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Wednesday, April 29, 2020

Digits in 2n

Let an = number of odd digits in the base-10 expansion of 2n and let bn = total number of digits in 2n.

For eg, 28=256 and so a8=1 (5 is the only odd digit) and b8=3 (it is a 3 digit number).

Three problems:

A) Show that n=1an2n=19

B) Show that S=n=1bn2n is an irrational number.

C) Show that S as defined in B) above satisfies S>11691023


[Solution]

Thursday, April 2, 2020

Sum of squares <= Sum of cubes

a,b,c>0 and abc=1

Show that

a2+b2+c2a3+b3+c3

[Solution]