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Part of 2015 VJIMC

Problems(2)

VJIMC 2015 Category I, Problem 1

Source: VJIMC2015

8/10/2015
Problem 1 Let f:R→Rf : \mathbb{R} \rightarrow \mathbb{R} be differentiable on R\mathbb{R}. Prove that there exists x∈[0,1]x \in [0, 1] such that 4π(f(1)−f(0))=(1+x2)f′(x) .\frac{4}{\pi} ( f(1) - f(0) ) = (1+x^2) f'(x) \ .
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VJIMC 2015 Category II, Problem 1

Source: VJIMC2015

8/10/2015
Problem 1 Let AA and BB be two 3×33 \times 3 matrices with real entries. Prove that A−(A−1+(B−1−A)−1)−1=ABA , A-(A^{-1} +(B^{-1}-A)^{-1})^{-1} =ABA\ , provided all the inverses appearing on the left-hand side of the equality exist.
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