Olympiad problems

2008 Korean National Olympiad, 8

For fixed positive integers $s, t$, define $a_n$ as the following. $a_1 = s, a_2 = t$, and $\forall n \ge 1$, $a_{n+2} = \lfloor \sqrt{a_n+(n+2)a_{n+1}+2008} \rfloor$. Prove that the solution set of $a_n \not= n$, $n \in \mathbb{N}$ is finite.

MOAA Team Rounds, 2022.13

Tags: algebra , floor function

Determine the number of distinct positive real solutions to $$\lfloor x \rfloor ^{\{x\}} = \frac{1}{2022}x^2$$ . Note: $\lfloor x \rfloor$ is known as the floor function, which returns the greatest integer less than or equal to $x$. Furthermore, $\{x\}$ is defined as $x - \lfloor x \rfloor$.

2006 Stanford Mathematics Tournament, 14

Tags: floor function , factorial , function , modular arithmetic , number theory

Find the smallest nonnegative integer $n$ for which $\binom{2006}{n}$ is divisible by $7^3$.

2005 Indonesia MO, 5

Tags: number theory , floor function , function

For an arbitrary real number $ x$, $ \lfloor x\rfloor$ denotes the greatest integer not exceeding $ x$. Prove that there is exactly one integer $ m$ which satisfy $ \displaystyle m\minus{}\left\lfloor \frac{m}{2005}\right\rfloor\equal{}2005$.

2005 Harvard-MIT Mathematics Tournament, 4

Tags: algebra , inequalities , function , floor function , search

If $a,b,c>0$, what is the smallest possible value of $ \left\lfloor \dfrac {a+b}{c} \right\rfloor + \left\lfloor \dfrac {b+c}{a} \right\rfloor + \left\lfloor \dfrac {c+a}{b} \right\rfloor $? (Note that $ \lfloor x \rfloor $ denotes the greatest integer less than or equal to $x$.)

1999 AIME Problems, 6

Tags: geometry , floor function , inversion , analytic geometry

A transformation of the first quadrant of the coordinate plane maps each point $(x,y)$ to the point $(\sqrt{x},\sqrt{y}).$ The vertices of quadrilateral $ABCD$ are $A=(900,300), B=(1800,600), C=(600,1800),$ and $D=(300,900).$ Let $k$ be the area of the region enclosed by the image of quadrilateral $ABCD.$ Find the greatest integer that does not exceed $k.$

2010 Bundeswettbewerb Mathematik, 2

Tags: floor function , algebra , recurrence relation , sequence , radical

The sequence of numbers $a_1, a_2, a_3, ...$ is defined recursively by $a_1 = 1, a_{n + 1} = \lfloor \sqrt{a_1+a_2+...+a_n} \rfloor $ for $n \ge 1$. Find all numbers that appear more than twice at this sequence.

2014 China Girls Math Olympiad, 2

Tags: floor function , inequalities , ceiling function

Let $x_1,x_2,\ldots,x_n $ be real numbers, where $n\ge 2$ is a given integer, and let $\lfloor{x_1}\rfloor,\lfloor{x_2}\rfloor,\ldots,\lfloor{x_n}\rfloor $ be a permutation of $1,2,\ldots,n$. Find the maximum and minimum of $\sum\limits_{i=1}^{n-1}\lfloor{x_{i+1}-x_i}\rfloor$ (here $\lfloor x\rfloor $ is the largest integer not greater than $x$).

2013 APMO, 3

Tags: algebra , floor function , arithmetic sequence

For $2k$ real numbers $a_1, a_2, ..., a_k$, $b_1, b_2, ..., b_k$ define a sequence of numbers $X_n$ by \[ X_n = \sum_{i=1}^k [a_in + b_i] \quad (n=1,2,...). \] If the sequence $X_N$ forms an arithmetic progression, show that $\textstyle\sum_{i=1}^k a_i$ must be an integer. Here $[r]$ denotes the greatest integer less than or equal to $r$.

