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7 problems match “Diophantine equations”. Clear

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1. 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?
2. Largest number of jogs ●●○○○ 20 ptsBill can buy jags, jigs and jogs for $1, $2 and $7 each, respectively. What is the largest number of jogs he can purchase if he must buy at least one of each item and will spend exactly $50?
3. 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.
4. Sum of variables in a month-length equation ●●●○○ 50 ptsLet $x$, $y$, $z$ be natural numbers satisfying $29x+30y+31z=366$. Find $x+y+z$.
5. Least n with many unit-fraction solutions ●●●○○ 50 ptsLet $x$, $y$ and $n$ be positive integers, and count the solutions $(x, y)$ of $\frac{1}{x}+\frac{1}{y}=\frac{1}{n}$ with $x \le y$. For $n=4$ there are exactly three such solutions. Find the least value of $n$ for which the number of solutions exceeds one thousand.
6. 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 $p$.
7. Product preserved after altering two numbers ●●●○○ 50 ptsLucas writes two distinct positive integers on a whiteboard. He decreases the smaller number by 20 and increases the larger number by 23, and finds that the product of the two original numbers equals the product of the two altered numbers. Compute the minimum possible sum of the original two numbers.