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2 problems match “optimization over integers”. Clear

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1. Largest sum for an involutory matrix ●●○○○ 20 ptsLet $a,$ $b,$ $c$ be integers such that $\mathbf{A} = \frac{1}{5} \begin{pmatrix} -3 & a \\ b & c \end{pmatrix}$ and $\mathbf{A}^2 = \mathbf{I}.$ Find the largest possible value of $a + b + c.$
2. Smallest n with fewer pencils after a gift ●●●○○ 50 ptsI have two cents and Bill has $n$ cents. Bill wants to buy pencils, which come in two packages: one package costs 6 cents for 7 pencils, and the other costs 10 cents for 12 pencils. Bill can spend all $n$ of his cents on some combination of these packages to get $P$ pencils. If I give my two cents to Bill, he can instead spend all $n+2$ of his cents on some combination of these packages to get fewer than $P$ pencils. Find the smallest value of $n$ for which this is possible.