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1. Cube volume equals six times edge sum ●●○○○ 20 ptsThe volume of a cube is numerically equal to six times the sum of the lengths of its edges. Find the volume of the cube, in cubic units, rounded to two decimal places.
2. Arithmetic then geometric progression integers ●●●○○ 50 ptsThree distinct integers $a$, $b$, $c$ satisfy the following conditions: - $abc = 17955$; - $a$, $b$, $c$ are three consecutive terms of an arithmetic sequence, in that order; - $3a+b$, $3b+c$, $3c+a$ are three consecutive terms of a geometric sequence, in that order. Find $a+b+c$.
3. Equation with cube roots ●●●○○ 50 ptsSolve the equation $\sqrt[3]{2x+1}+\sqrt[3]{4-x}=-\sqrt[3]{3}$ for $x$.
4. Exponential equation with a cube root ●●●●○ 100 ptsSolve the equation $27^{x}-7 \sqrt[3]{7 \cdot 3^{x}+6}=6$.