Geometric Problems

If we throw a heavy convex polyhedron with arbitrary interior mass distribution onto a horizontal floor then it will come to rest in a stable position on one of its faces. That is there exists for an arbitrary point $P$ lying in the interior of the convex polyhedron one face $F$ (at least) with the following property: The perpendicular dropped from $P$ onto the plane in which $F$ lies has its foot in the interior of the face $F$. Give a purely geometrical proof free from mechanical considerations for the existence of the face $F$.

The intersection of a plane $\pi$ with a circular cone is an ellipse. Let $S$ denote the area of the part of the conical surface between the ellipse and the vertex of the cone, $p$ and $q$ the distances of the vertex from the two endpoints of the major axis of the ellipse, and finally $\alpha$ the angle included between the axis of the cone and any generator. Prove that

The intersection of a plane $\pi$ with a circular cone is an ellipse. Let $S$ denote the area of the part of the conical surface between the ellipse and the vertex of the cone, $p$ and $q$ the distances of the vertex from the two endpoints of the major axis of the ellipse, and finally $\alpha$ the angle included between the axis of the cone and any generator. Prove that

Two spheres, one of radius $r$, the other of radius $R$, are externally tangent to each other. We denote by $B$ the smallest convex solid containing both spheres (a tight bag containing two balls). Let $V$ stand for the volume of $B$, $S$ for the area of its surface, and $M$ for its Minkowski constant (surface integral of the mean curvature). Prove that

Two spheres, one of radius $r$, the other of radius $R$, are externally tangent to each other. We denote by $B$ the smallest convex solid containing both spheres (a tight bag containing two balls). Let $V$ stand for the volume of $B$, $S$ for the area of its surface, and $M$ for its Minkowski constant (surface integral of the mean curvature). Prove that

A lampshade has the shape of a frustum of a right circular cone. Its perimeter is $P$ at the bottom, $p$ at the top, its slant height is $s$, and the inclination $i$ of the slant height to the bottom plane is $\geq 60°$; $P \geq p$. Show that such a lampshade can be cut out in one piece from a rectangular sheet of paper of dimensions $P$ and $s + \left[p(P-p)\right]/8s$. There will even be paper left over for a flap to glue the ends together, except in the extreme case where $P=p$, $i=\pi /2$, and no paper will be wasted.

A lampshade has the shape of a frustum of a right circular cone. Its perimeter is $P$ at the bottom, $p$ at the top, its slant height is $s$, and the inclination $i$ of the slant height to the bottom plane is $\geq 60°$; $P \geq p$. Show that such a lampshade can be cut out in one piece from a rectangular sheet of paper of dimensions $P$ and $s + \left[p(P-p)\right]/8s$. There will even be paper left over for a flap to glue the ends together, except in the extreme case where $P=p$, $i=\pi /2$, and no paper will be wasted.

The numbers $a$ and $b$ are positive and $f(x)$ is a non-linear function such that

in the interval $[0, a]$. Prove that

(The inequality becomes intuitively obvious when both sides are interpreted as areas of surfaces of revolution.)

The numbers $a$ and $b$ are positive and $f(x)$ is a non-linear function such that

in the interval $[0, a]$. Prove that

(The inequality becomes intuitively obvious when both sides are interpreted as areas of surfaces of revolution.)

If on a circular field a given quantity of wheat is piled up, then the maximum slope of the resulting heap is smallest when it forms a right circular cone. Prove the analytic formulation of this fact, i.e. the following theorem:

Let the function $f(x, y)$ possess both the partial derivatives

If $f(x, y) = 0$ on the boundary of the unit circle $x^2+y^2 = 1$, then there exists a point $\xi$, $\eta$ in the unit circle such that

where the double integral is extended over the unit circle.

If a point mass traverses a unit length in unit time moving from rest to rest, then at some point between the two rest points it must experience an acceleration of magnitude greater than 4. Demonstrate this fact analytically.

If a curve viewed from the point $O$ appears everywhere convex, then it subtends a visual angle $< 180°$; if it appears everywhere concave and extends to infinity it subtends a visual angle $> 180°$.

