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4 problems match “integer optimization”. Clear

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1. Greatest integer in a set of five ●○○○○ 10 ptsFive positive integers have an average of 15. What is the greatest possible value of one of these integers?
2. Greatest third term of an arithmetic sequence ●●○○○ 20 ptsThe sum of four positive integers that form an arithmetic sequence is 46. Of all such possible sequences, what is the greatest possible third term?
3. Smallest entry sum for a conjugating matrix ●●●○○ 50 ptsFor some matrix $\mathbf{P} = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$ with nonzero integer entries, $\begin{pmatrix} 4 & 3 \\ -1 & 0 \end{pmatrix} = \mathbf{P}^{-1} \begin{pmatrix} 1 & 0 \\ 0 & 3 \end{pmatrix} \mathbf{P}$. Find the smallest possible value of $|a| + |b| + |c| + |d|$.
4. Largest edge of a box ●●●○○ 50 ptsAn $m\times n\times p$ rectangular box has half the volume of an $(m + 2)\times(n + 2)\times(p + 2)$ rectangular box, where $m, n,$ and $p$ are positive integers with $m\le n\le p$. Find the largest possible value of $p$.