Putnam 1938

1. A solid is bounded by two bases in the horizontal planes $z=h / 2$ and $z$ $=-h / 2$, and by such a surface that the area of every section in a horizontal plane is given by a formula of the sort $$\text { Area }=a_{0} z^{3}+a_{1} z^{2}+a_{2} z+a_{3}$$ (where as special cases some of the coefficients may be 0 ). Show that the volume is given by the formula $$V=\frac{1}{6} h\left[B_{1}+B_{2}+4 M\right]$$ where $B_{1}$ and $B_{2}$ are the areas of the bases, and $M$ is the area of the middle horizontal section. Show that the formulas for the volume of a cone and of a sphere can be included in this formula when $a_{0}=0$.

2. A can buoy is to be made of three pieces, namely, a cylinder and two equal cones, the altitude of each cone being equal to the altitude of the cylinder. For a given area of surface, what shape will have the greatest volume?

3. If a particle moves in the plane, we may express its coordinates $x$ and $y$ as functions of the time $t$. If $x=t^{3}-t$ and $y=t^{4}+t$, show that the curve has a point of inflection at $t=0$ and that the velocity of the moving particle has a maximum at $t=0$.

4. A lumberman wishes to cut down a tree whose trunk is cylindrical and whose material is uniform. He will cut a notch, the two sides of which will be planes intersecting at a dihedral angle $\theta$ along a horizontal line through the axis of the cylinder. If $\theta$ is given, show that the least volume of material is cut out when the plane bisecting the dihedral angle is horizontal.

5. Evaluate the following limits: (i) $\lim _{n \rightarrow \infty} \frac{n^{2}}{e^{n}}$. (ii) $\lim _{x \rightarrow 0} \frac{1}{x} \int_{0}^{x}(1+\sin 2 t)^{1 / t} d t$.

6. A swimmer stands at one corner of a square swimming pool and wishes to reach the diagonally opposite corner. If $w$ is his walking speed and $s$ is his swimming speed $(s<w)$, find his path for shortest time. [Consider two cases: (i) $w / s<\sqrt{2}$, and (ii) $w / s>\sqrt{2}$.]

8. Take either (i) or (ii). (i) Let $A_{i k}$ be the cofactor of $a_{i k}$ in the determinant $$\boldsymbol{d}=\left|\begin{array}{llll} a_{11} & a_{12} & a_{13} & a_{14} \\ a_{21} & a_{22} & a_{23} & a_{24} \\ a_{31} & a_{32} & a_{33} & a_{34} \\ a_{41} & a_{42} & a_{43} & a_{44} \end{array}\right|$$

Let $D$ be the corresponding determinant with $a_{i k}$ replaced by $A_{i k}$. Prove $D=$ $d^{3}$. (page 86) (ii) Let $P(y)=A y^{2}+B y+C$ be a quadratic polynomial in $y$. If the roots of the quadratic equation $P(y)-y=0$ are $a$ and $b(a \neq b)$, show that $a$ and $b$ are roots of the biquadratic equation $P[P(y)]-y=0$. Hence write down a quadratic equation which will give the other two roots, $c$ and $d$, of the biquadratic. Apply this result to solving the following biquadratic equation: $$\left(y^{2}-3 y+2\right)^{2}-3\left(y^{2}-3 y+2\right)+2-y=0$$

9. Find all the solutions of the equation $$y y^{\prime \prime}-2\left(y^{\prime}\right)^{2}=0$$ which pass through the point $x=1, y=1$.

10. A horizontal disc of diameter 3 inches is rotating at 4 revolutions per minute. A light is shining at a distant point in the plane of the disc. An insect is placed at the edge of the disc furthest from the light, facing the light. It at once starts crawling, and crawls so as always to face the light, at 1 inch per second. Set up the differential equation of motion, and find at what point the insect again reaches the edge of the disc.

11. Given the parabola $y^{2}=2 m x$, what is the length of the shortest chord that is normal to the curve at one end?

12. From the center of a rectangular hyperbola a perpendicular is dropped upon a variable tangent. Find the locus of the foot of the perpendicular. Obtain the equation of the locus in polar coordinates, and sketch the curve.

13. Find the shortest distance between the plane $A x+B y+C z+1=0$ and the ellipsoid $x^{2} / a^{2}+y^{2} / b^{2}+z^{2} / c^{2}=1$. (For brevity, let $$\left.h=1 / \sqrt{A^{2}+B^{2}+C^{2}} \text { and } m=\sqrt{a^{2} A^{2}+b^{2} B^{2}+c^{2} C^{2}} .\right)$$

State algebraically the condition that the plane shall lie outside the ellipsoid.