Subcontests
(12)Putnam 1950 B6
Consider the closed plane curves Ci and Co, their respective lengths ∣Ci∣ and ∣Co∣, the closed surfaces Si and So, and their respective areas ∣Si∣ and ∣So∣. Assume that Ci lies inside Co and Si inside So. (Subscript i stands for "inner," o for "outer.") Prove the correct assertions among the following four, and disprove the others.(i) If Ci is convex, ∣Ci∣≤∣Co∣.
(ii) If Si is convex, ∣Si∣≤∣So∣.
(iii) If Co is the smallest convex curve containing Ci, then ∣Co∣≤∣Ci∣.
(iv) If So is the smallest convex surface containing Si, then ∣So∣≤∣Si∣.You may assume that Ci and Co are polygons and Si and So polyhedra. Putnam 1950 B4
The cross-section of a right cylinder is an ellipse, with semi-axes a and b, where a>b. The cylinder is very long, made of very light homogeneous material. The cylinder rests on the horizontal ground which it touches along the straight line joining the lower endpoints of the minor axes of its several cross-sections. Along the upper endpoints of these minor axes lies a very heavy homogeneous wire, straight and just as long as the cylinder. The wire and the cylinder are rigidly connected. We neglect the weight of the cylinder, the breadth of the wire, and the friction of the ground.The system described is in equilibrium, because of its symmetry. This equilibrium seems to be stable when the ratio b/a is very small, but unstable when this ratio comes close to 1. Examine this assertion and find the value of the ratio b/a which separates the cases of stable and unstable equilibrium.
Putnam 1950 A5
A function D(n) of the positive integral variable n is defined by the following properties: D(1)=0,D(p)=1 if p is a prime, D(uv)=uD(v)+vD(u) for any two positive integers u and v. Answer all three parts below.(i) Show that these properties are compatible and determine uniquely D(n). (Derive a formula for D(n)/n, assuming that n=p1α1p2α2⋯pkαk where p1,p2,…,pk are different primes.)(ii) For what values of n is D(n)=n?(iii) Define D2(n)=D[D(n)], etc., and find the limit of Dm(63) as m tends to ∞. Putnam 1950 A3
The sequence x0,x1,x2,… is defined by the conditions x0=a,x1=b,xn+1=2nxn−1+(2n−1) xn for n≥1, where a and b are given numbers. Express limn→∞xn concisely in terms of a and b.