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Part of 1997 Putnam

Problems(2)

Putnam 1997 A4

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5/30/2014
Let GG be group with identity ee and ϕ:G→G\phi :G\to G be a function such that : ϕ(g1)⋅ϕ(g2)⋅ϕ(g3)=ϕ(h1)⋅ϕ(h2)⋅ϕ(h3) \phi(g_1)\cdot \phi(g_2)\cdot \phi(g_3)=\phi(h_1)\cdot \phi(h_2)\cdot \phi(h_3) Whenever g1⋅g2⋅g3=e=h1⋅h2⋅h3g_1\cdot g_2\cdot g_3=e=h_1\cdot h_2\cdot h_3 Show there exists a∈Ga\in G such that ψ(x)=aϕ(x)\psi(x)=a\phi(x) is a homomorphism. (that is ψ(x⋅y)=ψ(x)⋅ψ(y)\psi(x\cdot y)=\psi (x)\cdot \psi(y) for all x,y∈Gx,y\in G )
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Putnam 1997 B4

Source:

5/30/2014
Let am,na_{m,n} denote the coefficient of xnx^n in the expansion (1+x+x2)n(1+x+x^2)^n. Prove the inequality for all integers k≥0k\ge 0 : 0≤∑ℓ=0⌊2k3⌋(−1)ℓak−ℓ,ℓ≤1 0\le \sum_{\ell=0}^{\left\lfloor{\frac{2k}{3}}\right\rfloor} (-1)^{\ell} a_{k-\ell,\ell}\le 1
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