Putnam 1982

Let $V$ be the region in the Cartesian plane consisting of all points $(x,y)$ satisfying the simultaneous conditions $$|x|\le y\le|x|+3\text{ and }y\le4.$$ Find the centroid of $V$ .

For positive real $x$ , let $$B_n(x)=1^x+2^x+\ldots+n^x.$$ Prove or disprove the convergence of $$\sum_{n=2}^\infty\frac{B_n(\log_n2)}{(n\log_2n)^2}.$$

Evaluate $$\int^\infty_0\frac{\operatorname{arctan}(\pi x)-\operatorname{arctan}(x)}xdx.$$

Assume that the system of differential equations $y'=-z^3$ , $z'=y^3$ with the initial conditions $y(0)=1$ , $z(0)=0$ has a unique solution $y=f(x)$ , $z=g(x)$ defined for real $x$ . Prove that there exists a positive constant $L$ such that for all real $x$ , $$f(x+L)=f(x),\enspace g(x+L)=g(x).$$

Let $\sigma$ be a bijection on the positive integers. Let $x_1,x_2,x_3,\ldots$ be a sequence of real numbers with the following three properties: $(\text i)$ $|x_n|$ is a strictly decreasing function of $n$ ; $(\text{ii})$ $|\sigma(n)-n|\cdot|x_n|\to0$ as $n\to\infty$ ; $(\text{iii})$ $\lim_{n\to\infty}\sum_{k=1}^nx_k=1$ .

Prove or disprove that these conditions imply that $$\lim_{n\to\infty}\sum_{k=1}^nx_{\sigma(k)}=1.$$

Let $M$ be the midpoint of side $BC$ of $\triangle ABC$ . Using the smallest possible $n$ , described a method for cutting $\triangle AMB$ into $n$ triangles which can be reassembled to form a triangle congruent to $\triangle AMC$ .

Let $A(x,y)$ be the number of points $(m,n)$ in the plane with integer coordinates $m$ and $n$ satisfying $m^2+n^2\le x^2+y^2$ . Let $g=\sum_{k=1}^\infty e^{-k^2}$ . Express $$\int^\infty_{-\infty}\int^\infty_{-\infty}A(x,y)e^{-x^2-y^2}dxdy$$ as a polynomial in $g$ .

Let $p_n$ be the probability that $c+d$ is a perfect square when the integers $c$ and $d$ are selected independently at random from the set $\{1,2,\ldots,n\}$ . Show that $\lim_{n\to\infty}p_n\sqrt n$ exists and express this limit in the form $r(\sqrt s-t)$ , where $s$ and $t$ are integers and $r$ is a rational number.

Let $n_1,n_2,\ldots,n_s$ be distinct integers such that $$(n_1+k)(n_2+k)\cdots(n_s+k)$$ is an integral multiple of $n_1n_2\cdots n_s$ for every integer $k$ . For each of the following assertions give a proof or a counterexample: $(\text a)$ $|n_i|=1$ for some $i$ $(\text b)$ If further all $n_i$ are positive, then $$\{n_1,n_2,\ldots,n_2\}=\{1,2,\ldots,s\}.$$

For each $x>e^e$ define a sequence $S_x=u_0,u_1,\ldots$ recursively as follows: $u_0=e$ , and for $n\ge0$ , $u_{n+1}=\log_{u_n}x$ . Prove that $S_x$ converges to a number $g(x)$ and that the function $g$ defined in this way is continuous for $x>e^e$ .