A sequence of positive integers with 2020 terms is called an FT sequence if each term after the second is the sum of the previous two terms. For example, if the first two terms of an FT sequence are 8 and 7, the sequence would begin 8,7,15,22,37, ... For some positive integer m, there are exactly 2415 FT sequences where the first two terms are each less than 2m and the number of odd-valued terms is more than twice the number of even-valued terms. What is the value of m?

Step-by-step explanation:

Consider the FT 0, 1, 1, 2, 3, 5, 8, 13, 21, 24,45, ...

The numbers are arranged in order of even, odd, odd, even, odd, odd, even, odd, odd, even, ...

Hence, the loop contains 3 elements. If the number of terms is 2020 terms, then we have 673 loops + 1 element. (That is 3 * 673 + 1 = 2020) The last element will start the new loop and it is an even number.

In the other hand with 2019 terms, we have number of odd = 2 * number of even. But the last term is even. That makes number of odd < twice number of even and it contradicts with the condition.

Therefore, the first term must be an odd number.

Also, the loop is either (odd, odd, even) or (odd, even, odd).

if m = 1, only (odd, odd, even) satisfies the condition of the first 2 terms each < 2m.

If m = 2, both satisfy.

But we have exactly 2415 FT sequence, m must be 1073 (half of 2415 + 0.5)