8. Find the sum of all integer values of c for which the equation $10|p-3|+|2 p-| p+c \mid|=6$ has at least one root with respect to p

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# Solution:

Consider the function $\mathrm{f}(\mathrm{r})=10|\mathrm{p}-3|+|2 \mathrm{p}-| \mathrm{p}+\mathrm{c}||-6 \mathrm{p}$. The coefficient of the first modulus in absolute value is greater than the sum of the other coefficients of $\mathrm{p}$. $10>2+1+6$. Therefore, on all intervals up to $\mathrm{p}=3$, the coefficient of the linear increment is negative, and after $\mathrm{p}=3$ it is positive. $\mathrm{p}=3$ is a point of minimum. For the equation $\mathrm{f}(\mathrm{p})=0$ to have at least one root, it is necessary and sufficient that the inequality:

$\mathrm{f}(3) \leq 0$

$\Rightarrow$ Let $|c+3|=\mathrm{t} ;|6-\mathrm{t}|-18 \leq 0$

$(6-\mathrm{t})^{2}-18^{2} \geq 0$

$(24-t)(t+12) \geq 0$

$\mathrm{t} \in[-12 ; 24]$

$|c+3| \leq 24$

Answer: -147.