Putnam 1947

If $(a_n)$ is a sequence of real numbers such that for $n \geq 1$ $$(2-a_n )a_{n+1} =1,$$ prove that $\lim_{n\to \infty} a_n =1.$

A real valued continuous function $f$ satisfies for all real $x$ and $y$ the functional equation $$f(\sqrt{x^2 +y^2 })= f(x)f(y).$$ Prove that $$f(x) =f(1)^{x^{2}}.$$

Given a triangle $ABC$ with an interior point $P$ and points $Q_1 , Q_2$ not lying on any of the segments $AB , AC ,BC,$ $AP ,BP ,CP,$ show that there does not exist a polygonal line $K$ joining $Q_1$ and $Q_2$ such that i) $K$ crosses each segment exactly once, ii) $K$ does not intersect itself iii) $K$ does not pass through $A, B , C$ or $P.$

A coast artillery gun can fire at every angle of elevation between $0^{\circ}$ and $90^{\circ}$ in a fixed vertical plane. If air resistance is neglected and the muzzle velocity is constant ( $=v_0$ ), determine the set $H$ of points in the plane and above the horizontal which can be hit.

Let $a_1 , b_1 , c_1$ be positive real numbers whose sum is $1,$ and for $n=1, 2, \ldots$ we define $$a_{n+1}= a_{n}^{2} +2 b_n c_n, \;\; b_{n+1}= b_{n}^{2} +2 a_n c_n, \;\; c_{n+1}= c_{n}^{2} +2 a_n b_n.$$ Show that $a_n , b_n ,c_n$ approach limits as $n\to \infty$ and find those limits.

A $3\times 3$ matrix has determinant $0$ and the cofactor of any element is equal to the square of that element. Show that every element in the matrix is $0.$

Let $f(x)$ be a function such that $f(1)=1$ and for $x \geq 1$ $$f'(x)= \frac{1}{x^2 +f(x)^{2}}.$$ Prove that $$\lim_{x\to \infty} f(x)$$ exists and is less than $1+ \frac{\pi}{4}.$

Let $f(x)$ be a differentiable function defined on the interval $(0,1)$ such that $|f'(x)| \leq M$ for $0<x<1$ and a positive real number $M.$ Prove that $$\left| \int_{0}^{1} f(x) dx - \frac{1}{n} \sum_{k=1}^{n} f\left(\frac{k}{n} \right) \right | \leq \frac{M}{n}.$$

Given $P(z)= z^2 +az +b,$ where $a,b \in \mathbb{C}.$ Suppose that $|P(z)|=1$ for every complex number $z$ with $|z|=1.$ Prove that $a=b=0.$

Let $a,b,c,d$ be distinct integers such that $$(x-a)(x-b)(x-c)(x-d) -4=0$$ has an integer root $r.$ Show that $4r=a+b+c+d.$

Let $OX, OY, OZ$ be mutually orthogonal lines in space. Let $C$ be a fixed point on $OZ$ and $U,V$ variable points on $OX, OY,$ respectively. Find the locus of a point $P$ such that $PU, PV, PC$ are mutually orthogonal.