Proofs From The Book
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Contents:

Number Theory
1  Six proofs of the infinity of primes
2  Bertrand’s postulate
3  Binomial coefficients are (almost) never powers
4  Representing numbers as sums of two squares
5  The law of quadratic reciprocity
6  Every finite division ring is a field
7  Some irrational numbers
8  Three times $\pi^2/6$ 
Geometry
9  Hilbert’s third problem: decomposing polyhedra
10  Lines in the plane and decompositions of graphs
11  The slope problem
12  Three applications of Euler’s formula
13  Cauchy’s rigidity theorem
14  Touching simplices
15  Every large point set has an obtuse angle
16  Borsuk’s conjecture 
Analysis
17  Sets, functions, and the continuum hypothesis
18  In praise of inequalities
19  The fundamental theorem of algebra
20  One square and an odd number of triangles
21  A theorem of Polya on polynomials
22  On a lemma of Littlewood and Offord
23  Cotangent and the Herglotz trick
24  Buffon’s needle problem 
Combinatorics
25  Pigeonhole and double countring
26  Tiling rectangles
27  Three famous theorems on finite sets
28  Shuffling cards
29  Lattice paths and determinants
30  Cayley’s formula for the number of trees
31  Identities versus bijections
32  Completing Latin squares 
Graph Theory
33  The Dinitz problem
34  Fivecoloring plane graphs
35  How to guard a museum
36  Turan’s graph theorem
37  Communicating without errors
38  The chromatic number of Kneser graphs
39  Of friends and politicians
40  Probability makes countring (sometimes) easy