Sunday 19 May 2019

ISI 2019 SOLUTIONS

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Two solid cylinders are mathematically similar. The sum of their heights is 1. The sum of their surface areas is 8π. The sum of their volumes is 2π. Find all possibilities for the dimensions of each cylinder.


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Ares multiplies two integers which differ by 9. Grace multiplies two integers which differ by 6. They obtain the same product T. Determine all possible values of T.


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A list of five two-digit positive integers is written in increasing order on a blackboard. Each of the five integers is a multiple of 3, and each digit 0,1,2,3,4,5,6,7,8,9 appears exactly once on the blackboard. In how many ways can this be done? Note that a two-digit number cannot begin with the digit 0.


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Define \(f: \mathbb{R} \rightarrow \mathbb{R}\) by \[f(x)= \begin{cases}(1-\cos x) \sin \left(\frac{1}{x}\right), & x \neq 0 \\ 0, &a...