PUMaC 2020 · 代数（B 组） · 第 4 题

PUMaC 2020 — Algebra (Division B) — Problem 4

Let C denote the curve y = . The points ( , a ) , ( b, c ) , and (24 , d ) lie on C and 6 2 2 2 are collinear, and ad < 0 . Given that b, c are rational numbers, find 100 b + c .

解析

Let C denote the curve y = . The points ( , a ) , ( b, c ) , and (24 , d ) lie on C and 6 2 2 2 are collinear, and ad < 0 . Given that b, c are rational numbers, find 100 b + c . Proposed by: Sunay Joshi Answer: 101 1 1 1 By plugging x = into the equation for C , we find a = ∓ . Similarly, d = ± 70. By geometric 2 2 1 1 intuition (?), there are only two possible pairs ( a, d ), namely ( a, d ) = ( − , 70) or ( , − 70). 2 2 1 1 1 Suppose ( a, d ) = ( − , 70). Then the equation of the line through ( , − ) and (24 , 70) is 2 2 2 x ( x +1)(2 x +1) 2 y = 3 x − 2. Plugging this into the equation for C , we find (3 x − 2) = . Simplifying, 6 3 2 we find 2 x − 51 x + . . . = 0. 1 1 At this point, instead of solving this equation explicitly, we use a trick. Since ( , − ) and 2 2 1 (24 , − 70) lie on this line, x = and x = 24 are roots of this cubic. Thus, the remaining root 2 1 51 x = b must satisfy Vieta’s Formula for the sum of roots! We get b + + 24 = , thus b = 1. 2 2 Plugging this into the equation of our line, we find c = 1, hence ( b, c ) = (1 , 1). By the symmetry of C across the x axis, the other case yields ( b, c ) = (1 , − 1). In either case, 2 2 we find an answer of 100 · 1 + 1 = 101 .