The ABC conjecture is a powerful statement about positive integers which, if proved, would imply a large number of other significant results, among them Fermat's Last Theorem. As of 2012, a proof by the Japanese mathematician Shinichi Mochizuki is being checked by members of the mathematical community. But we don't have to wait for the check to explore some ramifications of the conjecture. Read about its exact statement, about how it implies the truth of Fermat's Last Theorem, and about several other results it implies.

The 2012 Pan-African Mathematical Olympiad was held in Tunis, Tunisia in September. These are two of the problems set by the jury for the students participating from 12 African countries. For the first time, a US team participated (unofficially). Three of the team members performed on the level of a medalist (one gold, one silver, and one bronze). The solutions provided here are intended to show students not just a mathematically correct solution, but to indicate at least one path to the solution for students unused to solving Olympiad problems.

In the summer of 2012, the AAAS (supported by a generous grant from the Alfred P. Sloan Foundation) gathered 20 students, mostly from backgrounds under-represented in the mathematical sciences, for a training session in Washington, D.C. The students practiced solving Olympiad-level mathematics problems. Four of them were then selected to represent the US in the Pan-African Mathematical Olympiad. Four more were selected to attend the Mexican Mathematical Olympiad. These problems are typical of those the students worked on. We give formal solutions, but also hints and discussions about how to find a pathway to the solution.

This file contains a guide to a solution to IMO 2012 problem 1, a geometry problem. The Olympiad-level problem is broken down into fourteen easier problems. An average (successful) student of the first year of geometry will be able to work most of these problems. The student will need to know how to measure angles inscribed in circles. For the very last stage of the solution, the student needs to know that opposite angles of a cyclic quadrilateral are supplementary. But no more advanced knowledge is needed. In working the problem this way, students will begin to understand how a mathematical palace can be built out toothpicks--tiny slivers of information.