Putnam 1941

Prove that the polynomial $$(a-x)^6 -3a(a-x)^5 +\frac{5}{2} a^2 (a-x)^4 -\frac{1}{2} a^4 (a-x)^2$$ takes only negative values for $0<x<a$ .

Find the $n$ -th derivative with respect to $x$ of $$\int_{0}^{x} \left(1+\frac{x-t}{1!}+\frac{(x-t)^{2}}{2!}+\ldots+\frac{(x-t)^{n-1}}{(n-1)!}\right)e^{nt} dt.$$

A circle of radius $a$ rolls in the plane along the $x$ -axis. Show that the envelope of a diameter is a cycloid, coinciding with the cycloid traced out by a point on the circumference of a circle of diameter $a$ , likewise rolling in the plane along the $x$ -axis.

Let the roots $a,b,c$ of $$f(x)=x^3 +p x^2 + qx+r$$ be real, and let $a\leq b\leq c$ . Prove that $f'(x)$ has a root in the interval $\left[\frac{b+c}{2}, \frac{b+2c}{3}\right]$ . What will be the form of $f(x)$ if the root in question falls at either end of the interval?

The line $L$ is parallel to the plane $y=z$ and meets the parabola $2x=y^2 ,z=0$ and the parabola $3x=z^2, y=0$ . Prove that if $L$ moves freely subject to these constraints then it generates the surface $x=(y-z)\left(\frac{y}{2}-\frac{z}{3}\right)$ .

If the $x$ -coordinate $\overline{x}$ of the center of mass of the area lying between the $x$ -axis and the curve $y=f(x)$ with $f(x)>0$ , and between the lines $x=0$ and $x=a$ is given by $$\overline{x}=g(a),$$ show that $$f(x)=A\cdot \frac{g'(x)}{(x-g(x))^{2}} \cdot e^{\int \frac{1}{t-g(t)} dt},$$ where $A$ is a positive constant.

A particle $(x,y)$ moves so that its angular velocities about $(1,0)$ and $(-1,0)$ are equal in magnitude but opposite in sign. Prove that $$y(x^2 +y^2 +1) dx= x(x^2 +y^2 -1) dy,$$ and verify that this is the differential equation of the family of rectangular hyperbolas passing through $(1,0)$ and $(-1,0)$ and having the origin as center.

Find (i) $\lim_{n\to \infty} \sum_{i=1}^{n} \frac{1}{\sqrt{n^2 +i^{2}}}$ . (ii) $\lim_{n\to \infty} \sum_{i=1}^{n} \frac{1}{\sqrt{n^2 +i}}$ . (iii) $\lim_{n\to \infty} \sum_{i=1}^{n^{2}} \frac{1}{\sqrt{n^2 +i}}$ .

Let $y_1$ and $y_2$ be two linearly independent solutions of the equation $$y''+P(x)y'+Q(x)=0.$$ Find the differential equation satisfied by the product $z=y_1 y_2$ .

Given two perpendicular diameters $AB$ and $CD$ of an ellipse, we say that the diameter $A'B'$ is conjugate to $AB$ if $A'B'$ is parallel to the tangent to the ellipse at $A$ . Let $A'B'$ be conjugate to $AB$ and $C'D'$ be conjugate to $CD$ . Prove that the rectangular hyperbola through $A', B', C'$ and $D'$ passes through the foci of the ellipse.

A car is being driven so that its wheels, all of radius $a$ feet, have an angular velocity of $\omega$ radians per second. A particle is thrown off from the tire of one of these wheels, where it is supposed that $a \omega^{2} >g$ . Neglecting the resistance of the air, show that the maximum height above the roadway which the particle can reach is $$\frac{(a \omega+g \omega^{-1})^{2}}{2g}.$$

Assuming that $f(x)$ is continuous in the interval $(0,1)$ , prove that $$\int_{x=0}^{x=1} \int_{y=x}^{y=1} \int_{z=x}^{z=y} f(x)f(y)f(z) dz dy dx= \frac{1}{6}\left(\int_{0}^{1} f(t) dt\right)^{3}.$$