2007 Mathematics for Its Sake, 3

Tags: algebra , floor function , equation

Solve in the real numbers the equation $ \lfloor ax \rfloor -\lfloor (1+a)x \rfloor = (1+a)(1-x) . $ [i]Dumitru Acu[/i]

1997 Romania National Olympiad, 4

Tags: algebra , floor function , equation

Consider the numbers $a,b, \alpha, \beta \in \mathbb{R}$ and the sets $$A=\left \{x \in \mathbb{R} : x^2+a|x|+b=0 \right \},$$ $$B=\left \{ x \in \mathbb{R} : \lfloor x \rfloor^2 + \alpha \lfloor x \rfloor + \beta = 0\right \}.$$ If $A \cap B$ has exactly three elements, prove that $a$ cannot be an integer.

1999 Singapore Team Selection Test, 2

Tags: floor function , algebra

Find all possible values of $$ \lfloor \frac{x - p}{p} \rfloor + \lfloor \frac{-x-1}{p} \rfloor $$ where $x$ is a real number and $p$ is a nonzero integer. Here $\lfloor z \rfloor$ denotes the greatest integer less than or equal to $z$.

2011 Balkan MO Shortlist, C1

Tags: ratio , number theory , floor function , modular arithmetic

Let $S$ be a finite set of positive integers which has the following property:if $x$ is a member of $S$,then so are all positive divisors of $x$. A non-empty subset $T$ of $S$ is [i]good[/i] if whenever $x,y\in T$ and $x<y$, the ratio $y/x$ is a power of a prime number. A non-empty subset $T$ of $S$ is [i]bad[/i] if whenever $x,y\in T$ and $x<y$, the ratio $y/x$ is not a power of a prime number. A set of an element is considered both [i]good[/i] and [i]bad[/i]. Let $k$ be the largest possible size of a [i]good[/i] subset of $S$. Prove that $k$ is also the smallest number of pairwise-disjoint [i]bad[/i] subsets whose union is $S$.

2010 HMNT, 6

Tags: algebra , floor function

What is the sum of the positive solutions to $2x^2 -\lfloor x \rfloor = 5$, where $\lfloor x \rfloor$ is the largest integer less than or equal to $x$?

2015 Brazil Team Selection Test, 2

Tags: floor function , number theory , sequence , parity

Let $n > 1$ be a given integer. Prove that infinitely many terms of the sequence $(a_k )_{k\ge 1}$, defined by \[a_k=\left\lfloor\frac{n^k}{k}\right\rfloor,\] are odd. (For a real number $x$, $\lfloor x\rfloor$ denotes the largest integer not exceeding $x$.) [i]Proposed by Hong Kong[/i]

1990 Baltic Way, 17

Tags: floor function

There are two piles with $72$ and $30$ candies. Two students alternate taking candies from one of the piles. Each time the number of candies taken from a pile must be a multiple of the number of candies in the other pile. Which student can always assure taking the last candy from one of the piles?

2019 HMNT, 5

Tags: floor function , ceiling function , algebra

Compute the sum of all positive real numbers $x \le 5$ satisfying $$x =\frac{ \lceil x^2 \rceil + \lceil x \rceil \cdot \lfloor x \rfloor}{ \lceil x\rceil + \lfloor x \rfloor}$$

2016 PUMaC Individual Finals A, 2

Tags: number theory , floor function

Let $m, k$, and $c$ be positive integers with $k > c$, and let $\lambda$ be a positive, non-integer real root of the equation $\lambda^{m+1} - k \lambda^m - c = 0$. Let $f : Z^+ \to Z$ be defined by $f(n) = \lfloor \lambda n \rfloor$ for all $n \in Z^+$. Show that $f^{m+1}(n) \equiv cn - 1$ (mod $k$) for all $n \in Z^+$. (Here, $Z^+$ denotes the set of positive integers, $ \lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$, and $f^{m+1}(n) = f(f(... f(n)...))$ where $f$ appears $m + 1$ times.)

1998 IberoAmerican, 3

Tags: algebra , floor function , modular arithmetic

Let $\lambda$ the positive root of the equation $t^2-1998t-1=0$. It is defined the sequence $x_0,x_1,x_2,\ldots,x_n,\ldots$ by $x_0=1,\ x_{n+1}=\lfloor\lambda{x_n}\rfloor\mbox{ for }n=1,2\ldots$ Find the remainder of the division of $x_{1998}$ by $1998$. Note: $\lfloor{x}\rfloor$ is the greatest integer less than or equal to $x$.