OTIS Unit Wiki

Hard Ineq

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A unit featuring some of the hardest gems from the golden age of inequalities.

Hard Inequalities is a unit featuring difficult inequalities which can be solved by various (advanced) techniques.

Content

With regards to content, the Reading summarises the OTIS philosophy on inequalities, which itself boils down to sharpness (being precise in your estimates) and tractability (deciding to what extent brute-force is a feasible approach).

Here is the set of standard tools at your disposal:

  • Substitutions and homogenisation (either ridding you of conditions or for building intuition);

  • AM-GM / Power Mean / Muirhead (the standard batch);

  • Cauchy / Hölder (efficient counters to roots of any order);

  • Jensen / Karamata / tangent line trick / $n-1$ EV (functional approaches);

  • Isolated fudging (comparing terms of a sum with like expressions, so that summing gives the conclusion);

  • Smoothing (the process of bringing your variables closer to optimum, often with a combinatorial feel);

  • SOS and other tricks.

This is all the theory required to tackle most easy inequalities, which is why they’ve become almost obsolete on modern contests (with rare but notable exceptions). Their repetitiveness and lack of mathematical substance add on to this fact (but this still doesn’t universally apply). The difficulty of hard inequalities often lies in their presentation (often involving weird conditions and setups, tricky manipulations or combinatorial and analytic ideas).

Notable Problems

Some poster examples include:

  • USAMO 2000/6: A walkthrough featuring the art of smoothing.

  • Shortlist 2019 A2: A cute and satisfying solve.

  • Brandon Wang: Wait, what?

  • USAMO 2020/6: A crazy and intriguing finale to the 2020 USAMO.

  • Bulgaria 2018/3: An iconic and unconventional required problem, which took Evan almost 2 hours on stream.

Recommendations

This is fairly easy for a Z-level unit, especially if you’ve done a lot of inequalities before. This should be unlocked after completing its standard (or basic) and functional counterparts, since the techniques developed in those units become common knowledge here.

Otherwise, if you come from the Balkans (where inequalities are still a big deal) or if you’re very experienced, unlocking can still be a good idea.