Lagrange multiplier method can solve the extreme value problem of multivariate functions, but it is under good restrictions (the function is derivable, the domain is open set, etc.). ). Sometimes Lagrange multiplier method can solve the extreme value problem of multivariate function quickly, but only if you can judge this problem in this way, and this judgment requires you to know many concepts of mathematical analysis (such as implicit function theorem), and it is not worth the candle to study mathematical analysis thoroughly. According to my experience, if the competition problems encountered in high school are really difficult inequality problems, even using Lagrange multiplier method is difficult to solve, and the constraints are all in line with the requirements (if used, you may have to solve a high-order multivariate equation), but it can be skillfully solved by using elementary methods.

I think you should master basic calculus and derivatives. They are very useful for solving inequalities.

Projective geometry and affine transformation can solve many early mathematical competition problems, and some even winter camp problems are made by projective geometry in one or two steps, but now it is estimated that those experts will not solve problems by projective geometry in one or two steps, considering that we will be opportunistic. If you have time, you can teach yourself and improve your intuitive understanding of geometry, but don't try to take advantage, because it is very interesting and exercisable to think about those plane geometry problems with elementary methods.

Higher algebra (matrix determinant) has little influence. If you want to use it to solve the problem of math contest, you must study it very deeply.

My advice to you is to master the basic knowledge of calculus (especially derivatives) and read some books on combinatorial number theory (these university courses don't need any foundation, and they are quite interesting and highly related to competitions). Of course, the most important thing is to think more about yourself at ordinary times. Some questions are worth thinking about even for a whole day, because your thinking ability will be improved imperceptibly, which is the real purpose of the math competition.