PUMaC 2023 · 代数（B 组） · 第 1 题

PUMaC 2023 — Algebra (Division B) — Problem 1

Consider the equations x + y = 16 and xy = . Find the sum, over all ordered pairs ( x, y ) 2 satisfying these equations, of | x + y | .

解析

Consider the equations x + y = 16 and xy = . Find the sum, over all ordered pairs ( x, y ) 2 satisfying these equations, of | x + y | . Proposed by Frank Lu Answer: 20 9 We first observe that x = 0 , y = 0 are not valid values, so we can then write x = . Hence, 2 y 81 81 2 4 2 we have that y + = 16 , or that y must satisfy y − 16 y + = 0 . We can see that the 2 4 y 4 2 discriminant of this equation is 16 − 81 > 0 , and so we have 4 distinct real solutions to this equation for y. 2 2 2 9 Finally, observe that for each pair ( x, y ) , we require that ( x + y ) = x + y +2 xy = 16+2 · = 25 , 2 or that | x + y | = 5 for each solution. Therefore, our total sum is just the number of solutions times 5 , or 20 .