Showing what's behind the curtain

Finding Nim positions consisting of heap sizes n, 2n and 3n for n ≤ 2^{30} that result in a losing game.

Determining the value of infinite polynomial series for which the coefficients are Fibonacci numbers.

Investigating isosceles triangle for which the height and base length differ by one.

Find the sum of all the even-valued terms in the Fibonacci sequence which do not exceed four million.

What is the first term in the Fibonacci sequence to contain 1000 digits?