#### TheGraphMethod

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Let $n$ be a positive integer, prove that in this series $$\Big\lfloor{\frac{n}{1}}\Big\rfloor, \Big\lfloor{\frac{n}{2}}\Big\rfloor, \Big\lfloor{\frac{n}{3}}\Big\rfloor \cdots \Big\lfloor{\frac{n}{n}}\Big\rfloor$$, there are less than $2\sqrt{n}$ integers distinct.

For each integer $n \ge 2$, let $A(n)$ be the area of the region in the coordinate plane defined by the inequalities $1\le x \le n$ and $0\le y \le x \left\lfloor \sqrt x \right\rfloor$, where $\left\lfloor \sqrt x \right\rfloor$ is the greatest integer not exceeding $\sqrt x$. Find the number of values of $n$ with $2\le n \le 1000$ for which $A(n)$ is an integer.

This graph shows the water temperature T degrees at time t minutes for a pot of water placed on a stove and heated to 100 degrees. On average, how many degrees did the temperature increase each minute during the first $8$ minutes? For how many integer values of $k$ do the graphs of $x^2+y^2=k^2$ and $xy = k$ not intersect?

The graph shows the price of five gallons of gasoline during the first ten months of the year. By what percent is the highest price more than the lowest price? The graph shows the number of minutes studied by both Asha (black bar) and Sasha (grey bar) in one week. On the average, how many more minutes per day did Sasha study than Asha? In square units, what is the area of the region bounded by the graph of |x \u2013 y| + |x + y| = 6 ?

The graph of the function $f$ is shown below. How many solutions does the equation $f(f(x))=6$ have?

Find the minimal value of $\sqrt{x^2 - 4x + 5} + \sqrt{x^2 +4x +8}$.