MathDB

2

Part of 2009 IMC

Problems(2)

IMC 2009 Day 1 P2

Source:

7/15/2014
Let A,B,CA,B,C be real square matrices of the same order, and suppose AA is invertible. Prove that (A−B)C=BA−1  ⟹  C(A−B)=A−1B (A-B)C=BA^{-1}\implies C(A-B)=A^{-1}B
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IMC 2009 Day 2 P2

Source:

7/16/2014
Suppose f:R→Rf:\mathbb{R}\to \mathbb{R} is a two times differentiable function satisfying f(0)=1,f′(0)=0f(0)=1,f^{\prime}(0)=0 and for all x∈[0,∞)x\in [0,\infty), it satisfies f′′(x)−5f′(x)+6f(x)≥0 f^{\prime \prime}(x)-5f^{\prime}(x)+6f(x)\ge 0 Prove that, for all x∈[0,∞)x\in [0,\infty), f(x)≥3e2x−2e3x f(x)\ge 3e^{2x}-2e^{3x}
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