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Part of 2014 Putnam

Problems(2)

Putnam 2014 A1

Source:

12/7/2014
Prove that every nonzero coefficient of the Taylor series of (1−x+x2)ex(1-x+x^2)e^x about x=0x=0 is a rational number whose numerator (in lowest terms) is either 11 or a prime number.
Putnamfunctionrelatively primeprime factorizationPutnam calculusPutnam 2014
Putnam 2014 B1

Source:

12/8/2014
A base 10 over-expansion of a positive integer NN is an expression of the form N=dk10k+dk−110k−1+⋯+d0100N=d_k10^k+d_{k-1}10^{k-1}+\cdots+d_0 10^0 with dk≠0d_k\ne 0 and di∈{0,1,2,…,10}d_i\in\{0,1,2,\dots,10\} for all i.i. For instance, the integer N=10N=10 has two base 10 over-expansions: 10=10⋅10010=10\cdot 10^0 and the usual base 10 expansion 10=1⋅101+0⋅100.10=1\cdot 10^1+0\cdot 10^0. Which positive integers have a unique base 10 over-expansion?
Putnammodular arithmeticinductioncollege contestsPutnam 2014