Putnam 1946

Suppose that the function $f(x)=a x^2 +bx+c$ , where $a,b,c$ are real, satisfies the condition $|f(x)|\leq 1$ for $|x|\leq1$ . Prove that $|f'(x)|\leq 4$ for $|x|\leq1$ .

If $a(x), b(x), c(x)$ and $d(x)$ are polynomials in $x$ , show that $$\int_{1}^{x} a(x) c(x) dx \cdot \int_{1}^{x} b(x) d(x) dx - \int_{1}^{x} a(x) d(x) dx \cdot \int_{1}^{x} b(x) c(x) dx$$ is divisible by $(x-1)^4.$

A projectile in flight is observed simultaneously from four radio stations which are situated at the corners of a square of side $b$ . The distances of the projectile from the four stations, taken in order around the square, are found to be $R_1 , R_2 , R_3$ and $R_4$ . Show that $$R_{1}^{2}+ R_{3}^{2}= R_{2}^{2}+ R_{4}^{2}.$$ Show also that the height $h$ of the projectile above the ground is given by $$h^{2}=- \frac{1}{2} b^2 +\frac{1}{4}(R_{1}^{2}+R_{2}^{2}+R_{3}^{2}+R_{4}^{2}) -\frac{1}{8 b^{2}}(R_{1}^{4}+R_{2}^{4}+R_{3}^{4}+R_{4}^{4}- 2 R_{1}^{2}R_{3}^{2} -2 R_{2}^{2} R_{4}^{2}).$$

Let $g(x)$ be a function that has a continuous first derivative $g'(x)$ . Suppose that $g(0)=0$ and $|g'(x)| \leq |g(x)|$ for all values of $x.$ Prove that $g(x)$ vanishes identically.

Find the smallest volume bounded by the coordinate planes and by a tangent plane to the ellipsoid $$\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}}=1.$$

A particle of unit mass moves on a straight line under the action of a force which is a function $f(v)$ of the velocity $v$ of the particle, but the form of the function is not known. A motion is observed, and the distance $x$ covered in time $t$ satisfies the formula $x= at^2 + bt+c$ , where $a,b,c$ have numerical values determined by observation of the motion. Find the function $f(v)$ for the range of $v$ covered by the experiment.

Let $K$ denote the circumference of a circular disk of radius $1$ , and let $k$ denote a circular arc that joins two points $a,b$ on $K$ and lies otherwise in the given circular disc. Suppose that $k$ divides the circular disk into two parts of equal area. Prove that the length of $k$ exceeds $2.$

Let $A, B$ be two variable points on a parabola $P_{0}$ , such that the tangents at $A$ and $B$ are perpendicular to each other. Show that the locus of the centroid of the triangle formed by $A,B$ and the vertex of $P_0$ is a parabola $P_1 .$ Apply the same process to $P_1$ and repeat the process, obtaining the sequence of parabolas $P_1, P_2 , \ldots, P_n$ . If the equation of $P_0$ is $y=m x^2$ , find the equation of $P_n .$

In a solid sphere of radius $R$ the density $\rho$ is a function of $r$ , the distance from the center of the sphere. If the magnitude of the gravitational force of attraction due to the sphere at any point inside the sphere is $k r^2$ , where $k$ is a constant, find $\rho$ as a function of $r.$ Find also the magnitude of the force of attraction at a point outside the sphere at a distance $r$ from the center.

For each positive integer $n$ , put $$p_n =\left(1+\frac{1}{n}\right)^{n}, P_n =\left(1+\frac{1}{n}\right)^{n+1}, h_n = \frac{2 p_n P_{n}}{ p_n + P_n }.$$ Prove that $h_1 < h_2 < h_3 <\ldots$

Show that $\lceil (\sqrt{3}+1)^{2n})\rceil$ is divisible by $2^{n+1}.$

A particle moves on a circle with center $O$ , starting from rest at a point $P$ and coming to rest again at a point $Q$ , without coming to rest at any intermediate point. Prove that the acceleration vector of the particle does not vanish at any point between $P$ and $Q$ and that, at some point $R$ between $P$ and $Q$ , the acceleration vector points in along the radius $RO.$