dd7a05161bb6fcdfbf8735f5d.png)
![\[\partial u/\partial n=0\text{ for }x^2+y^2=1\]](http://latex.artofproblemsolving.com/3/7/e/37e7c501a38b08a15b78d5ab6589584591ac190c.png)
![\[\inf\iint_{x^2+y^2\le1}\left(\frac{\partial u}{\partial x}\right)^2+\left(\frac{\partial u}{\partial y}\right)^2dxdy\]](http://latex.artofproblemsolving.com/6/b/4/6b4a970d4998619e811845471c0ea9fd3c4bd86e.png)
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
Calculate the mean value of the solid angle by which the disc 
lying in the plane 
is seen from points of the sphere 
.
lying in the plane 
is seen from points of the sphere 
.71
Calculate the charge density on the conducting boundary 
of a cavity in which a charge 
is placed at distance 
from the centre.
of a cavity in which a charge 
is placed at distance 
from the centre.72
Calculate to the first order in the effect that the influence of the flattening of the earth (
) on the gravitational field of the earth has on the distance of the moon (assuming the earth to be homogeneous).
the effect that the influence of the flattening of the earth (
) on the gravitational field of the earth has on the distance of the moon (assuming the earth to be homogeneous).73
Find (to the first order in ) the influence of the imperfection of an almost spherical capacitor 
on its capacity.
) the influence of the imperfection of an almost spherical capacitor 
on its capacity.
,
,![\[\frac{\partial u}{\partial t}=\frac{\partial^2 u}{\partial x^2},\;u|_{t=0}=x^2,\;u|_{x^2=x}=x^2\]](http://latex.artofproblemsolving.com/3/8/9/3898356c110428ba1eedadc2e0ad5fea1dd0d0c2.png)
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?
of the solution of the problem![\[u_t+(u\sin x)_x=\epsilon u_{xx},\;u|_{t=0}=1,\;\epsilon\ll1\]](http://latex.artofproblemsolving.com/e/a/6/ea6ec3b741bacd394e8cbfa9decddbe6801deb83.png)
77
Find the eigenvalues and their multiplicities of the Laplace operator 
on a sphere of radius 
in Euclidean space of dimension 
.
on a sphere of radius 
in Euclidean space of dimension 
.![\[\frac{\partial ^2A}{\partial t^2}=9\frac{\partial^2 A}{\partial x^2}-2B,\;\frac{\partial^2 B}{\partial t^2}=6\frac{\partial^2 B}{\partial x^2}-2A\]](http://latex.artofproblemsolving.com/e/6/6/e66fbe8588d2cb4ebf4d28e56b21dd923bf196de.png)
![\[A|_{t=0}=\cos x,\; B|_{t=0}=0,\; \left.\frac{\partial A}{\partial t}\right|_{t=0}=\left.\frac{\partial B}{\partial t}\right|_{t=0}=0\]](http://latex.artofproblemsolving.com/d/f/7/df7e0fad249e4534c4de7cbec60d0531e978a94c.png)
![\[u_{xx}+\lambda u=\sin x,\;u(0)=u(\pi)=0\]](http://latex.artofproblemsolving.com/d/d/8/dd88e6c03609bc0c32d7d124f771fce60fa2acc7.png)
![\[\int_0^1(x+y)^2u(x)dx=\lambda u(y)+1\]](http://latex.artofproblemsolving.com/b/9/5/b95c453e0fb252355c109a86c9f9544d6b3fee73.png)
and solve the equation![\[\int_{-\infty}^{+\infty}e^{-|x-y|}u(y)dy=e^{-x^2}\]](http://latex.artofproblemsolving.com/b/4/f/b4f4ac328ddcce92156a42f18027ff066b081f78.png)
82
For what values of the velocity does the equation 
have a solution in the form of a traveling wave 
, 
, 
, 
?
does the equation 
have a solution in the form of a traveling wave 
, 
, 
, 
?
in the form of a traveling wave 
,
.84
Find the number of positive and negative squares in the canonical form of the quadratic form 
in variables. The same for the form 
.
in variables. The same for the form 
.![\[\sum_{i\le j}x_i x_j=1\]](http://latex.artofproblemsolving.com/0/f/5/0f516c235b5e7c5f577a1060fc2dd5a19cdf367c.png)
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
Find the derivatives of the lengths of the semiaxes of the ellipsoid 
with respect to at 
.
with respect to at 
.88
How many figures can be obtained by intersecting the infinite-dimensional cube 
, 
with a two-dimensional plane?
