Putnam 1981

Let $E(n)$ denote the largest integer $k$ such that $5^k$ divides $1^{1}\cdot 2^{2} \cdot 3^{3} \cdot \ldots \cdot n^{n}.$ Calculate $$\lim_{n\to \infty} \frac{E(n)}{n^2 }.$$

Two distinct squares of the $8\times8$ chessboard $C$ are said to be adjacent if they have a vertex or side in common. Also, $g$ is called a $C$ -gap if for every numbering of the squares of $C$ with all the integers $1, 2, \ldots, 64$ there exist twoadjacent squares whose numbers differ by at least $g$ . Determine the largest $C$ -gap $g$ .

Find $$\lim_{t\to \infty} e^{-t} \int_{0}^{t} \int_{0}^{t} \frac{e^x -e^y }{x-y} dx dy,$$ or show that the limit does not exist.

A point $P$ moves inside a unit square in a straight line at unit speed. When it meets a corner it escapes. When it meets an edge its line of motion is reflected so that the angle of incidence equals the angle of reflection. Let $N( t)$ be the number of starting directions from a fixed interior point $P_0$ for which $P$ escapes within $t$ units of time. Find the least constant $a$ for which constants $b$ and $c$ exist such that $$N(t) \leq at^2 +bt+c$$ for all $t>0$ and all initial points $P_0 .$

Let $P(x)$ be a polynomial with real coefficients and form the polynomial $$Q(x) = ( x^2 +1) P(x)P'(x) + x(P(x)^2 + P'(x)^2 ).$$ Given that the equation $P(x) = 0$ has $n$ distinct real roots exceeding $1$ , prove or disprove that the equation $Q(x)=0$ has at least $2n - 1$ distinct real roots.

Suppose that each of the vertices of $ABC$ is a lattice point in the $xy$ -plane and that there is exactly one lattice point $P$ in the interior of the triangle. The line $AP$ is extended to meet $BC$ at $E$ . Determine the largest possible value for the ratio of lengths of segments $$\frac{|AP|}{|PE|}.$$

Find $$\lim_{n\to \infty} \frac{1}{n^5 } \sum_{h=1}^{n} \sum_{k=1}^{n} (5h^4 -18h^2 k^2 +5k^4).$$

Determine the minimum value of $$(r-1)^2 + \left(\frac{s}{r}-1 \right)^2 + \left(\frac{t}{s}-1 \right)^{2} + \left( \frac{4}{t} -1 \right)^2$$ for all real numbers $1\leq r \leq s \leq t \leq 4.$

Let $V$ be a set of $5\times7$ matrices, with real entries and closed under addition and scalar multiplication. Prove or disprove the following assertion: If $V$ contains matrices of ranks $0, 1, 2, 4,$ and $5$ , then it also contains a matrix of rank $3$ .

Let $B(n)$ be the number of ones in the base two expression for the positive integer $n.$ Determine whether $$\exp \left( \sum_{n=1}^{\infty} \frac{ B(n)}{n(n+1)} \right)$$ is a rational number.

Let $C$ be a fixed unit circle in the cartesian plane. For any convex polygon $P$ , each of whose sides is tangent to $C$ , let $N( P, h, k)$ be the number of points common to $P$ and the unit circle with center at $(h, k).$ Let $H(P)$ be the region of all points $(x, y)$ for which $N(P, x, y) \geq 1$ and $F(P)$ be the area of $H(P).$ Find the smallest number $u$ with $$\frac{1}{F(P)} \int \int N(P,x,y) dx dy <u$$ for all polygons $P$ , where the double integral is taken over $H(P).$