Putnam 1940

Prove that if $f(x)$ is a polynomial with integer coefficients and there exists an integer $k$ such that none of $f(1),\ldots,f(k)$ is divisible by $k$ , then $f(x)$ has no integral root.

Let $A,B$ be two fixed points on the curve $y=f(x)$ , $f$ is continuous with continuous derivative and the arc $\widehat{AB}$ is concave to the chord $AB$ . If $P$ is a point on the arc $\widehat{AB}$ for which $AP+PB$ is maximal, prove that $PA$ and $PB$ are equally inclined to the tangent to the curve $y=f(x)$ at $P$ .

Let $a$ be a real number. Find all real-valued functions $f$ such that $$\int f(x)^{a} dx=\left( \int f(x) dx \right)^{a}$$ when constants of integration are suitably chosen.

Let $p$ be a real constant. The parabola $y^2=-4px$ rolls without slipping around the parabola $y^2=4px$ . Find the equation of the locus of the vertex of the rolling parabola.

Prove that the simultaneous equations $$x^4 -x^2 =y^4 -y^2 =z^4 -z^2$$ are satisfied by the points of $4$ straight lines and $6$ ellipses, and by no other points.

Let $f(x)$ be a polynomial of degree $n$ such that $f(x)^{p}$ is divisible by $f'(x)^{q}$ for some positive integers $p,q$ . Prove that $f(x)$ is divisible by $f'(x)$ and that $f(x)$ has a single root of multiplicity $n$ .

A projectile, thrown with initial velocity $v_0$ in a direction making angle $\alpha$ with the horizontal, is acted on by no force except gravity. Find the lenght of its path until it strikes a horizontal plane through the starting point. Show that the flight is longest when $$\sin \alpha \log(\sec \alpha+ \tan \alpha)=1.$$

A cylindrical hole of radius $r$ is bored through a cylinder of radiues $R$ ( $r\leq R$ ) so that the axes intersect at right angles. i) Show that the area of the larger cylinder which is inside the smaller one can be expressed in the form $$S=8r^2\int_{0}^{1} \frac{1-v^{2}}{\sqrt{(1-v^2)(1-m^2 v^2)}} dv,\; \text{where} \; m=\frac{r}{R}.$$ ii) If $K=\int_{0}^{1} \frac{1}{\sqrt{(1-v^2)(1-m^2 v^2)}} dv$ and $E=\int_{0}^{1} \sqrt{\frac{1-m^2 v^2}{1-v^2 }}dv$ . show that $$S=8[R^2 E - (R^2 - r^2 )K].$$

Let $p>0$ be a real constant. From any point $(a,b)$ in the cartesian plane, show that i) Three normals, real or imaginary, can be drawn to the parabola $y^2=4px$ . ii) These are real and distinct if $4(2-p)^3 +27pb^2<0$ . iii) Two of them coincide if $(a,b)$ lies on the curve $27py^2=4(x-2p)^3$ . iv) All three coincide only if $a=2p$ and $b=0$ .

Prove that the locus of the point of intersection of three mutually perpendicular planes tangent to the surface $$ax^2 + by^2 +cz^2 =1\;\; (\text{where}\; abc \ne 0)$$ is the sphere $$x^2 +y^2 +z^2 =\frac{1}{a}+\frac{1}{b}+\frac{1}{c}.$$

Prove that the determinant of the matrix $$\begin{pmatrix} a_{1}^{2}+k & a_1 a_2 & a_1 a_3 &\ldots & a_1 a_n a_2 a_1 & a_{2}^{2}+k & a_2 a_3 &\ldots & a_2 a_n \ldots & \ldots & \ldots & \ldots & \ldots a_n a_1& a_n a_2 & a_n a_3 & \ldots & a_{n}^{2}+k \end{pmatrix}$$ is divisible by $k^{n-1}$ and find its other factor.