, 
with a two-dimensional plane?![$[[x, y], z] + [[y, z], x] + [[z, x], y]$](http://latex.artofproblemsolving.com/3/b/e/3be5723f06eef9e84ed88ac3bb0ace18859c6f93.png)
![$[A, [B, C]] + [B, [C, A]] + [C, [A, B]]$](http://latex.artofproblemsolving.com/7/3/f/73fedc321bd27613248b38461d36ec03af9eaa4c.png)
,![$[A, B] = AB-BA$](http://latex.artofproblemsolving.com/3/4/1/3419bfa515723e990d8e401c91fc86d7f9df475a.png)
91
Find the Jordan normal form of the operator 
in the space of quasi-polynomials 
where the degree of the polynomial 
is less than , and of the operator 
, ![$B\mapsto [A, B]$](//latex.artofproblemsolving.com/3/2/3/323643ca52715cb281a030c7b8406466a78d8c0e.png)
, in the space of 
matrices 
, where is a diagonal matrix.
in the space of quasi-polynomials 
where the degree of the polynomial 
is less than , and of the operator 
, ![$B\mapsto [A, B]$](http://latex.artofproblemsolving.com/3/2/3/323643ca52715cb281a030c7b8406466a78d8c0e.png)
, in the space of 
matrices 
, where is a diagonal matrix.92
Find the orders of the subgroups of the group of rotations of the cube, and find its normal subgroups.
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
Decompose a -dimensional real linear space into the irreducible invariant subspaces of the group generated by cyclic permutations of the basis vectors.
-dimensional real linear space into the irreducible invariant subspaces of the group generated by cyclic permutations of the basis vectors.95
Decompose the space of homogeneous polynomials of degree in 
into irreducible subspaces invariant with respect to the rotation group 
.
in 
into irreducible subspaces invariant with respect to the rotation group 
.96
Each of 
subscribers of a telephone exchange calls it once an hour on average. What is the probability that in a given second or more calls are received? Estimate the mean interval of time between such seconds 
.
subscribers of a telephone exchange calls it once an hour on average. What is the probability that in a given second or more calls are received? Estimate the mean interval of time between such seconds 
.97
A particle performing a random walk on the integer points of the semi-axis 
moves a distance to the right with probability 
, and to the left with probability 
, and stands still in the remaining cases (if 
, it stands still instead of moving to the left). Determine the steady-state probability distribution, and also the expectation of and 
over a long time, if the particle starts at the point 
.
moves a distance to the right with probability 
, and to the left with probability 
, and stands still in the remaining cases (if 
, it stands still instead of moving to the left). Determine the steady-state probability distribution, and also the expectation of and 
over a long time, if the particle starts at the point 
.98
In the game of "Fingers", 
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 be for a suitably chosen group of 
players to contain a winner with probability at least 
? How does the probability that the leader wins behave as 
?
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 be for a suitably chosen group of 
players to contain a winner with probability at least 
? How does the probability that the leader wins behave as 
?99
One player conceals a or 
copeck coin, and the other guesses its value. If he is right he gets the coin, if wrong he pays 
copecks. Is this a fair game? What are the optimal mixed strategies for both players?
or 
copeck coin, and the other guesses its value. If he is right he gets the coin, if wrong he pays 
copecks. Is this a fair game? What are the optimal mixed strategies for both players?100
Find the mathematical expectation of the area of the projection of a cube with edge of length onto a plane with an isotropically distributed random direction of projection.
onto a plane with an isotropically distributed random direction of projection.
