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Found problems: 2

2019 Ramnicean Hope, 3
For this exercise, $ \{\} $ denotes the fractional part. [b]a)[/b] Let be a natural number $ n. $ Compare $ \left\{ \sqrt{n+1} -\sqrt{n} \right\} $ with $ \left\{ \sqrt{n} -\sqrt{n-1} \right\} . $ [b]b)[/b] Show that there are two distinct natural numbers $ a,b, $ such that $ \left\{ \sqrt{a} -\sqrt{b} \right\} =\left\{ \sqrt{b} -\sqrt{a} \right\} . $ [i]Traian Preda[/i]
equalities inequalities fractional part algebra fractional parts

2004 District Olympiad, 2
Find all natural numbers for which there exist that many distinct natural numbers such that the factorial of one of these is equal to the product of the factorials of the rest of them.
equalities factorial algebra