Putnam 1942

A square of side $2a$ , lying always in the first quadrant of the $xy$ -plane, moves so that two consecutive vertices are always on the $x$ - and $y$ -axes respectively. Find the locus of the midpoint of the square.

If a polynomial $f(x)$ is divided by $(x-a)^{2} (x-b)$ , where $a\ne b$ , derive a formula for the remainder.

Is the series $$\sum_{n=0}^{\infty} \frac{n!}{(n+1)^{n}}\cdot \left(\frac{19}{7}\right)^{n}$$ convergent or divergent?

Find the orthogonal trajectories of the family of conics $(x+2y)^{2} = a(x+y)$ . At what angle do the curves of one family cut the curves of the other family at the origin?

A circle of radius $a$ is revolved through $180^{\circ}$ about a line in its plane, distant $b$ from the center of the circle, where $b>a$ . For what value of the ratio $\frac{b}{a}$ does the center of gravity of the solid thus generated lie on the surface of the solid?

Any circle in the $xy$ -plane is "represented" by a point on the vertical line through the center of the circle and at a distance "above" the plane of the circle equal to the radius of the circle. Show that the locus of the representations of all the circles which cut a fixed circle at a constant angle is a portion of a one-sheeted hyperboloid. By consideration of a suitable family of circles in the plane, demonstrate the existence of two families of rulings on the hyperboloid.

A square of side $2a$ , lying always in the first quadrant of the $xy$ -plane, moves so that two consecutive vertices are always on the $x$ - and $y$ -axes respectively. Prove that a point within or on the boundary of the square will in general describe a portion of a conic. For what points of the square does this locus degenerate?

For the family of parabolas $$y= \frac{ a^3 x^{2}}{3}+ \frac{ a^2 x}{2}-2a$$ (i) find the locus of vertices, (ii) find the envelope, (iii) sketch the envelope and two typical curves of the family.

Given $x=\phi(u,v)$ and $y=\psi(u,v)$ , where $\phi$ and $\psi$ are solutions of the partial differential equation $$(1) \;\,\; \frac{ \partial \phi}{\partial u} \frac{\partial \psi}{ \partial v} - \frac{ \partial \phi}{\partial v} \frac{\partial \psi}{ \partial u}=1.$$ By assuming that $x$ and $y$ are the independent variables, show that $(1)$ may be transformed to $$(2) \;\,\; \frac{ \partial y}{ \partial v} =\frac{ \partial u}{\partial x}.$$ Integrate $(2)$ and show how this effects in general the solution of $(1)$ . What other solutions does $(1)$ possess?

A particle moves under a central force inversely proportional to the $k$ -th power of the distance. If the particle describes a circle ( the central force proceeding from a point on the circumference of the circle ), find $k$ .

Sketch the curve $$y= \frac{x}{1+x^6 (\sin x)^{2}},$$ and show that $$\int_{0}^{\infty} \frac{x}{1+x^6 (\sin x)^{2}} dx$$ exists.