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6

Part of 2000 Putnam

Problems(2)

Putnam 2000 A6

Source:

9/6/2011
Let f(x)f(x) be a polynomial with integer coefficients. Define a sequence a0,a1,⋯a_0, a_1, \cdots of integers such that a0=0a_0=0 and an+1=f(an)a_{n+1}=f(a_n) for all n≥0n \ge 0. Prove that if there exists a positive integer mm for which am=0a_m=0 then either a1=0a_1=0 or a2=0a_2=0.
Putnamalgebrapolynomialmodular arithmeticinductionRational Root Theoremfunction
Putnam 2000 B6

Source:

9/6/2011
Let BB be a set of more than 2n+1n\tfrac{2^{n+1}}{n} distinct points with coordinates of the form (±1,±1,⋯ ,±1)(\pm 1, \pm 1, \cdots, \pm 1) in nn-dimensional space with n≥3n \ge 3. Show that there are three distinct points in BB which are the vertices of an equilateral triangle.
Putnaminequalitiescollege contestscombinatoricscombinatorial geometryProbabilistic Method