Subcontests
(12)Putnam 1978 B4
Prove that for every real number N the equation
x12+x22+x32+x42=x1x2x3+x1x2x4+x1x3x4+x2x3x4
has an integer solution (x1,x2,x3,x4) for which x1,x2,x3 and x4 are all larger than N. Putnam 1978 B6
Let p and n be positive integers. Suppose that the numbers chk (h=1,2,…,n ; k=1,2,…,ph) satisfy 0≤chk≤1. Prove that
(∑hchk)2≤2p∑chk,
where each summation is over all admissible ordered pairs (h,k). Putnam 1978 A4
A bypass operation on a set S is a mapping B:S×S→S with the property B(B(w,x),B(y,z))=B(w,z) for all w,x,y,z∈S.
(a) Prove that B(a,b)=c implies B(c,c)=c when B is a bypass.
(b) Prove that B(a,b)=c implies B(a,x)=B(c,x) for all x∈S when B is a bypass.
(c) Construct a bypass operation B on a finite set S with the following three properties[*] B(x,x)=x for all x∈S.
[*] There exist d and e in S with B(d,e)=d=e.
[*] There exist f and g in S with B(f,g)=f.
Putnam 1978 A2
Let a,b,p1,p2,…,pn be real numbers with a=b. Define f(x)=(p1−x)(p2−x)⋯(pn−x). Show that
detp1bb⋮bap2b⋮baap3⋮b⋯⋯⋯⋱⋯aaa⋮pn=b−abf(a)−af(b).
Putnam 1978 A1
Let A be any set of 20 distinct integers chosen from the arithmetic progression 1,4,7,…,100. Prove that there must be two distinct integers in A whose sum is 104.