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Good solid algebra. Yes a bit slow, but for someone like myself who has not done much serious math since student days (half a century ago now), better to include all the steps rather than skip some. And yes, the obvious real solutions from simple inspection are x=2, y=1 and x=1, y=2, however that's not the point of the exercise. Whether it's a good Math Olympiad problem or not, I'm afraid I can't judge.
While it might be nice to see how to work out another solution, it's possible any additional solution would be extraneous, or a duplicate of an existing solution. But most importantly, any valid solution for the problem as a whole, has to work in both equations, and another solution to the "x⁵ + y⁵=33" equation, if it exists, might not work in the "x + y = 3" equation.
Who is your audience? Your audience is people who can appreciate and learn what you are showing them. Such people must have mastered high school algebra, and have presumably mastered grade school arithmetic, preparing them to learn the more advanced techniques and principles you are showing them. They--we--don't need to see basic numerical division belabored. They--we--don't need the detailed steps of factoring a quadratic. With the factors given, we need about three seconds to verify them, then we can move on. The value in these solutions is the overall scheme, recognizing the larger patterns that we can use to pry the problem apart. But you bury those steps in the minutia that we have already mastered, making it hard to find the precious grains of gold mixed with the dross of first year algebra and fourth grade arithmetic. These are brilliant solutions. The presentations would be so much better if you demonstrated the brilliance and moved routinely through the routine mechanics.
If you set x = 3-y using equation 2 then the highest term of x^5 is - y^5, which cancels out with y^5 in equation 1. This reduces the problem to solving a quartic and results in four roots.
Evaluate before doing all of your shit. If x+y = integers. and x⁵+y⁵= integer then x and y is integer then evaluate b²-4ac, if it's negative, stop. No point of continuing, i^odd integers. Will always has I as result.
Sorry, but I find your video disappointing for 3 reasons: 1. You did not specify in which domain you are solving, integers? real numbers? complex numbers?. You solved for complex, but you did not specify in advance. In math alympiads, problems are always specified more precisely. You omitted important information. This is an error often commited by a number of people doing this kind of videos, and I find it hugly annoying. 2. Your solution is correct but the problem can be solved more easily. In this kind of problems, there are often more efficient approaches than the brute force method. The colculations get easier when you realise that that x5+x5=(x+y)(x4-x3y+x2y2-xy3+y4) and continue from that. You then only need 4th and second powers of (x+y) which makes the fomulas a little bit simpler and thus less error prone. 3. This comment is mino: state your sources. Like xxx math olympiad problem from yyyy where xxx is the orangisation/country or whatever and dddd is the year. For poitn 3, I have the impression that many youtubers just copy the problems from other youtube videos as while there are so many interessing math olympiad problems, you oftend find the same problems over and over again.
