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Part of 2014 Putnam

Problems(2)

Putnam 2014 A4

Source:

12/7/2014
Suppose XX is a random variable that takes on only nonnegative integer values, with E[X]=1,E[X]=1, E[X2]=2,E[X^2]=2, and E[X3]=5.E[X^3]=5. (Here E[Y]E[Y] denotes the expectation of the random variable Y.Y.) Determine the smallest possible value of the probability of the event X=0.X=0.
Putnamprobabilityinequalitiesalgebrapolynomialexpected valuePutnam 2014
Putnam 2014 B4

Source:

12/8/2014
Show that for each positive integer n,n, all the roots of the polynomial ∑k=0n2k(n−k)xk\sum_{k=0}^n 2^{k(n-k)}x^k are real numbers.
PutnamalgebrapolynomialinequalitiescalculusPutnam 2014