smc

For $i\in \{1,2,3,\dots ,10\},$ suppose Person $i$ picks the number $a_i$ and announces the number $i.$ We wish to find $a_6.$ Taking the indices modulo $10,$ we are given that $\frac{a_{i-1}+a_{i+1}}{2}=i,$ from which $a_{i-1}+a_{i+1}=2i.$ We have ten equations: five with odd-numbered indices and five with even-numbered indices. Note that these two sets of equations are independent. The set that involves $a_6$ is $$\begin{array}{rlrl}{a}_2+a_4& =6,& & (1)\\ a_4+a_6& =10,& & (2)\\ a_6+a_8& =14,& & (3)\\ a_8+a_{10}& =18,& & (4)\\ a_{10}+a_2& =2.& & (5)\end{array}$$ Summing these five equations, we get $2(a_2+a_4+a_6+a_8+a_{10})=50,$ from which $$a_2+a_4+a_6+a_8+a_{10}=25.(★)$$ Subtracting $(1)+(4)$ from $(★),$ we obtain $a_6=\boxed{1}.$