Happy Pi Day! I have an economics midterm on Wednesday, so here is my attempt at studying. 1. Mechanisms The idea is as follows. We have $latex {N}&fg=000000$ people and a seller who wants to auction off a power drill. The $latex {i}&fg=000000$th person has a private value of at most $latex {\$1000}&fg=000000$ on the… Continue reading Mechanism Design and Revenue Equivalence
Category: Elementary Math
Writing Olympiad Geometry Problems
You can use a wide range of wild, cultivated or supermarket greens in this recipe. Consider nettles, beet tops, turnip tops, spinach, or watercress in place of chard. The combination is also up to you so choose the ones you like most. --- Y. Ottolenghi. Plenty More In this post I'll describe how I come… Continue reading Writing Olympiad Geometry Problems
The Mixtilinear Incircle
This blog post corresponds to my newest olympiad handout on mixtilinear incircles. My favorite circle associated to a triangle is the $latex {A}&fg=000000$-mixtilinear incircle. While it rarely shows up on olympiads, it is one of the richest configurations I have seen, with many unexpected coincidences showing up, and I would be overjoyed if they become… Continue reading The Mixtilinear Incircle
Cauchy’s Functional Equation and Zorn’s Lemma
This is a draft of an appendix chapter for my Napkin project. In the world of olympiad math, there's a famous functional equation that goes as follows: $latex \displaystyle f : {\mathbb R} \rightarrow {\mathbb R} \qquad f(x+y) = f(x) + f(y). &fg=000000$ Everyone knows what its solutions are! There's an obvious family of solutions… Continue reading Cauchy’s Functional Equation and Zorn’s Lemma
Three Properties of Isogonal Conjugates
In this post I'll cover three properties of isogonal conjugates which were only recently made known to me. These properties are generalization of some well-known lemmas, such as the incenter/excenter lemma and the nine-point circle. 1. Definitions Let $latex {ABC}&fg=000000$ be a triangle with incenter $latex {I}&fg=000000$, and let $latex {P}&fg=000000$ be any point in… Continue reading Three Properties of Isogonal Conjugates
Why do roots come in conjugate pairs?
This is an expanded version of an answer I gave to a question that came up while I was assisting the 2014-2015 WOOT class. It struck me as an unusually good way to motivate higher math using stuff that people notice in high school but for some reason decide to not think about. In high… Continue reading Why do roots come in conjugate pairs?
Power Overwhelming 