MathDB

Putnam 1939 B7

Source:

August 20, 2021

Putnam

Problem Statement

Do either

(1)

or

(2)

:

(1)

Let

ai = \sum_{n=0}^{\infty} \dfrac{x^{3n+i}}{(3n+i)!}

Prove that

a_0^3 + a_1^3 + a_2^3 - 3 a_0a_1a_2 = 1.

(2)

Let

O

be the origin,

\lambda

a positive real number,

C

be the conic

ax^2 + by^2 + cx + dy + e = 0,

and

C\lambda

the conic

ax^2 + by^2 + \lambda cx + \lambda dy + \lambda 2e = 0.

Given a point

P

and a non-zero real number

k,

define the transformation

D(P,k)

as follows. Take coordinates

(x',y')

with

P

as the origin. Then

D(P,k)

takes

(x',y')

to

(kx',ky').

Show that

D(O,\lambda)

and

D(A,-\lambda)

both take

C

into

C\lambda,

where

A

is the point

(\dfrac{-c \lambda} {(a(1 + \lambda))}, \dfrac{-d \lambda} {(b(1 + \lambda))})

. Comment on the case

\lambda = 1.

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