91Ó°ÊÓ

Let \((a, b, c, d) \in \mathrm{R}^{4}\). Prove that $$ || a-c|-| b-c|| \leq|a-b| \leq|a-c|+|b-c| . $$

Given that 1002004008016032 has a prime factor \(\boldsymbol{p}>250000\), find it.

Let \(A, B\) be finite sets with \(\operatorname{card}(A)=n\) and \(\operatorname{card}(B)=m .\) Prove that \- The number of functions from \(A\) to \(B\) is \(m^{n}\). \- If \(n \leq m\), the number of injective functions from \(A\) to \(B\) is \(m(m-1)(m-2) \cdots(m-n+1) .\) If \(n>m\) there are no injective functions from \(A\) to \(B\).

(Putnam 1948) If \(n\) is a positive integer, demonstrate that $$ \lfloor\sqrt{n}+\sqrt{n+1}\rfloor=\lfloor\sqrt{4 n+2} \rrbracket $$

Let \(a_{k}\) be a sequence of pairwise distinct positive integers. Prove that $$ \sum_{k=1}^{n} \frac{a_{k}}{k^{2}} \geq \sum_{k=1}^{n} \frac{1}{k} $$