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Found problems: 1

2024 Benelux, 1
Tags: sequence , algebra inequality , algebra , inequalities
Let $a_0,a_1,\dots,a_{2024}$ be real numbers such that $\left|a_{i+1}-a_i\right| \le 1$ for $i=0,1,\dots,2023$. a) Find the minimum possible value of $$a_0a_1+a_1a_2+\dots+a_{2023}a_{2024}$$ b) Does there exist a real number $C$ such that $$a_0a_1-a_1a_2+a_2a_3-a_3a_4+\dots+a_{2022}a_{2023}-a_{2023}a_{2024} \ge C$$ for all real numbers $a_0,a_1,\dots,a_2024$ such that $\left|a_{i+1}-a_i\right| \le 1$ for $i=0,1,\dots,2023$.