Math Core

The floor function turns up all over the AMC 10 and 12: in equations like $x \lfloor x \rfloor = 50$ , in long sums like $\lfloor \sqrt 1 \rfloor + \lfloor \sqrt 2 \rfloor + \dots + \lfloor \sqrt{100} \rfloor$ , and in counting how many different values an expression takes. The tools are few: split $x$ into integer and fractional parts, group terms where the floor is constant, and pair terms that add up nicely.

Integer and fractional parts

Definition

Floor and fractional part

$\lfloor x \rfloor$ is the greatest integer less than or equal to $x$ . The fractional part is $\{x\} = x - \lfloor x \rfloor$ , so $0 \le \{x\} \lt 1$ and $x = \lfloor x \rfloor + \{x\}$ .

Watch negatives: $\lfloor -2.3 \rfloor = -3$ , not $-2$ . The single most useful restatement is

$\lfloor x \rfloor = n \iff n \le x \lt n + 1 \qquad (n \text{ an integer}).$

To solve an equation with a floor, name the floor: set $n = \lfloor x \rfloor$ , solve for $x$ in terms of $n$ , and then impose $n \le x \lt n + 1$ to find which $n$ are allowed.

A few facts you'll use constantly:

$\lfloor x + m \rfloor = \lfloor x \rfloor + m$ for any integer $m$ .
$\left\lfloor \dfrac{n}{d} \right\rfloor$ is the number of multiples of $d$ from $1$ to $n$ (you used this in Legendre's formula).
$\lfloor x \rfloor + \lfloor y \rfloor \le \lfloor x + y \rfloor \le \lfloor x \rfloor + \lfloor y \rfloor + 1$ .

Sums of floors

Group by value. In $\sum \lfloor \sqrt{k} \rfloor$ , the value is $m$ for every $k$ from $m^2$ to $(m+1)^2 - 1$ , which is $2m + 1$ values of $k$ . So instead of adding $100$ terms, you add about $10$ blocks.

Pair up. If $\gcd(a, n) = 1$ and $1 \le k \le n - 1$ , then $\dfrac{ak}{n}$ is never an integer, and

$\left\lfloor \frac{ak}{n} \right\rfloor + \left\lfloor \frac{a(n-k)}{n} \right\rfloor = a - 1,$

because the two fractions add up to exactly $a$ and neither is an integer. Pairing $k$ with $n - k$ gives the following.

Two floor-sum tools

Grouping: $\lfloor f(k) \rfloor$ is constant on blocks of $k$ ; count the size of each block and multiply.
Pairing: if $\gcd(a, n) = 1$ , then $\displaystyle\sum_{k=1}^{n-1} \left\lfloor \frac{ak}{n} \right\rfloor = \frac{(a-1)(n-1)}{2}$ .

Counting distinct values

How many different values does $\left\lfloor \dfrac{k^2}{N} \right\rfloor$ take? While consecutive values of $\dfrac{k^2}{N}$ differ by less than $1$ (that is, $2k + 1 \lt N$ ), the floor can't skip any integer, so every integer in the range appears. Once they differ by at least $1$ , every new $k$ gives a new value. Split the range at that point and count each part.

Worked example: An equation with a floor

Find the positive real number $x$ with $x \lfloor x \rfloor = 27$ .

If $0 \lt x \lt 1$ the product is $0$ , so $x \ge 1$ . Let $n = \lfloor x \rfloor \ge 1$ . Then $x = \dfrac{27}{n}$ , and you need $n \le \dfrac{27}{n} \lt n + 1$ , that is $n^2 \le 27 \lt n^2 + n$ . Only $n = 5$ works: $25 \le 27 \lt 30$ . So $x = \dfrac{27}{5} = 5.4$ . Check: $5.4 \cdot 5 = 27$ .

Worked example: Grouping

Find $\lfloor \sqrt{1} \rfloor + \lfloor \sqrt{2} \rfloor + \dots + \lfloor \sqrt{100} \rfloor$ .

For $m = 1, \dots, 9$ , the value $m$ appears $2m + 1$ times, and $\lfloor \sqrt{100} \rfloor = 10$ appears once. So the sum is

$\sum_{m=1}^{9} m(2m + 1) + 10 = 2 \cdot 285 + 45 + 10 = 625,$

using $1^2 + \dots + 9^2 = 285$ and $1 + \dots + 9 = 45$ .

Worked example: Pairing

Find $\displaystyle\sum_{k=1}^{30} \left\lfloor \frac{7k}{31} \right\rfloor$ .

$\gcd(7, 31) = 1$ , so pairing $k$ with $31 - k$ gives $15$ pairs, each adding to $7 - 1 = 6$ . The sum is $15 \cdot 6 = 90$ , matching $\dfrac{(7-1)(31-1)}{2} = 90$ .

Worked example: Counting distinct values

How many distinct numbers are in the list $\left\lfloor \dfrac{1^2}{100} \right\rfloor, \left\lfloor \dfrac{2^2}{100} \right\rfloor, \dots, \left\lfloor \dfrac{100^2}{100} \right\rfloor$ ?

Consecutive terms of $\dfrac{k^2}{100}$ differ by $\dfrac{2k + 1}{100}$ , which is less than $1$ for $k \le 49$ . So from $k = 1$ to $k = 50$ the floors climb from $0$ to $25$ without skipping: $26$ values. From $k = 50$ on the gaps are at least $1.01$ , so $k = 51, \dots, 100$ each give a new, larger value: $50$ more. The total is $76$ .

Common mistake

Two classic traps. First, $\lfloor -x \rfloor \ne -\lfloor x \rfloor$ unless $x$ is an integer: $\lfloor -5.4 \rfloor = -6$ . Second, after naming $n = \lfloor x \rfloor$ and solving, you must check $n \le x \lt n + 1$ . Most candidate values of $n$ fail that check.

Practice

Practice 1

What is $\left\lfloor \sqrt{2026} \right\rfloor + \left\lfloor \sqrt[3]{2026} \right\rfloor$ ?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 2

What is the positive real number $x$ with $x \lfloor x \rfloor = 50$ ?

Practice 3

For how many positive integers $n \le 100$ is $\left\lfloor \dfrac{n}{2} \right\rfloor + \left\lfloor \dfrac{n}{3} \right\rfloor + \left\lfloor \dfrac{n}{6} \right\rfloor = n$ ?

Practice 4

Find $\displaystyle\sum_{k=1}^{100} \left\lfloor \frac{k}{3} \right\rfloor$ .

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 5

What is $\displaystyle\sum_{k=1}^{40} \left\lfloor \frac{13k}{41} \right\rfloor$ ?

Practice 6

For how many positive integers $n \le 100$ is $\lfloor \sqrt{n} \rfloor$ a divisor of $n$ ?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 7

How many distinct numbers are in the list

$\left\lfloor \frac{1^2}{200} \right\rfloor, \left\lfloor \frac{2^2}{200} \right\rfloor, \left\lfloor \frac{3^2}{200} \right\rfloor, \dots, \left\lfloor \frac{200^2}{200} \right\rfloor?$

Practice 8

Find $\displaystyle\sum_{k=1}^{1000} \lfloor \log_2 k \rfloor$ .

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Real contest practice

2020 AMC 10A, Problem 22: when a sum of three floors of $\dfrac{998}{n}, \dfrac{999}{n}, \dfrac{1000}{n}$ is a multiple of $3$ .
2020 AMC 10B, Problem 24: an equation mixing $n$ and $\lfloor \sqrt{n} \rfloor$ , solved by naming the floor.