Olympiad problems

2015 Romania Team Selection Tests, 4

Tags: algebra , density , kronecker , floor function , number theory

Let $k$ be a positive integer congruent to $1$ modulo $4$ which is not a perfect square and let $a=\frac{1+\sqrt{k}}{2}$. Show that $\{\left \lfloor{a^2n}\right \rfloor-\left \lfloor{a\left \lfloor{an}\right \rfloor}\right \rfloor : n \in \mathbb{N}_{>0}\}=\{1 , 2 , \ldots ,\left \lfloor{a}\right \rfloor\}$.

2004 Italy TST, 3

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

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

2012 ELMO Shortlist, 2

Tags: floor function , algorithm , logarithm , ceiling function , combinatorics

Determine whether it's possible to cover a $K_{2012}$ with a) 1000 $K_{1006}$'s; b) 1000 $K_{1006,1006}$'s. [i]David Yang.[/i]

2013 Online Math Open Problems, 18

Tags: floor function , absolute value , trigonometry

Determine the absolute value of the sum \[ \lfloor 2013\sin{0^\circ} \rfloor + \lfloor 2013\sin{1^\circ} \rfloor + \cdots + \lfloor 2013\sin{359^\circ} \rfloor, \] where $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$. (You may use the fact that $\sin{n^\circ}$ is irrational for positive integers $n$ not divisible by $30$.) [i]Ray Li[/i]

2011 Germany Team Selection Test, 2

Tags: number theory , floor function

Let $n$ be a positive integer prove that $$6\nmid \lfloor (\sqrt[3]{28}-3)^{-n} \rfloor.$$

2012 Harvard-MIT Mathematics Tournament, 5

Tags: hmmt , floor function

Find all ordered triples $(a,b,c)$ of positive reals that satisfy: $\lfloor a\rfloor bc=3,a\lfloor b\rfloor c=4$, and $ab\lfloor c\rfloor=5$, where $\lfloor x\rfloor$ denotes the greatest integer less than or equal to $x$.

2019 Silk Road, 4

Tags: sum , even , combinatorics , function , algebra , floor function

The sequence $ \{a_n \} $ is defined as follows: $ a_0 = 1 $ and $ {a_n} = \sum \limits_ {k = 1} ^ {[\sqrt n]} {{a_ {n - {k ^ 2 }}}} $ for $ n \ge 1. $ Prove that among $ a_1, a_2, \ldots, a_ {10 ^ 6} $ there are at least $500$ even numbers. (Here, $ [x] $ is the largest integer not exceeding $ x $.)

1982 IMO Longlists, 7

Tags: algorithm , euclidean algorithm , greatest common divisor , floor function , number theory , modular arithmetic

Find all solutions $(x, y) \in \mathbb Z^2$ of the equation \[x^3 - y^3 = 2xy + 8.\]

1981 Dutch Mathematical Olympiad, 1

Tags: algebra , floor function

$f(x) = [x] + [2x] + [3x] + [4x] + [5x] + [6x]$. What values does $f$ take?

2019 District Olympiad, 4

Tags: floor function , integer part , algebra

Solve the equation in the set of real numbers: $$\left[ x+\frac{1}{x} \right] = \left[ x^2+\frac{1}{x^2} \right]$$ where $[a]$, represents the integer part of the real number $a$.

2004 National Olympiad First Round, 20

Tags: floor function , inequalities

What is the largest real number $C$ that satisfies the inequality $x^2 \geq C \lfloor x \rfloor (x-\lfloor x \rfloor)$ for every real $x$? $ \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 4 \qquad\textbf{(D)}\ 9 \qquad\textbf{(E)}\ 25 $

1996 AIME Problems, 2

Tags: logarithm , floor function

For each real number $x,$ let $\lfloor x\rfloor$ denote the greatest integer that does not exceed $x.$ For how many positive integers $n$ is it true that $n<1000$ and that $\lfloor \log_2 n\rfloor$ is a positive even integer.

1978 Germany Team Selection Test, 3

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

Let $n$ be an integer greater than $1$. Define \[x_1 = n, y_1 = 1, x_{i+1} =\left[ \frac{x_i+y_i}{2}\right] , y_{i+1} = \left[ \frac{n}{x_{i+1}}\right], \qquad \text{for }i = 1, 2, \ldots\ ,\] where $[z]$ denotes the largest integer less than or equal to $z$. Prove that \[ \min \{x_1, x_2, \ldots, x_n \} =[ \sqrt n ]\]

2021 Taiwan TST Round 1, N

Tags: sequence , number theory , floor function

For each positive integer $n$, define $V_n=\lfloor 2^n\sqrt{2020}\rfloor+\lfloor 2^n\sqrt{2021}\rfloor$. Prove that, in the sequence $V_1,V_2,\ldots,$ there are infinitely many odd integers, as well as infinitely many even integers. [i]Remark.[/i] $\lfloor x\rfloor$ is the largest integer that does not exceed the real number $x$.

