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Part of 1976 Putnam

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Putnam 1976 A2

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4/18/2022
Let P(x,y)=x2y+xy2P(x,y)=x^2y+xy^2 and Q(x,y)=x2+xy+y2.Q(x,y)=x^2+xy+y^2. For n=1,2,3,…,n=1,2,3,\dots, let \begin{align*}F_n(x,y)=(x+y)^n-x^n-y^n \text{ and,}\\ G_n(x,y)=(x+y)^n+x^n+y^n. \end{align*} One observes that G2=2Q,F3=3P,G4=2Q2,F5=5PQ,G6=2Q3+3P2.G_2=2Q, F_3=3P, G_4=2Q^2, F_5=5PQ, G_6=2Q^3+3P^2. Prove that, in fact, for each nn either FnF_n or GnG_n is expressible as a polynomial in PP and QQ with integer coefficients.
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Putnam 1976 B2

Source:

4/19/2022
Suppose that GG is a group generated by elements AA and BB, that is, every element of GG can be written as a finite "word" An1Bn2An3…Bnk,A^{n_1}B^{n_2}A^{n_3}\dots B^{n_k}, where n1,…nkn_1,\dots n_k are any integers, and A0=B0=1A^0=B^0=1 as usual. Also suppose that A4=B7=ABA−1B=1,A2≠1,A^4=B^7=ABA^{-1}B=1, A^2\neq 1, and B≠1.B\neq 1.
(a) How many elements of GG are of the form C2C^2 with CC in GG? (b) Write each such square as a word in AA and B.B.
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