Olympiad problems

2018 China Team Selection Test, 4

Let $k, M$ be positive integers such that $k-1$ is not squarefree. Prove that there exist a positive real $\alpha$, such that $\lfloor \alpha\cdot k^n \rfloor$ and $M$ are coprime for any positive integer $n$.

PEN D Problems, 12

Tags: congruence , modular arithmetic , floor function , abstract algebra , number theory , relatively prime , group theory

Suppose that $m>2$, and let $P$ be the product of the positive integers less than $m$ that are relatively prime to $m$. Show that $P \equiv -1 \pmod{m}$ if $m=4$, $p^n$, or $2p^{n}$, where $p$ is an odd prime, and $P \equiv 1 \pmod{m}$ otherwise.

2010 Romania Team Selection Test, 5

Tags: floor function , number theory

Let $a$ and $n$ be two positive integer numbers such that the (positive) prime factors of $a$ be all greater than $n$. Prove that $n!$ divides $(a - 1)(a^2 - 1)\cdots (a^{n-1} - 1)$. [i]AMM Magazine[/i]

PEN S Problems, 19

Tags: floor function , calculus , integration , induction

Determine all pairs $(a, b)$ of real numbers such that $a\lfloor bn\rfloor =b\lfloor an\rfloor$ for all positive integer $n$.

1999 Baltic Way, 6

Tags: induction , floor function , ceiling function , combinatorics

What is the least number of moves it takes a knight to get from one corner of an $n\times n$ chessboard, where $n\ge 4$, to the diagonally opposite corner?

2012 Math Prize For Girls Problems, 3

Tags: floor function

What is the least positive integer $n$ such that $n!$ is a multiple of $2012^{2012}$?

2003 Vietnam Team Selection Test, 3

Tags: floor function , calculus , integration , induction , algebra

Let $f(0, 0) = 5^{2003}, f(0, n) = 0$ for every integer $n \neq 0$ and \[\begin{array}{c}\ f(m, n) = f(m-1, n) - 2 \cdot \Bigg\lfloor \frac{f(m-1, n)}{2}\Bigg\rfloor + \Bigg\lfloor\frac{f(m-1, n-1)}{2}\Bigg\rfloor + \Bigg\lfloor\frac{f(m-1, n+1)}{2}\Bigg\rfloor \end{array}\] for every natural number $m > 0$ and for every integer $n$. Prove that there exists a positive integer $M$ such that $f(M, n) = 1$ for all integers $n$ such that $|n| \leq \frac{(5^{2003}-1)}{2}$ and $f(M, n) = 0$ for all integers n such that $|n| > \frac{5^{2003}-1}{2}.$

2006 USAMO, 1

Tags: inequalities , floor function , modular arithmetic , pigeonhole principle , number theory

Let $p$ be a prime number and let $s$ be an integer with $0 < s < p.$ Prove that there exist integers $m$ and $n$ with $0 < m < n < p$ and \[ \left \{\frac{sm}{p} \right\} < \left \{\frac{sn}{p} \right \} < \frac{s}{p} \] if and only if $s$ is not a divisor of $p-1$. Note: For $x$ a real number, let $\lfloor x \rfloor$ denote the greatest integer less than or equal to $x$, and let $\{x\} = x - \lfloor x \rfloor$ denote the fractional part of x.

2014 PUMaC Number Theory B, 8

Tags: floor function , number theory , greatest common divisor , calculus , integration , modular arithmetic

Find the number of positive integers $n \le 2014$ such that there exists integer $x$ that satisfies the condition that $\frac{x+n}{x-n}$ is an odd perfect square.

1993 Balkan MO, 2

Tags: calculus , integration , ceiling function , floor function , inequalities , combinatorics

A positive integer given in decimal representation $\overline{ a_na_{n-1} \ldots a_1a_0 }$ is called [i]monotone[/i] if $a_n\leq a_{n-1} \leq \cdots \leq a_0$. Determine the number of monotone positive integers with at most 1993 digits.

