Find all pairs of consecutive even positive integers,both of which are larger than $5$,such that their sum is less than $23$.

Solve for $x$ in the following inequality: $\frac{4}{x+1} \leq 3 \leq \frac{6}{x+1}$ where $x > 0$.

The longest side of a triangle is twice the shortest side and the third side is $2 \text{ cm}$ longer than the shortest side. If the perimeter of the triangle is more than $166 \text{ cm}$, then find the minimum length of the shortest side. (in $\text{ cm}$)

The number of points $P(x, y)$ with natural numbers as coordinates that lie inside the quadrilateral formed by the lines $2x + y = 2$,$x = 0$,$y = 0$,and $x + y = 5$ is

The origin belongs to a region between the lines $x+2y-5=0$ and $3x-4y+5=0$. The number of points in that region of the form $((\alpha-1)^2, \alpha)$,where $\alpha \in \mathbb{Z}$,is

The cost and revenue functions of a product are given by $C(x) = 20x + 4000$ and $R(x) = 60x + 2000$ respectively,where $x$ is the number of items produced and sold. The value of $x$ to earn profit is