3rd and final installment peeps!!!!!!!!!!!!!!!!!!!!!!!!!?

Question

In this problem, the notation A / B means A divided by B.

Jim recently bought a rather odd computer game, which works as follows. First of all, Jim picks a positive whole number A. The computer then must pick a positive whole number B which is strictly less than A (0 < B < A) and so:

(A / B) + (30 / A) is the same as (B / A) + (30 / B).


If it is possible to pick such a number, the computer will do so and the computer then wins. Otherwise, if the computer cannot find any such whole number B, Jim wins the game.

Jim played the game 20 times, using all the whole numbers between 1 and 20 inclusive. The first time, he chose the number 1, the second time the number 2, and so on up to the twentieth time, when he chose 20. How many times did Jim win?

Answer 1

A/B + 30/A = B/A + 30/B

multiply the entire equation by AB:

A^2 + 30B = B^2 + 30A
A^2 - B^2 = 30A - 30B
(A-B)(A+B) = 30(A-B)
A+B = 30

So for integers 1-15, there can be no number less that would add to 30. However, for 16-20, the computer could pick 14-10, thus the computer wins 5 times and Jim wins 15 times.

Best of luck!