26. Consider rational numbers x, y, and z such that the number a being their sum and the number b being the sum of their reciprocals are both integers. Then what is the smallest possible value of a squared plus b squared? We can come up with an example where a and b are integers, in fact any positive integer. We can let k be some non-zero number and then set x equal to y equal to two-thirds times this number and set also z equal to negative one-third times this number. Then we have a is equal to k and b is equal to zero so a squared plus b squared is going to be equal to k squared. Now as long as k is not equal to zero we can then make a squared plus b squared equal to any positive number but we also demand that both a and b be integers so with k equal to one we have a squared plus b squared is equal to one which is choice b but we also have choice a and we could set k is equal to zero here and we run into some trouble with k equal to zero. Certainly we obtain a smaller answer for a squared plus b squared but with k is equal to zero we also end up using x, y, and z equal to zero with this change of variable here so we can't right away discount the answer in a we have to check separately that it is not possible so that's what we do next we let we suppose a squared plus b squared is equal to zero then in that case both a and b have to be equal to zero there are no other solutions and with a and b equal to zero we have zero is equal to x plus y plus z and zero is also equal to the sum of the reciprocals of x y and z so we can solve for x in both equations x is negative y minus z one over one over x would be equal to negative one over y minus one over z and so we have this system of two equations we can eliminate one of the variables by multiplying them together on the left hand side x times one over x gives us one and on the right hand side after simplifying we obtain a y over x plus x over y so we try to solve this equation we have one is equal to the product y squared plus x squared over x times y and this is equivalent after bringing everything to one side x squared minus x y plus y squared and there are several ways to check that this equation here doesn't have any solutions one of them is in the suggested solutions another way is to note that here on the right hand side this is a rotated it conic section in fact that's sort of the case anytime we have a term of x y here and that doesn't allow us to complete the square and in fact this would be an ellipse on the left hand side we have a zero so this is degenerate so we have a degenerate conic section means that it doesn't actually contain any points the graph of this degenerate conic section so there are no solutions here other than of course x is equal to zero and y is equal to zero which we discount they're not in the domain so it is not possible to have a is equal to zero after all we choose b is equal to one as our answer

smallest value

rational numbers

sum reciprocals

integer constraints

optimization problem