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Part of 2012 Putnam

Problems(2)

Putnam 2012 A5

Source:

12/3/2012
Let Fp\mathbb{F}_p denote the field of integers modulo a prime p,p, and let nn be a positive integer. Let vv be a fixed vector in Fpn,\mathbb{F}_p^n, let MM be an n×nn\times n matrix with entries in Fp,\mathbb{F}_p, and define G:Fpn→FpnG:\mathbb{F}_p^n\to \mathbb{F}_p^n by G(x)=v+Mx.G(x)=v+Mx. Let G(k)G^{(k)} denote the kk-fold composition of GG with itself, that is, G(1)(x)=G(x)G^{(1)}(x)=G(x) and G(k+1)(x)=G(G(k)(x)).G^{(k+1)}(x)=G(G^{(k)}(x)). Determine all pairs p,np,n for which there exist vv and MM such that the pnp^n vectors G(k)(0),G^{(k)}(0), k=1,2,…,pnk=1,2,\dots,p^n are distinct.
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Putnam 2012 B5

Source:

12/3/2012
Prove that, for any two bounded functions g1,g2:R→[1,∞),g_1,g_2 : \mathbb{R}\to[1,\infty), there exist functions h1,h2:R→Rh_1,h_2 : \mathbb{R}\to\mathbb{R} such that for every x∈R,x\in\mathbb{R},sup⁡s∈R(g1(s)xg2(s))=max⁡t∈R(xh1(t)+h2(t)).\sup_{s\in\mathbb{R}}\left(g_1(s)^xg_2(s)\right)=\max_{t\in\mathbb{R}}\left(xh_1(t)+h_2(t)\right).
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