MathDB

B7

Part of Putnam 1939

Problems(1)

Putnam 1939 B7

Source:

8/20/2021
Do either (1)(1) or (2)(2):
(1)(1) Let ai=∑n=0∞x3n+i(3n+i)!ai = \sum_{n=0}^{\infty} \dfrac{x^{3n+i}}{(3n+i)!} Prove that a03+a13+a23−3a0a1a2=1.a_0^3 + a_1^3 + a_2^3 - 3 a_0a_1a_2 = 1.
(2)(2) Let OO be the origin, λ\lambda a positive real number, CC be the conic ax2+by2+cx+dy+e=0,ax^2 + by^2 + cx + dy + e = 0, and CλC\lambda the conic ax2+by2+λcx+λdy+λ2e=0.ax^2 + by^2 + \lambda cx + \lambda dy + \lambda 2e = 0. Given a point PP and a non-zero real number k,k, define the transformation D(P,k)D(P,k) as follows. Take coordinates (x′,y′)(x',y') with PP as the origin. Then D(P,k)D(P,k) takes (x′,y′)(x',y') to (kx′,ky′).(kx',ky'). Show that D(O,λ)D(O,\lambda) and D(A,−λ)D(A,-\lambda) both take CC into Cλ,C\lambda, where AA is the point (−cλ(a(1+λ)),−dλ(b(1+λ)))(\dfrac{-c \lambda} {(a(1 + \lambda))}, \dfrac{-d \lambda} {(b(1 + \lambda))}) . Comment on the case λ=1.\lambda = 1.
Putnam