SINGAPORE MATH OLYMPIAD 2009 Problem | Algebra Challenge

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SINGAPORE MATH OLYMPIAD 2009 Problem | Algebra Challenge


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Комментарии
Автор

You can just square the target expression

(x + y + 1)^2 = x^2 + y^2 + 1 + 2(x +xy + y)
= 6 + 1 + 4 + 6 sqrt(2)
= 11 + 6 sqrt(2)
= (3 + sqrt(2))^2

|x + y + 1| = 3 + sqrt(2)

pwmiles
Автор

I just used symmetry of the equations to form two possible outcomes of the first equation, depending on whether you assume both X and Y are surds or only 1 of them

In the case they are both surds then from the first equation XY = 2, and X + Y = 3sqrt(2) which cannot be solved.

Which means that only one of them is a surd so X + XY = 3sqrt(2), Y = 2. Which can easily by solved to find X = sqrt(2). So you can solve the entire equation without even using the second equation.

Then X + Y + 1 = sqrt(2) + 2 + 1 which is the given answer

There is probably other solutions to X and Y using negative values or imaginary numbers but the answer only wants the absolute value of them so this working gets the correct answer.

Alternatively you can find the same working from the second equation. Once you realise that there is a surd somewhere as shown in the first equation, you can rewrite the 6 in the second equation as 3x2 or sqrt(3)^2x2 or sqrt(2)^2x3 which gives answers for X and Y. Then plug them in the first equation to see which is correct.

MyNameIsSalo
Автор

Выделение полного квадрата всегда помогает!

AAZ