Let $O$ be the origin of a system of polar coordinates $r$, $\varphi$ and $r = 1/f(\varphi )$ the equation of the curve, where $f(x)$ is twice differentiable. In calculating the radius of curvature we are led to the following two theorems:

1. If the function $f(x) > 0$ in the interval $a \leq x \leq a+\pi$, then there exists a point $\xi$, $a < \xi < a+\pi$, such that

   $$f(\xi ) + f''(\xi ) > 0 .$$

2. If $f(x) > 0$, $f(x)+f''(x) > 0$ for $a < x < b$ and $f(a) = f(b) = 0$, then $b-a > \pi$. Prove these theorems by purely analytical methods.

On a closed convex curve with continuous curvature there are always at least three different pairs of points with the following property: The tangents at the two points of a pair are parallel and the curvature of the curve at the two points is the same.

If a convex curve of circumference $l$ and of area $a$ can roll unimpeded in the interior of another convex curve of circumference $L$ and area $A$, then we have

Denote the volume, the surface area and the integral of the mean curvature of two convex surfaces by $v$, $s$, $m$ and $V$, $S$, $M$ respectively. If the first body can roll unimpeded in the interior of the second, then we have the inequalities

Every closed surface is in equilibrium under the action of uniform pressure applied from all sides. In order to show this, prove that the following formulae hold for every closed surface:

Here $\cos \alpha$, $\cos \beta$, $\cos \gamma$ are the direction cosines of the outward normal and $dS$ is the element of surface area.

If on every element $dS$ of a closed, rigid surface with continuous curvature there acts a force of magnitude $\tfrac 12(1/R_1 + 1/R_2) dS$ in the direction of the inward normal, where $R_1$ and $R_2$ are the two principal radii of curvature, then the surface is in equilibrium.

(continued). If the magnitude of the force acting on $dS$ in the direction of the inward normal is $(1/R_1 R_2) \, dS$ (i.e. proportional to the Gaussian curvature rather than to the mean curvature as in the preceding problem), then the surface is also in equilibrium.

At every edge $e$ of a rigid polyhedron there acts a force $F$. The point of application of $F$ is the mid-point of $e$. $F$ is perpendicular to $e$ and lies in the plane that bisects the interior angle included by the two faces of the polyhedron meeting in $e$. The magnitude of $F$ is $l \left|\cos (\alpha /2)\right|$, where $l$ denotes the length of $e$ and $\alpha$ the interior angle at the edge $e$ considered above. Finally $F$ is directed inwards or outwards according as the edge $e$ is convex or re-entrant, i.e. according to the sign of $\cos (\alpha /2)$. Under the system of all the forces $F$ the polyhedron is in equilibrium.

If at every element $ds$ of a closed, rigid space curve with continuous curvature there acts a force in the direction of the positive principal normal of magnitude $ds/r$, where $r$ denotes the radius of curvature, then the curve is in equilibrium.

Imagine a unit vector drawn from the origin of a coordinate system in the direction of the tangent at each point of a continuously differentiable space curve. The end-points of these unit vectors generate the spherical image of the space curve. If the space curve is closed, then its spherical image will be intersected by every great circle of the unit sphere.

Let the function $f(x)$ be defined in the interval $a \leq x \leq b$ and satisfy the following conditions:

1. $f(x)$ is positive in the interior and $=0$ at the end-points of the interval $a$, $b$;

2. $f(x)$ is twice continuously differentiable for $a < x < b$;

3. $f'(x)$ converges to $+\infty$ as $x \to a$, and to $-\infty$ as $x \to b$. Under these conditions the function

cannot be monotonic in the whole interval $a < x < b$, except in the case that

On a surface of revolution with continuous curvature there are always two different parallel circles with the same Gaussian curvature.

Let $a$, $b$, $c$ be the sides, and $\alpha$, $\beta$, $\gamma$ the angles of a triangle in radian measure, $a$ lying opposite to $\alpha$, $b$ to $\beta$ and $c$ to $\gamma$. Then we have

Equality holds in the first inequality if and only if the triangle is equilateral, and in the second inequality if and only if the triangle degenerates into a straight line counted twice.

