Putnam 1962

Find every real-valued function $f$ whose domain is an interval $I$ (finite or infinite) having $0$ as a left-hand endpoint, such that for every positive $x\in I$ the average of $f$ over the closed interval $[0,x]$ is equal to $\sqrt{ f(0) f(x)}.$

In a triangle $ABC$ , let $A'$ be a point on the segment $BC$ , $B'$ be a point on the segment $CA$ and $C'$ a point on the segment $AB$ such that $$\frac{AB'}{B'C}= \frac{BC'}{C'A} =\frac{CA'}{A'B}=k,$$ where $k$ is a positive constant. Let $\triangle$ be the triangle formed by the interesctions of $AA'$ , $BB'$ and $CC'$ . Prove that the areas of $\triangle$ and $ABC$ are in the ratio $$\frac{(k-1)^{2}}{k^2 +k+1}.$$

Assume that $|f(x)|\leq 1$ and $|f''(x)|\leq 1$ for all $x$ on an interval of length at least $2.$ Show that $|f'(x)|\leq 2$ on the interval.

Evaluate $$\sum_{k=0}^{n} \binom{n}{k}k^{2}.$$

Let $S$ be a set of rational numbers such that whenever $a$ and $b$ are members of $S$ , so are $ab$ and $a+b$ , and having the property that for every rational number $r$ exactly one of the following three statements is true: $$r\in S,\; -r\in S,\; r =0.$$ Prove that $S$ is the set of all positive rational numbers.

Let $x^{(n)}=x(x-1)\cdots (x-n+1)$ for $n$ a positive integer and let $x^{(0)}=1.$ Prove that $$(x+y)^{(n)}= \sum_{k=0}^{n} \binom{n}{k} x^{(k)} y^{(n-k)}.$$

Let $S$ be the set of all subsets of the positive integers. Construct a function $f \colon \mathbb{R} \rightarrow S$ such that $f(a)$ is a proper subset of $f(b)$ whenever $a <b.$

Let $S$ be a convex region in the euclidean plane containing the origin. Assume that every ray from the origin has at least one point outside $S$ . Prove that $S$ is bounded.

The euclidean plane is divided into regions by drawing a finite number of circles. Show that it is possible to color each of these regions either red or blue in such a way that no two adjacent regions have the same color.

Prove that for every integer $n$ greater than $1:$ $$\frac{3n+1}{2n+2} < \left( \frac{1}{n} \right)^{n} + \left( \frac{2}{n} \right)^{n}+ \ldots+\left( \frac{n}{n} \right)^{n} <2.$$

Let $$f(x) =\sum_{k=0}^{n} a_{k} \sin kx +b_{k} \cos kx,$$ where $a_k$ and $b_k$ are constants. Show that if $|f(x)| \leq 1$ for $x \in [0, 2 \pi]$ and there exist $0\leq x_1 < x_2 <\ldots < x_{2n} < 2 \pi$ with $|f(x_i )|=1,$ then $f(x)= \cos(nx +a)$ for some constant $a.$