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Algebra Difficulty 5.2 AIME, harder Find the answer

4. (7 points) The numbers a,b,c,da, b, c, d belong to the interval [−7.5;7.5][-7.5 ; 7.5]. Find the maximum value of the expression a+2b+c+2d−ab−bc−cd−daa+2 b+c+2 d-a b-b c-c d-d a.

A number or a short expression. Spacing and $ signs are ignored.

Solution

Answer: 240

Solution: Note that a+2b+c+2d−ab−bc−cd−da=(a+c)+2(b+d)−(a+a+2 b+c+2 d-a b-b c-c d-d a=(a+c)+2(b+d)-(a+ c)(b+d)c)(b+d). Let x=a+c,y=b+dx=a+c, y=b+d, then we will find the maximum value of the expression x+2y−xy=(y−1)(2−x)+2x+2 y-x y=(y-1)(2-x)+2, where −16⩽y−1⩽14-16 \leqslant y-1 \leqslant 14 and −13⩽2−x⩽17-13 \leqslant 2-x \leqslant 17. Therefore, (y−1)(2−x)+2⩽14⋅17+2=240(y-1)(2-x)+2 \leqslant 14 \cdot 17+2=240. The maximum value of the expression is achieved when a=c=−7.5,b=d=7.5a=c=-7.5, b=d=7.5.

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.