Let $k_1, k_2, \ldots , k_6$ be the lengths of the six edges of a tetrahedron and $\alpha_1, \alpha_2, \ldots , \alpha_6$ the radian measures of the angles formed by the faces of the tetrahedron that meet at the respective edges. Then we have

The given bounds are the best possible.

Consider a triangle. Let $a$, $b$ and $c$ denote its sides and $A$ its area. Prove that

with equality only for equilateral triangles.

Consider a triangle. Let $a$, $b$ and $c$ denote its sides and $A$ its area. Prove that

with equality only for equilateral triangles.

(continued). If $p > 0$

with equality only for equilateral triangles.

(continued). If $p > 0$

with equality only for equilateral triangles.

(continued). Let $r$ and $R$ denote the radii of the inscribed and the circumscribed circle of a triangle, respectively. Prove that

with equality only for equilateral triangles.

(continued). Let $r$ and $R$ denote the radii of the inscribed and the circumscribed circle of a triangle, respectively. Prove that

with equality only for equilateral triangles.

Formulate a proposition that has an analogous relation to 17.1 as 17 has to 16.

Formulate a proposition that has an analogous relation to 17.1 as 17 has to 16.

(continued). Try to find simple particular cases for which the proposition formulated above can be conveniently tested.

(continued). Try to find simple particular cases for which the proposition formulated above can be conveniently tested.

Let $P_0 P_1 P_2$ be a triangle right-angled at $P_2$, $P_2 P_3$ the perpendicular from $P_2$ to $P_0 P_1$, similarly $P_3 P_4 \perp P_1 P_2$, $P_4 P_5 \perp P_2 P_3$, etc. Find the point $\lim_{n \to \infty } P_n$.

Let $K$ and $K'$ be two circles in space, each given as the intersection of a sphere and a plane

Find a rational entire function of the 16 real coefficients $a, b, \ldots , d, A, \ldots , D, a', \ldots , d', A', \ldots , D'$, that is $<0$ if and only if the two circles are interlocked i.e. their relative position is such that they have no common point, and, if conceived of as being made of wire, say, they cannot be taken apart without tearing.

Let $X_1$, $X_2$, $X_3$ be the projective coordinates of a variable point in the plane and let $X_0$ be defined by $X_0+X_1+X_2+X_3 = 0$. In the triply infinite family of conics

there are doubly infinitely many that degenerate to pairs of straight lines; and moreover every point of the plane is the singular point (intersection) of such a pair of straight lines. Find the integral curves of these pairs of straight lines. (At every point $P$ of the required curve the tangent coincides with one straight line of the pair of straight lines whose point of intersection is $P$.)

Let the point $P$ describe a plane curve. Let $\rho$ be the radius of curvature at $P$. Let the segment of the normal at $P$ lying between the two rectangular coordinate axes be of length $\nu$. Determine those curves for which the ratio of $\rho$ and $\nu$ is fixed:

The constant $n$ may be supposed positive, since $n$ is transformed into $-n$ if $x$ and $y$ are interchanged.---For integral $n$ there is a particular solution in the form of a rational curve of order $2n$ that has a point of intersection of multiplicity $2n$ with the line at infinity.

The family of surfaces of second order

with parameter $t$ has an enveloping surface $H$ of order 10 and class 4. Determine the lines of curvature of $H$.

(continued). Deduce a parametric representation of $H$ with the lines of curvature as parameter lines. The latter are algebraic curves of 12th order.

(continued). The central surface and the parallel surfaces of $H$ may be regarded as envelopes of a suitably chosen family of surfaces of second order.

Let $F(x, y)$ be a continuous function, periodic with period 1 in the two variables $x$, $y$; i.e.

Furthermore let $F(x, y)$ have the property (e.g. let it satisfy a Lipschitz condition) that through every point of the plane there passes one and only one infinitely continuable integral curve of the differential equation

Then there exists a number $\omega$ associated with the function $F(x, y)$ such that for every integral curve $y = f(x)$ the difference $f(x) - \omega x$ is bounded for all values of $x$.