Olympiad Inequalities Problem


#1

Inequalities has been a favourite Olympiad topic of mine, so this piqued interest. As I have limited experience with LSTMs, wanted to pass on a possible first angle of attack.

There is a Vietnamese book that advocates a very organic approach to solving inequalities through rewriting as sum of squares “Diamonds in Mathematical Inequalities”.

This could be used to attack a subset of the problem.

  1. Reformulate the problem to prove LHS(x, y, z, …) >= 0
  2. Generate a large set of expressions of sum of squares like so (a-b)^2, (a-2b+c)^2+b^2, etc.
  3. Calculate the expanded form of these expressions, e.g., a^2-2ab+b^2
  4. Train an LSTM to solve the inverse problem, e.g.,
    LSTM(“a^2+b^2+c^2-ab-bc-ca”)=“0.5(a-b)^2+0.5(b-c)^2+0.5(c-a)^2”
  5. To see if an inequality is solvable with this approach, see if the neural network output expression is a sum of squares and expands to the original - if so => inequality proved.

Thoughts?