Let {q_{n}} be an enumeration of the rationals, and for each n, let U_{n} be an open interval of length 1/2^{n} centered at q_{n}. Denote the union of all U_{n} by U. Then U is a dense open . . . [Read More]


Let {q_{n}} be an enumeration of the rationals, and for each n, let U_{n} be an open interval of length 1/2^{n} centered at q_{n}. Denote the union of all U_{n} by U. Then U is a dense open . . . [Read More] This is a classic introductory result in combinatorics. Suppose that six people are gathered at a dinner party. Then there is a group of three people at the party who are either all mutual acquaintances or all mutual strangers. To formalize . . . [Read More] Let k ≥ 0, and let n = k(2k + 1). Then we have: n^{2} + (n+1)^{2} + … + (n+k)^{2} = (n+k+1)^{2} + … + (n+2k)^{2}. For example, 

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