Subcontests
(12)Putnam 2019 B4
Let F be the set of functions f(x,y) that are twice continuously differentiable for x≥1, y≥1 and that satisfy the following two equations (where subscripts denote partial derivatives):
xfx+yfy=xyln(xy), x2fxx+y2fyy=xy.
For each f∈F, let
m(f)=s≥1min(f(s+1,s+1)−f(s+1,s)−f(s,s+1)+f(s,s)).
Determine m(f), and show that it is independent of the choice of f. Putnam 2019 B1
Denote by Z2 the set of all points (x,y) in the plane with integer coordinates. For each integer n≥0, let Pn be the subset of Z2 consisting of the point (0,0) together with all points (x,y) such that x2+y2=2k for some integer k≤n. Determine, as a function of n, the number of four-point subsets of Pn whose elements are the vertices of a square. Putnam 2019 A4
Let f be a continuous real-valued function on R3. Suppose that for every sphere S of radius 1, the integral of f(x,y,z) over the surface of S equals zero. Must f(x,y,z) be identically zero? Putnam 2019 A3
Given real numbers b0,b1,…,b2019 with b2019=0, let z1,z2,…,z2019 be the roots in the complex plane of the polynomial
P(z)=k=0∑2019bkzk.
Let μ=(∣z1∣+⋯+∣z2019∣)/2019 be the average of the distances from z1,z2,…,z2019 to the origin. Determine the largest constant M such that μ≥M for all choices of b0,b1,…,b2019 that satisfy
1≤b0<b1<b2<⋯<b2019≤2019. Putnam 2019 B5
Let Fm be the m'th Fibonacci number, defined by F1=F2=1 and Fm=Fm−1+Fm−2 for all m≥3. Let p(x) be the polynomial of degree 1008 such that p(2n+1)=F2n+1 for n=0,1,2,…,1008. Find integers j and k such that p(2019)=Fj−Fk.