Putnam 1983

How many positive integers $n$ are there such that $n$ is an exact divisors of at least one of the numbers $10^{40}$ and $20^{30}$ ?

Let $p$ be an odd prime and let $$F(n)=1+2n+3n^2+\ldots+(p-1)n^{p-2}.$$ Prove that if $a$ and $b$ are distinct integers in $\{0,1,2,\ldots,p-1\}$ then $F(a)$ and $F(b)$ are not congruent modulo $p$ .

Let $k$ be a positive integer and let $m=6k-1$ . Let $$S(m)=\sum_{j=1}^{2k-1}(-1)^{j+1}\binom m{3j-1}.$$ Prove that $S(m)$ is never zero.

Prove or disprove that there exists a positive real $u$ such that $\lfloor u^n\rfloor-n$ is an even integer for all positive integers $n$ .

Let $$F(x)=\frac{x^4}{\exp(x^3)}\int^x_0\int^{x-u}_0\exp(u^3+v^3)dvdu.$$ Find $\lim_{x\to\infty}F(x)$ or prove that it does not exist.

Let $v$ be a vertex of a cube $C$ with edges of length $4$ . Let $S$ be the largest sphere that can be inscribed in $C$ . Let $R$ be the region consisting of all points $p$ between $S$ and $C$ such that $p$ is closer to $v$ than to any other vertex of the cube. Find the volume of $R$ .

For positive integers $n$ , let $C(n)$ be the number of representation of $n$ as a sum of nonincreasing powers of $2$ , where no power can be used more than three times. For example, $C(8)=5$ since the representations of $8$ are: $$8,4+4,4+2+2,4+2+1+1,\text{ and }2+2+2+1+1.$$ Prove or disprove that there is a polynomial $P(x)$ such that $C(n)=\lfloor P(n)\rfloor$ for all positive integers $n$ .

Assume that the differential equation $$y'''+p(x)y''+q(x)y'+r(x)y=0$$ has solutions $y_1(x)$ , $y_2(x)$ , $y_3(x)$ on the real line such that $$y_1(x)^2+y_2(x)^2+y_3(x)^2=1$$ for all real $x$ . Let $$f(x)=y_1'(x)^2+y_2'(x)^2+y_3'(x)^2.$$ Find constants $A$ and $B$ such that $f(x)$ is a solution to the differential equation $$y'+Ap(x)y=Br(x).$$

Problem. Let $f:\mathbb{R}_0^+\rightarrow\mathbb{R}_0^+$ be a function defined as $$f(n)=n+\lfloor\sqrt{n}\rfloor~\forall~n\in\mathbb{R}_0^+.$$ Prove that for any positive integer $m,$ the sequence $$m,f(m),f(f(m)),f(f(f(m))),\ldots$$ contains a perfect square.

Let $\lVert u\rVert$ denote the distance from the real number $u$ to the nearest integer. For positive integers $n$ , let $$a_n=\frac1n\int^n_1\left\lVert\frac nx\right\rVert dx.$$ Determine $\lim_{n\to\infty}a_n$ .