Floor and Ceiling Functions

How many distinct values of $x$ satisfy $\lfloor x \rfloor^2 - 3x + 2 = 0$, where $\lfloor x \rfloor$ denotes the largest integer less than or equal to $x$?

The graph of

is symmetric about which of the following? (Here $\lfloor x \rfloor$ is the greatest integer not exceeding $x$.)

Cozy the Cat and Dash the Dog are going up a staircase with a certain number of steps. However, instead of walking up the steps one at a time, both Cozy and Dash jump. Cozy goes two steps up with each jump (though if necessary, he will just jump the last step). Dash goes five steps up with each jump (though if necessary, he will just jump the last steps if there are fewer than $5$ steps left). Suppose Dash takes $19$ fewer jumps than Cozy to reach the top of the staircase. Let $s$ denote the sum of all possible numbers of steps this staircase can have. What is the sum of the digits of $s$?

For how many positive integers $n \le 1000$ is

not divisible by $3$? (Recall that $\lfloor x \rfloor$ is the greatest integer less than or equal to $x$.)

Suppose $r$ is a real number for which

Find $\lfloor 100r \rfloor$. (For real $x$, $\lfloor x \rfloor$ is the greatest integer less than or equal to $x$.)

How many positive integers $N$ less than $1000$ are there such that the equation $x^{\lfloor x \rfloor} = N$ has a solution for $x$?

Given a nonnegative real number $x$, let $\langle x \rangle$ denote the fractional part of $x$; that is, $\langle x \rangle = x - \lfloor x \rfloor$, where $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$. Suppose that $a$ is positive, $\langle a^{-1} \rangle = \langle a^2 \rangle$, and $2 < a^2 < 3$. Find the value of $a^{12} - 144a^{-1}$.

Find the least positive integer $k$ for which the equation $\left\lfloor \dfrac{2002}{n} \right\rfloor = k$ has no integer solutions for $n$. (The notation $\lfloor x \rfloor$ means the greatest integer less than or equal to $x$.)

Find the number of positive integers $n$ less than $1000$ for which there exists a positive real number $x$ such that $n = x\lfloor x \rfloor$. Note: $\lfloor x \rfloor$ is the greatest integer less than or equal to $x$.

For a real number $a$, let $\lfloor a \rfloor$ denote the greatest integer less than or equal to $a$. Let $\mathcal{R}$ denote the region in the coordinate plane consisting of points $(x, y)$ such that $\lfloor x \rfloor^2 + \lfloor y \rfloor^2 = 25$. The region $\mathcal{R}$ is completely contained in a disk of radius $r$ (a disk is the union of a circle and its interior). The minimum value of $r$ can be written as $\frac{\sqrt{m}}{n}$, where $m$ and $n$ are integers and $m$ is not divisible by the square of any prime. Find $m + n$.

A sample of $121$ integers is given, each between $1$ and $1000$ inclusive, with repetitions allowed. The sample has a unique mode (most frequent value). Let $D$ be the difference between the mode and the arithmetic mean of the sample. What is the largest possible value of $\lfloor D \rfloor$? (For real $x$, $\lfloor x \rfloor$ is the greatest integer less than or equal to $x$.)

Given a real number $x$, let $\lfloor x \rfloor$ denote the greatest integer less than or equal to $x$. For a certain integer $k$, there are exactly $70$ positive integers $n_1, n_2, \ldots, n_{70}$ such that

and $k$ divides $n_i$ for all $i$ such that $1 \le i \le 70$. Find the maximum value of $\dfrac{n_i}{k}$ for $1 \le i \le 70$.

How many positive integers $n$ satisfy

(Recall that $\lfloor x \rfloor$ is the greatest integer not exceeding $x$.)

Find the number of positive integers $n \le 600$ whose value can be uniquely determined when the values of $\left\lfloor \frac{n}{4} \right\rfloor$, $\left\lfloor \frac{n}{5} \right\rfloor$, and $\left\lfloor \frac{n}{6} \right\rfloor$ are given, where $\lfloor x \rfloor$ denotes the greatest integer less than or equal to the real number $x$.