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Part of 2013 Putnam

Problems(2)

Putnam 2013 A3

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12/9/2013
Suppose that the real numbers a0,a1,…,ana_0,a_1,\dots,a_n and x,x, with 0<x<1,0<x<1, satisfy a01−x+a11−x2+⋯+an1−xn+1=0.\frac{a_0}{1-x}+\frac{a_1}{1-x^2}+\cdots+\frac{a_n}{1-x^{n+1}}=0. Prove that there exists a real number yy with 0<y<10<y<1 such that a0+a1y+⋯+anyn=0.a_0+a_1y+\cdots+a_ny^n=0.
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Putnam 2013 B3

Source:

12/9/2013
Let PP be a nonempty collection of subsets of {1,…,n}\{1,\dots,n\} such that:
(i) if S,S′∈P,S,S'\in P, then S∪S′∈PS\cup S'\in P and S∩S′∈P,S\cap S'\in P, and (ii) if S∈PS\in P and S≠∅,S\ne\emptyset, then there is a subset T⊂ST\subset S such that T∈PT\in P and TT contains exactly one fewer element than S.S.
Suppose that f:P→Rf:P\to\mathbb{R} is a function such that f(∅)=0f(\emptyset)=0 and f(S∪S′)=f(S)+f(S′)−f(S∩S′) for all S,S′∈P.f(S\cup S')= f(S)+f(S')-f(S\cap S')\text{ for all }S,S'\in P. Must there exist real numbers f1,…,fnf_1,\dots,f_n such that f(S)=∑i∈Sfif(S)=\sum_{i\in S}f_i for every S∈P?S\in P?
Putnamfunctioninductionlinear algebramatrixcollege contestsPutnam 2013