Olympiad problems

2003 Gheorghe Vranceanu, 1

Solve in $ \mathbb{R}^2 $ the equation $ \lfloor x/y-y/x \rfloor =x^2/y+y/x^2. $

1962 Miklós Schweitzer, 4

Tags: symmetry , floor function , quadratic , advanced fields , modular arithmetic

Show that \[ \prod_{1\leq x < y \leq \frac{p\minus{}1}{2}} (x^2\plus{}y^2) \equiv (\minus{}1)^{\lfloor\frac{p\plus{}1}{8}\rfloor} \;(\textbf{mod}\;p\ ) \] for every prime $ p\equiv 3 \;(\textbf{mod}\;4\ )$. [J. Suranyi]

2005 AIME Problems, 9

Tags: trigonometry , floor function , modular arithmetic , complex number

For how many positive integers $n$ less than or equal to $1000$ is \[(\sin t + i \cos t)^n=\sin nt + i \cos nt\] true for all real $t$?

2021 Peru Cono Sur TST., P1

Tags: number theory , floor function

Find the set of all possible values of the expression $\lfloor m^2+\sqrt{2} n \rfloor$, where $m$ and $n$ are positive integers. Note: The symbol $\lfloor x\rfloor$ denotes the largest integer less than or equal to $x$.

2012 ELMO Shortlist, 4

Tags: floor function , induction , inequalities , pigeonhole principle , combinatorics , ceiling function

A tournament on $2k$ vertices contains no $7$-cycles. Show that its vertices can be partitioned into two sets, each with size $k$, such that the edges between vertices of the same set do not determine any $3$-cycles. [i]Calvin Deng.[/i]

2004 Uzbekistan National Olympiad, 1

Tags: floor function , algebra , function

Solve the equation: $[\sqrt x+\sqrt{x+1}]+[\sqrt {4x+2}]=18$

2012 Indonesia TST, 1

Tags: function , floor function , algebra , induction

Suppose a function $f : \mathbb{Z}^+ \rightarrow \mathbb{Z}^+$ satisfies $f(f(n)) + f(n+1) = n+2$ for all positive integer $n$. Prove that $f(f(n)+n) = n+1$ for all positive integer $n$.

2010 ELMO Shortlist, 4

Tags: floor function , algebra , number theory , polynomial

Let $r$ and $s$ be positive integers. Define $a_0 = 0$, $a_1 = 1$, and $a_n = ra_{n-1} + sa_{n-2}$ for $n \geq 2$. Let $f_n = a_1a_2\cdots a_n$. Prove that $\displaystyle\frac{f_n}{f_kf_{n-k}}$ is an integer for all integers $n$ and $k$ such that $0 < k < n$. [i]Evan O' Dorney.[/i]

2012 Romanian Master of Mathematics, 5

Tags: combinatorics , geometry , floor function , rectangle , double counting , ceiling function

Given a positive integer $n\ge 3$, colour each cell of an $n\times n$ square array with one of $\lfloor (n+2)^2/3\rfloor$ colours, each colour being used at least once. Prove that there is some $1\times 3$ or $3\times 1$ rectangular subarray whose three cells are coloured with three different colours. [i](Russia) Ilya Bogdanov, Grigory Chelnokov, Dmitry Khramtsov[/i]

2004 Italy TST, 3

Tags: algorithm , floor function , function , algebra , induction , quadratic

Find all functions $f:\mathbb{N}\rightarrow \mathbb{N}$ such that for all $m,n\in\mathbb{N}$, \[(2^m+1)f(n)f(2^mn)=2^mf(n)^2+f(2^mn)^2+(2^m-1)^2n. \]

1999 Federal Competition For Advanced Students, Part 2, 3

Tags: probability , floor function , hyperbola , parabola , conic , function

Find all pairs $(x, y)$ of real numbers such that \[y^2 - [x]^2 = 19.99 \text{ and } x^2 + [y]^2 = 1999\] where $f(x)=[x]$ is the floor function.

PEN P Problems, 3

Tags: floor function , algebra , function , additive number theory

Prove that infinitely many positive integers cannot be written in the form \[{x_{1}}^{3}+{x_{2}}^{5}+{x_{3}}^{7}+{x_{4}}^{9}+{x_{5}}^{11},\] where $x_{1}, x_{2}, x_{3}, x_{4}, x_{5}\in \mathbb{N}$.

