HMMT 二月 2015 · 冲刺赛 · 第 4 题

HMMT February 2015 — Guts Round — Problem 4

[ 4 ] Consider the function z ( x, y ) describing the paraboloid 2 2 z = (2 x − y ) − 2 y − 3 y. Archimedes and Brahmagupta are playing a game. Archimedes first chooses x . Afterwards, Brah- magupta chooses y . Archimedes wishes to minimize z while Brahmagupta wishes to maximize z . Assuming that Brahmagupta will play optimally, what value of x should Archimedes choose? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . HMMT FEBRUARY 2015, 21 FEBRUARY 2015 — GUTS ROUND Organization Team Team ID#

解析

[ 4 ] Consider the function z ( x, y ) describing the paraboloid 2 2 z = (2 x − y ) − 2 y − 3 y. Archimedes and Brahmagupta are playing a game. Archimedes first chooses x . Afterwards, Brah- magupta chooses y . Archimedes wishes to minimize z while Brahmagupta wishes to maximize z . Assuming that Brahmagupta will play optimally, what value of x should Archimedes choose? 3 Answer: − Viewing x as a constant and completing the square, we find that 8 2 2 2 z = 4 x − 4 xy + y − 2 y − 3 y 2 2 = − y − (4 x + 3) y + 4 x ( ) ( ) 2 2 4 x + 3 4 x + 3 2 = − y + + + 4 x . 2 2 4 x +3 Bhramagupta wishes to maximize z , so regardless of the value of x , he will pick y = − . The 2 expression for z then simplifies to 9 2 z = 8 x + 6 x + . 4 Guts Archimedes knows this and will therefore pick x to minimize the above expression. By completing the 3 square, we find that x = − minimizes z . 8 Alternatively, note that z is convex in x and concave in y , so we can use the minimax theorem to switch y the order of moves. If Archimedes goes second, he will set x = to minimize z , so Brahmagupta will 2 2 3 3 maximize − 2 y − 3 y by setting y = − . Thus Archimedes should pick x = − , as above. 4 8