Olympiad problems

2024 Spain Mathematical Olympiad, 6

Let $a$, $b$ and $n$ be positive integers, satisfying that $bn$ divides $an-a+1$. Let $\alpha=a/b$. Prove that, when the numbers $\lfloor\alpha\rfloor,\lfloor2\alpha\rfloor,\dots,\lfloor(n-1)\alpha\rfloor$ are divided by $n$, the residues are $1,2,\dots,n-1$, in some order.

VMEO IV 2015, 11.3

Tags: number theory , divisible , floor function

How many natural number $n$ less than $2015$ that is divisible by $\lfloor\sqrt[3]{n}\rfloor$ ?

1973 Miklós Schweitzer, 3

Tags: floor function , prime number , number theory , logarithm

Find a constant $ c > 1$ with the property that, for arbitrary positive integers $ n$ and $ k$ such that $ n>c^k$, the number of distinct prime factors of $ \binom{n}{k}$ is at least $ k$. [i]P. Erdos[/i]

2014 Indonesia MO Shortlist, A1

Tags: floor , algebra , floor function

Let $a, b$ be positive real numbers such that there exist infinite number of natural numbers $k$ such that $\lfloor a^k \rfloor + \lfloor b^k \rfloor = \lfloor a \rfloor ^k + \lfloor b \rfloor ^k$ . Prove that $\lfloor a^{2014} \rfloor + \lfloor b^{2014} \rfloor = \lfloor a \rfloor ^{2014} + \lfloor b \rfloor ^{2014}$

2001 Junior Balkan Team Selection Tests - Moldova, 2

Tags: algebra , floor function

Solve in $R$ equation $[x] \cdot \{x\} = 2001 x$, where$ [ .]$ and $\{ .\}$ represent respectively the floor and the integer functions.

2016 Belarus Team Selection Test, 1

Tags: floor function , number theory , sequence

Determine all positive integers $M$ such that the sequence $a_0, a_1, a_2, \cdots$ defined by \[ a_0 = M + \frac{1}{2} \qquad \textrm{and} \qquad a_{k+1} = a_k\lfloor a_k \rfloor \quad \textrm{for} \, k = 0, 1, 2, \cdots \] contains at least one integer term.

2002 National Olympiad First Round, 24

Tags: floor function , geometry , 3d geometry

How many positive integers $n$ are there such that the equation $\left \lfloor \sqrt[3] {7n + 2} \right \rfloor = \left \lfloor \sqrt[3] {7n + 3} \right \rfloor $ does not hold? $ \textbf{a)}\ 0 \qquad\textbf{b)}\ 1 \qquad\textbf{c)}\ 7 \qquad\textbf{d)}\ \text{Infinitely many} \qquad\textbf{e)}\ \text{None of above} $

1989 APMO, 4

Tags: floor function , inequalities , graph theory , combinatorics

Let $S$ be a set consisting of $m$ pairs $(a,b)$ of positive integers with the property that $1 \leq a < b \leq n$. Show that there are at least \[ 4m \cdot \dfrac{(m - \dfrac{n^2}{4})}{3n} \] triples $(a,b,c)$ such that $(a,b)$, $(a,c)$, and $(b,c)$ belong to $S$.

2010 Today's Calculation Of Integral, 557

Tags: calculus , integration , limit , trigonometry , floor function , function , analytic geometry

Find the folllowing limit. \[ \lim_{n\to\infty} \frac{(2n\plus{}1)\int_0^1 x^{n\minus{}1}\sin \left(\frac{\pi}{2}x\right)dx}{(n\plus{}1)^2\int_0^1 x^{n\minus{}1}\cos \left(\frac{\pi}{2}x\right)dx}\ \ (n\equal{}1,\ 2,\ \cdots).\]

2023 239 Open Mathematical Olympiad, 3

Tags: number theory , residue , floor function

Let $n>1$ be a natural number and $x_k{}$ be the residue of $n^2$ modulo $\lfloor n^2/k\rfloor+1$ for all natural $k{}$. Compute the sum \[\bigg\lfloor\frac{x_2}{1}\bigg\rfloor+\bigg\lfloor\frac{x_3}{2}\bigg\rfloor+\cdots+\left\lfloor\frac{x_n}{n-1}\right\rfloor.\]

1962 Dutch Mathematical Olympiad, 2

Tags: sequence , floor function

The $n^{th}$ term of a sequence is $t_n$. For $n \ge 1$, $t_n$ is given by the relation: $$t_n= n^3+\frac12 n^2+ \frac13 n + \frac14$$ The $n^{th}$ term of a second sequence $T_n$, where $T_n$ represents the smallest integer greater than $t_n$. Calculate: $$(T_1+T_2+...+T_{1014}) -(t_1+t_2+...+t_{1014}) $$

1990 Greece National Olympiad, 4

Tags: algebra , floor function , integer part , fractional part , integer and fractional part

Froa nay real $x$, we denote $[x]$, the integer part of $x$ and with $\{x\}$ the fractional part of $x$, such that $x=[x]+\{x\}$. a) Find at least one real $x$ such that$\{x\}+\left\{\frac{1}{x}\right\}=1$ b) Find all rationals $x$ such that $\{x\}+\left\{\frac{1}{x}\right\}=1$

2004 Bulgaria Team Selection Test, 2

Tags: floor function , number theory

Find all primes $p \ge 3$ such that $p- \lfloor p/q \rfloor q$ is a square-free integer for any prime $q<p$.

