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Advanced Problem Solving Module 19

Unit Interval
problem
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Unit Interval

Age
14 to 18
Challenge level
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Take any two numbers between 0 and 1. Prove that the sum of the numbers is always less than one plus their product?
Eyes Down
problem
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Eyes Down

Age
16 to 18
Challenge level
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The symbol [ ] means 'the integer part of'. Can the numbers [2x]; 2[x]; [x + 1/2] + [x - 1/2] ever be equal? Can they ever take three different values?
In Between
problem
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In Between

Age
16 to 18
Challenge level
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Can you find the solution to this algebraic inequality?