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Evaluate the sum of reciprocals of double factorial of even nonnegative integers.

x^3 + y^3 + z^3 = w^3.

The smallest set of positive integers x,y,z,w qualifying as a solution of the bolded equation above is: 3,4,5,6

a.Find some other sets of distinct integers, not necessarily positive.
b. Provide some comments on "The sum of 3 cubes is a cube" equation.

Find all solutions of
(x+y^2)*(x^2+y)=(x-y)^3 .
x and y have to be non-zero integers.

One Martian, one Venusian and one Human reside on Pluto.
One day they make the following conversation:
Martian: I have spent 1/12 of my life in Pluto.
Human: I also have.
Venusian: Me too.
Martian: But Venusian and I have spent much more time here than you, Human.
Human: However Venusian and I are of the same age.
Venusian: I have lived 300 Earth years.
Martian: Venusian and I have been on Pluto for the last 13 years.

It is known that Human and Martian together have lived for 104 Earth years.

Find their respective ages.

HINT: The numbers are not necessarily expressed in base 10 .
Source: 1971 IMO longlisted problem.

How many ones are used in writing all of the numbers from 0 to 2018 (inclusive) in binary form?

This problem was inspired by the infamous problem 6 in the 1988 Math Olympiad:

When is a²+b² divisible by (ab+1)² where a and b are non-negative integers?

Let S = a^5 + b^5 + c^5 + d^5, and
a,b,c,d are integers fulfilling a+b+c+d=0
Prove that S must be divisible by 10.