2007 Brazil National Olympiad, 3

Tags: floor function , inequalities , geometry , triangle inequality , angle bisector

Consider $ n$ points in a plane which are vertices of a convex polygon. Prove that the set of the lengths of the sides and the diagonals of the polygon has at least $ \lfloor n/2\rfloor$ elements.

2016 China Team Selection Test, 4

Tags: number theory , floor function

Set positive integer $m=2^k\cdot t$, where $k$ is a non-negative integer, $t$ is an odd number, and let $f(m)=t^{1-k}$. Prove that for any positive integer $n$ and for any positive odd number $a\le n$, $\prod_{m=1}^n f(m)$ is a multiple of $a$.

2022 BMT, 4

Tags: floor function , ceiling function

Find all real $x$ such that $$\lfloor x \lceil x \rceil \rfloor = 2022.$$ Express your answer in interval notation.

2005 France Team Selection Test, 4

Tags: floor function , number theory , induction , logarithm , inequalities , ceiling function

Let $X$ be a non empty subset of $\mathbb{N} = \{1,2,\ldots \}$. Suppose that for all $x \in X$, $4x \in X$ and $\lfloor \sqrt{x} \rfloor \in X$. Prove that $X=\mathbb{N}$.

1998 Portugal MO, 4

Tags: floor function , algebra

What is the largest integer less than or equal to $$\frac{3^{31}+2^{31}}{3^{29}+2^{29}} \,\,\, ?$$

1991 AIME Problems, 6

Tags: function , floor function , algebra , number theory , diophantine equation , inequalities , modular arithmetic

Suppose $r$ is a real number for which \[ \left\lfloor r + \frac{19}{100} \right\rfloor + \left\lfloor r + \frac{20}{100} \right\rfloor + \left\lfloor r + \frac{21}{100} \right\rfloor + \cdots + \left\lfloor r + \frac{91}{100} \right\rfloor = 546. \] Find $\lfloor 100r \rfloor$. (For real $x$, $\lfloor x \rfloor$ is the greatest integer less than or equal to $x$.)

2012 China Team Selection Test, 1

Tags: number theory , floor function , function , logarithm

Given an integer $n\ge 4$. $S=\{1,2,\ldots,n\}$. $A,B$ are two subsets of $S$ such that for every pair of $(a,b),a\in A,b\in B, ab+1$ is a perfect square. Prove that \[\min \{|A|,|B|\}\le\log _2n.\]

2014 Postal Coaching, 5

Tags: floor function , combinatorics , number theory

Let $A=\{1,2,3,\ldots,40\}$. Find the least positive integer $k$ for which it is possible to partition $A$ into $k$ disjoint subsets with the property that if $a,b,c$ (not necessarily distinct) are in the same subset, then $a\ne b+c$.

PEN I Problems, 9

Tags: floor function

Show that for all positive integers $m$ and $n$, \[\gcd(m, n) = m+n-mn+2\sum^{m-1}_{k=0}\left \lfloor \frac{kn}{m}\right \rfloor.\]

2007 Romania Team Selection Test, 1

Tags: floor function , function , factorial , inequalities , number theory

Prove that the function $f : \mathbb{N}\longrightarrow \mathbb{Z}$ defined by $f(n) = n^{2007}-n!$, is injective.

1985 USAMO, 4

Tags: floor function , algebra , ceiling function

There are $n$ people at a party. Prove that there are two people such that, of the remaining $n-2$ people, there are at least $\left\lfloor\frac{n}{2}\right\rfloor-1$ of them, each of whom either knows both or else knows neither of the two. Assume that knowing is a symmetric relation, and that $\lfloor x\rfloor$ denotes the greatest integer less than or equal to $x$.

2008 Bulgaria Team Selection Test, 3

Tags: floor function , algebra , polynomial , combinatorics , induction , ceiling function

Let $G$ be a directed graph with infinitely many vertices. It is known that for each vertex the outdegree is greater than the indegree. Let $O$ be a fixed vertex of $G$. For an arbitrary positive number $n$, let $V_{n}$ be the number of vertices which can be reached from $O$ passing through at most $n$ edges ( $O$ counts). Find the smallest possible value of $V_{n}$.

2008 All-Russian Olympiad, 4

Tags: floor function , combinatorics

There are several scientists collaborating in Niichavo. During an $ 8$-hour working day, the scientists went to cafeteria, possibly several times.It is known that for every two scientist, the total time in which exactly one of them was in cafeteria is at least $ x$ hours ($ x>4$). What is the largest possible number of scientist that could work in Niichavo that day,in terms of $ x$?