2008 Austria Beginners' Competition, 2

Tags: floor function , algebra

Determine all real numbers $x$ satisfying $$x \lfloor x \lfloor x \rfloor \rfloor =\sqrt2.$$

2005 France Team Selection Test, 4

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

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 $X=\mathbb{N}$.

2011 China Team Selection Test, 2

Tags: ceiling function , floor function , number theory

Let $a_1,a_2,\ldots,a_n,\ldots$ be any permutation of all positive integers. Prove that there exist infinitely many positive integers $i$ such that $\gcd(a_i,a_{i+1})\leq \frac{3}{4} i$.

1997 Romania Team Selection Test, 4

Tags: floor function , ceiling function , number theory , prime number , combinatorics , poset

Let $p,q,r$ be distinct prime numbers and let \[A=\{p^aq^br^c\mid 0\le a,b,c\le 5\} \] Find the least $n\in\mathbb{N}$ such that for any $B\subset A$ where $|B|=n$, has elements $x$ and $y$ such that $x$ divides $y$. [i]Ioan Tomescu[/i]

2000 IMO Shortlist, 2

Tags: floor function , algebra , fractional part

Let $ a, b, c$ be positive integers satisfying the conditions $ b > 2a$ and $ c > 2b.$ Show that there exists a real number $ \lambda$ with the property that all the three numbers $ \lambda a, \lambda b, \lambda c$ have their fractional parts lying in the interval $ \left(\frac {1}{3}, \frac {2}{3} \right].$

2009 Tuymaada Olympiad, 4

Tags: floor function , inequalities , function , algebra

The sum of several non-negative numbers is not greater than 200, while the sum of their squares is not less than 2500. Prove that among them there are four numbers whose sum is not less than 50. [i]Proposed by A. Khabrov[/i]

2010 Today's Calculation Of Integral, 602

Tags: calculus , integration , floor function , counting , derangement , function , logarithm

Prove the following inequality. \[\frac{e-1}{n+1}\leqq\int^e_1(\log x)^n dx\leqq\frac{(n+1)e+1}{(n+1)(n+2)}\ (n=1,2,\cdot\cdot\cdot) \] 1994 Kyoto University entrance exam/Science

2024 Philippine Math Olympiad, P6

Tags: floor function , number theory

The sequence $\{a_n\}_{n\ge 1}$ of real numbers is defined as follows: $$a_1=1, \quad \text{and}\quad a_{n+1}=\frac{1}{2\lfloor a_n \rfloor -a_n+1} \quad \text{for all} \quad n\ge 1$$ Find $a_{2024}$.

1997 AIME Problems, 9

Tags: floor function

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 $a^{12}-144a^{-1}.$

2013 Romanian Master of Mathematics, 5

Tags: floor function , induction , number theory

Given a positive integer $k\geq2$, set $a_1=1$ and, for every integer $n\geq 2$, let $a_n$ be the smallest solution of equation \[x=1+\sum_{i=1}^{n-1}\left\lfloor\sqrt[k]{\frac{x}{a_i}}\right\rfloor\] that exceeds $a_{n-1}$. Prove that all primes are among the terms of the sequence $a_1,a_2,\ldots$

2010 China National Olympiad, 1

Tags: inequalities , floor function , algebra

Let $m,n\ge 1$ and $a_1 < a_2 < \ldots < a_n$ be integers. Prove that there exists a subset $T$ of $\mathbb{N}$ such that \[|T| \leq 1+ \frac{a_n-a_1}{2n+1}\] and for every $i \in \{1,2,\ldots , m\}$, there exists $t \in T$ and $s \in [-n,n]$, such that $a_i=t+s$.

2007 AIME Problems, 7

Tags: geometry , 3d geometry , floor function , calculus , integration

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 $k=\lfloor\sqrt[3]{n_{1}}\rfloor = \lfloor\sqrt[3]{n_{2}}\rfloor = \cdots = \lfloor\sqrt[3]{n_{70}}\rfloor$ and $k$ divides $n_{i}$ for all $i$ such that $1 \leq i \leq 70.$ Find the maximum value of $\frac{n_{i}}{k}$ for $1\leq i \leq 70.$