Difference between revisions of "2022 SSMO Speed Round Problems/Problem 9"

@@ Line 1: / Line 1: @@
-==Problem==
-Find the sum of the maximum and minimum values of <math>8x^2+7xy+5y^2</math> under the constraint that <math>3x^2+5xy+3y^2 = 88.</math>
-==Solution==
-We want to maximize <math>k</math> such
-<cmath>
-x^2+7xy+5y^2 = \frac{k}{88}(3x^2+5xy+3y^2)
-</cmath>
-or, if <math>a = \frac{x}{y}</math>
-<cmath>
-a^2+7a+5 = \frac{k}{88}(3a^2+5a+3)
-</cmath>
-which has discriminant in <math>a</math> of
-<cmath>
-(-k^2 + 688k - 78144)
-</cmath>
-so the sum of the extremes of <math>k</math> are <math>\boxed{688}</math>.

Page