Olympiad problems

1989 Putnam, A6

Let $\alpha=1+a_1x+a_2x^2+\ldots$ be a formal power series with coefficients in the field of two elements. Let $$a_n=\begin{cases}1&\text{if every block of zeroes in the binary expansion of }n\text{ has an even number of zeroes}\\0&\text{otherwise}\end{cases}$$(For example, $a_{36}=1$ since $36=100100_2$) Prove that $\alpha^3+x\alpha+1=0$.

2021 ISI Entrance Examination, 3

Tags: power series , algebra , number theory

Prove that every positive rational number can be expressed uniquely as a finite sum of the form $$a_1+\frac{a_2}{2!}+\frac{a_3}{3!}+\dots+\frac{a_n}{n!},$$ where $a_n$ are integers such that $0 \leq a_n \leq n-1$ for all $n > 1$.

1992 Putnam, A2

Tags: integral , power series

Define $C(\alpha)$ to be the coefficient of $x^{1992}$ in the power series about $x = 0$ of $(1 + x)^{\alpha}$ . Evaluate $$\int_{0}^{1} \left( C(-y-1) \sum_{k=1}^{1992} \frac{1}{y+k} \right)\, dy.$$

1967 Putnam, A2

Tags: matrices , power series , generating functions

Define $S_0$ to be $1.$ For $n \geq 1 $, let $S_n $ be the number of $n\times n $ matrices whose elements are nonnegative integers with the property that $a_{ij}=a_{ji}$ (for $i,j=1,2,\ldots, n$) and where $\sum_{i=1}^{n} a_{ij}=1$ (for $j=1,2,\ldots, n$). Prove that a) $S_{n+1}=S_{n} +nS_{n-1}.$ b) $\sum_{n=0}^{\infty} S_{n} \frac{x^{n}}{n!} =\exp \left(x+\frac{x^{2}}{2}\right).$

1948 Putnam, A6

Tags: power series

Answer either (i) or (ii): (i) A force acts on the element $ds$ of a closed plane curve. The magnitude of this force is $r^{-1} ds$ where $r$ is the radius of curvature at the point considered, and the direction of the force is perpendicular to the curve, it points to the convex side. Show that the system of such forces acting on all elements of the curve keep it in equilibrium. (ii) Show that $$x+ \frac{2}{3}x^{3}+ \frac{2\cdot 4}{3\cdot 5} x^5 +\frac{2\cdot 4\cdot 6}{3\cdot 5\cdot 7} x^7 + \ldots= \frac{ \arcsin x}{\sqrt{1-x^{2}}}.$$

1994 Moldova Team Selection Test, 2

Tags: number theory , algebra , power series

Prove that every positive rational number can be expressed uniquely as a finite sum of the form $$a_1+\frac{a_2}{2!}+\frac{a_3}{3!}+\dots+\frac{a_n}{n!},$$ where $a_n$ are integers such that $0 \leq a_n \leq n-1$ for all $n > 1$.

1970 Putnam, A1

Tags: function , trigonometry , power series

Show that the power series for the function $$e^{ax} \cos bx,$$ where $a,b >0$, has either no zero coefficients or infinitely many zero coefficients.