Practice (138)
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 constant rate at which Suzanna rides her bike. If she rides a total of a half an hour at the same speed, how many miles would she have ridden?
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)
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 $\sqrt{x^2 - 4x + 5} + \sqrt{x^2 +4x +8}$.
Let $a$, $b$, $c$, $x$, $y$, and $z$ be positive numbers. Show that $$\sqrt{a^2+x^2} + \sqrt{b^2 + y^2}+\sqrt{c^2+z^2} \ge \sqrt{(a+b+c)^2 + (x+y+z)^2}$$
Let positive numbers $a$, $b$ and $c$ satisfy $a+b+c=8$. Find the minimal value of $\sqrt{a^2+1}+\sqrt{b^2+4}+\sqrt{c^2+9}$.