Subcontests
(6)Putnam 1976 A5
In the (x,y)−plane, if R is the set of points inside and on a convex polygon, let D(x,y) be the distance from (x,y) to the nearest point of R.(a) Show that there exists constants a,b,c, independent of R, such that ∫−∞∞∫−∞∞e−D(x,y)dxdy=a+bL+cA, where L is the perimeter of R and A is the area of R.(b) Find the values of a,b and c. Putnam 1976 A4
Let r be a root of P(x)=x3+ax2+bx−1=0 and r+1 be a root of y3+cy2+dy+1=0, where a,b,c and d are integers. Also let P(x) be irreducible over the rational numbers. Express another root s of P(x)=0 as a function of r which does not explicitly involve a,b,c or d. Putnam 1976 B3
Suppose that we have n events A1,…,An, each of which has probability at least 1−a of occuring, where a<1/4. Further suppose that Ai and Aj are mutually independent if ∣i−j∣>1. Assume as known that the recurrence uk+1=uk−auk−1,u0=1,u1=1−a, defines positive real numb uk for k=0,1,…. Show that the probability of all of A1,…,An occuring is at least un. Putnam 1976 A2
Let P(x,y)=x2y+xy2 and Q(x,y)=x2+xy+y2. For n=1,2,3,…, let \begin{align*}F_n(x,y)=(x+y)^n-x^n-y^n \text{ and,}\\ G_n(x,y)=(x+y)^n+x^n+y^n. \end{align*} One observes that G2=2Q,F3=3P,G4=2Q2,F5=5PQ,G6=2Q3+3P2. Prove that, in fact, for each n either Fn or Gn is expressible as a polynomial in P and Q with integer coefficients. Putnam 1976 B2
Suppose that G is a group generated by elements A and B, that is, every element of G can be written as a finite "word" An1Bn2An3…Bnk, where n1,…nk are any integers, and A0=B0=1 as usual. Also suppose that A4=B7=ABA−1B=1,A2=1, and B=1.(a) How many elements of G are of the form C2 with C in G?
(b) Write each such square as a word in A and B.