USA Math Olympiad Radical Equation. IMO Exam Quiz . Algebra Problem.

[@hstrinzel]

HELLISHLY complicated. Could have written a program that finds the solution by trial-and-error in 3 minutes or less. But that would have been a FLUNK in the Math Olympiad.

[@superacademy247]

That's the easiest method I know so far!

[@NadiehFan]

@hstrinzel I don't agree. Of course trial and error (whether by hand or by machine) is not a methodical solution. But this problem can be solved a lot easier. Super Academy sets up a cubic equation in (a + b) but that is quite unnecessary. We have the following system: a − b = 4 a³ + b³ = 370 From the first equation we get b = a − 4 and substituting that in the second equation we have a³ + (a − 4)³ = 370 which is a cubic equation in a. Now, first notice that the left hand side is strictly increasing, which means that this cubic has only a _single_ real solution. Also observe that for a = 2 the left hand side is 2³ + (−2)³ = 0, so the unique real solution we are looking for is greater than 2. Now expand the left hand side and bring the constant term 370 over to the left hand side to bring the equation into standard form, then we have a³ − 6a² + 24a − 217 = 0 Before attempting a formal solution of a cubic equation like this, we should always try to find rational roots because mostly Olympiad problems are created in such a way that they have 'nice' solutions. In this case, the _rational root theorem_ tells us that any rational solutions of this equation must be _integer_ solutions and that any such solution must be a factor of the constant term 217 = 7·31. So, since we already know that we must have a > 2, the smallest integer solution to try is a = 7, and indeed this is a solution of the equation, and therefore the _only_ real solution of the equation. Of course, the rest is now a piece of cake. From a = 7 and b = a − 4 we get b = 3, and a³ − b³ = 2k then gives us 2k = 7³ − 3³ = 343 − 27 = 316 so k = 158 and we are done.

[@superacademy247]

Great 👌 approach

[@rajesh-dh3dl]

Suggest to put value of a= b+4 in equation a power 3 + b power 3 = 370 Now solve for b .. will find (b-3) as factor and in quadric delta is negative When b=3 then by solving b power 3 = 27= 185-k then k = 153

[@geometer6121]

Excellent but at the end it should rather be k = 158.

[@StephenRayWesley]

3√185+k-3√185=4 =243 3√185)/=243. (243+243)= √486√. 6^8√3^2. √3^2 2^4 √1^1[√1^1 (x+2x-4)