2003 Argentina National Olympiad, 1

Tags: floor function , algebra , fractional part

Find all positive numbers $x$ such that:$$\frac{1}{[x]}-\frac{1}{[2x]}=\frac{1}{6\{x\}}$$ where $[x]$ represents the integer part of $x$ and $\{x\}=x-[x]$.

2012 Singapore MO Open, 4

Tags: modular arithmetic , floor function , relatively prime , number theory

Let $p$ be an odd prime. Prove that \[1^{p-2}+2^{p-2}+\cdots+\left(\frac{p-1}{2}\right)^{p-2}\equiv\frac{2-2^p}{p}\pmod p.\]

1975 IMO Shortlist, 3

Tags: integration , floor function , calculus , algebra , series summation , approximation

Find the integer represented by $\left[ \sum_{n=1}^{10^9} n^{-2/3} \right] $. Here $[x]$ denotes the greatest integer less than or equal to $x.$

PEN J Problems, 5

Tags: divisor function , floor function , relatively prime , number theory

If $n$ is composite, prove that $\phi(n) \le n- \sqrt{n}$.

2012 Portugal MO, 3

Tags: number theory , greatest common divisor , floor function , ceiling function , combinatorics

Isabel wants to partition the set $\mathbb{N}$ of the positive integers into $n$ disjoint sets $A_{1}, A_{2}, \ldots, A_{n}$. Suppose that for each $i$ with $1\leq i\leq n$, given any positive integers $r, s\in A_{i}$ with $r\neq s$, we have $r+s\in A_{i}$. If $|A_{j}|=1$ for some $j$, find the greatest positive integer that may belong to $A_{j}$.

1997 Canada National Olympiad, 3

Tags: induction , limit , floor function , inequalities

Prove that $\frac{1}{1999}< \prod_{i=1}^{999}{\frac{2i-1}{2i}}<\frac{1}{44}$.

2009 Olympic Revenge, 4

Tags: floor function , number theory

Let $d_i(k)$ the number of divisors of $k$ greater than $i$. Let $f(n)=\sum_{i=1}^{\lfloor \frac{n^2}{2} \rfloor}d_i(n^2-i)-2\sum_{i=1}^{\lfloor \frac{n}{2} \rfloor}d_i(n-i)$. Find all $n \in N$ such that $f(n)$ is a perfect square.

2006 MOP Homework, 4

Tags: floor function , number theory

Find all pairs $(a,b)$ of positive real numbers such that $\lfloor a \lfloor bn \rfloor \rfloor =n - 1$ for all positive integers $n$.

2004 AIME Problems, 12

Tags: floor function , inequalities , rectangle , number theory , relatively prime , logarithm , geometry

Let $S$ be the set of ordered pairs $(x, y)$ such that $0<x\le 1$, $0<y\le 1$, and $\left[\log_2{\left(\frac 1x\right)}\right]$ and $\left[\log_5{\left(\frac 1y\right)}\right]$ are both even. Given that the area of the graph of $S$ is $m/n$, where $m$ and $n$ are relatively prime positive integers, find $m+n$. The notation $[z]$ denotes the greatest integer that is less than or equal to $z$.

2008 ITest, 90

Tags: floor function , inequalities

For $a,b,c$ positive reals, let \[N=\dfrac{a^2+b^2}{c^2+ab}+\dfrac{b^2+c^2}{a^2+bc}+\dfrac{c^2+a^2}{b^2+ca}.\] Find the minimum value of $\lfloor 2008N\rfloor$.

2002 AMC 10, 22

Tags: factorial , floor function

In how many zeroes does the number $\dfrac{2002!}{(1001!)^2}$ end? $\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }200\qquad\textbf{(E) }400$

1999 Italy TST, 4

Tags: floor function , combinatorics

Let $X$ be an $n$-element set and let $A_1,\ldots ,A_m$ be subsets of $X$ such that i) $|A_i|=3$ for each $i=1,\ldots ,m$. ii) $|A_i\cap A_j|\le 1$ for any two distinct indices $i,j$. Show that there exists a subset of $X$ with at least $\lfloor\sqrt{2n}\rfloor$ elements which does not contain any of the $A_i$’s.