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Part of 1996 Putnam

Problems(2)

Putnam 1996 A4

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6/4/2014
SS be a set of ordered triples (a,b,c)(a,b,c) of distinct elements of a finite set AA. Suppose that
[*] (a,b,c)∈S  ⟺  (b,c,a)∈S(a,b,c)\in S\iff (b,c,a)\in S [*] (a,b,c)∈S  ⟺  (c,b,a)∉S(a,b,c)\in S\iff (c,b,a)\not\in S [*] (a,b,c),(c,d,a) both ∈S  ⟺  (b,c,d),(d,a,b) both ∈S(a,b,c),(c,d,a)\text{ both }\in S\iff (b,c,d),(d,a,b)\text{ both }\in S Prove there exists g:A→Rg: A\to \mathbb{R}, such that gg is one-one and g(a)<g(b)<g(c)  ⟹  (a,b,c)∈Sg(a)<g(b)<g(c)\implies (a,b,c)\in S
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Putnam 1996 B4

Source:

6/6/2014
For any square matrix A\mathcal{A} we define sin⁡A\sin {\mathcal{A}} by the usual power series. sin⁡A=∑n=0∞(−1)n(2n+1)!A2n+1 \sin {\mathcal{A}}=\sum_{n=0}^{\infty}\frac{(-1)^n}{(2n+1)!}\mathcal{A}^{2n+1} Prove or disprove : ∃2×2\exists 2\times 2 matrix A∈M2(R)A\in \mathcal{M}_2(\mathbb{R}) such that sin⁡A=(1199601) \sin{A}=\left(\begin{array}{cc}1 & 1996 \\0 & 1 \end{array}\right)
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