2017 BMT Spring, 8

Tags: floor function , integration , calculus , algebra

The numerical value of the following integral $$\int^1_0 (-x^2 + x)^{2017} \lfloor 2017x \rfloor dx$$ can be expressed in the form $a\frac{m!^2}{ n!}$ where a is minimized. Find $a + m + n$. (Note $\lfloor x\rfloor$ is the largest integer less than or equal to x.)

2011 Putnam, B1

Tags: absolute value , floor function , putnam analysis , inequalities , integration

Let $h$ and $k$ be positive integers. Prove that for every $\varepsilon >0,$ there are positive integers $m$ and $n$ such that \[\varepsilon < \left|h\sqrt{m}-k\sqrt{n}\right|<2\varepsilon.\]

1968 IMO Shortlist, 15

Tags: equation , number theory , summation , floor function

Let $n$ be a natural number. Prove that \[ \left\lfloor \frac{n+2^0}{2^1} \right\rfloor + \left\lfloor \frac{n+2^1}{2^2} \right\rfloor +\cdots +\left\lfloor \frac{n+2^{n-1}}{2^n}\right\rfloor =n. \] [hide="Remark"]For any real number $x$, the number $\lfloor x \rfloor$ represents the largest integer smaller or equal with $x$.[/hide]

2008 Romanian Master of Mathematics, 3

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

Let $ a>1$ be a positive integer. Prove that every non-zero positive integer $ N$ has a multiple in the sequence $ (a_n)_{n\ge1}$, $ a_n\equal{}\left\lfloor\frac{a^n}n\right\rfloor$.

2021 MIG, 14

Tags: floor function

The notation $\lfloor n \rfloor$ denotes the greatest integer less than or equal to $n$. Evaluate $\lfloor 2.1 \lfloor {-}4.3 \rfloor \rfloor$. $\textbf{(A) }{-}11\qquad\textbf{(B) }{-}10\qquad\textbf{(C) }{-}9\qquad\textbf{(D) }{-}8\qquad\textbf{(E) }{-}4$

2007 Regional Competition For Advanced Students, 2

Tags: function , floor function , algebra

Find all tuples $ (x_1,x_2,x_3,x_4,x_5)$ of positive integers with $ x_1>x_2>x_3>x_4>x_5>0$ and $ {\left \lfloor \frac{x_1+x_2}{3} \right \rfloor }^2 + {\left \lfloor \frac{x_2+x_3}{3} \right \rfloor }^2 + {\left \lfloor \frac{x_3+x_4}{3} \right \rfloor }^2 + {\left \lfloor \frac{x_4+x_5}{3} \right \rfloor }^2 = 38.$

2018 Bosnia And Herzegovina - Regional Olympiad, 3

Tags: equation , algebra , floor function , frac function

Solve equation $x \lfloor{x}\rfloor+\{x\}=2018$, where $x$ is real number

2016 AMC 10, 25

Tags: floor function

Let $f(x)=\sum_{k=2}^{10}(\lfloor kx \rfloor -k \lfloor x \rfloor)$, where $\lfloor r \rfloor$ denotes the greatest integer less than or equal to $r$. How many distinct values does $f(x)$ assume for $x \ge 0$? $\textbf{(A)}\ 32\qquad\textbf{(B)}\ 36\qquad\textbf{(C)}\ 45\qquad\textbf{(D)}\ 46\qquad\textbf{(E)}\ \text{infinitely many}$

2010 Math Prize for Girls Olympiad, 1

Tags: floor function , ceiling function

Let $S$ be a set of 100 integers. Suppose that for all positive integers $x$ and $y$ (possibly equal) such that $x + y$ is in $S$, either $x$ or $y$ (or both) is in $S$. Prove that the sum of the numbers in $S$ is at most 10,000.

2006 Estonia National Olympiad, 4

Tags: floor function , equation , algebra

Solve the equation $\left[\frac{x}{3}\right]+\left [\frac{2x}{3}\right]=x $

2002 Italy TST, 2

Tags: combinatorics , floor function

On a soccer tournament with $n\ge 3$ teams taking part, several matches are played in such a way that among any three teams, some two play a match. $(a)$ If $n=7$, find the smallest number of matches that must be played. $(b)$ Find the smallest number of matches in terms of $n$.