Olympiad problems

1992 Vietnam Team Selection Test, 1

Let two natural number $n > 1$ and $m$ be given. Find the least positive integer $k$ which has the following property: Among $k$ arbitrary integers $a_1, a_2, \ldots, a_k$ satisfying the condition $a_i - a_j$ ( $1 \leq i < j \leq k$) is not divided by $n$, there exist two numbers $a_p, a_s$ ($p \neq s$) such that $m + a_p - a_s$ is divided by $n$.

2006 Thailand Mathematical Olympiad, 9

Tags: algebra , floor function

Compute the largest integer not exceeding $$\frac{2549^3}{2547\cdot 2548} -\frac{2547^3}{2548\cdot 2549}$$

2008 Harvard-MIT Mathematics Tournament, 7

Tags: floor function , ratio , function , quadratic , calculus , integration , geometric series

Compute $ \sum_{n \equal{} 1}^\infty\sum_{k \equal{} 1}^{n \minus{} 1}\frac {k}{2^{n \plus{} k}}$.

1968 IMO Shortlist, 15

Tags: floor function , number theory , equation , summation

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]

2012 Indonesia TST, 4

Tags: floor function , number theory

The sequence $a_i$ is defined as $a_1 = 1$ and \[a_n = a_{\left\lfloor \dfrac{n}{2} \right\rfloor} + a_{\left\lfloor \dfrac{n}{3} \right\rfloor} + a_{\left\lfloor \dfrac{n}{4} \right\rfloor} + \cdots + a_{\left\lfloor \dfrac{n}{n} \right\rfloor} + 1\] for every positive integer $n > 1$. Prove that there are infinitely many values of $n$ such that $a_n \equiv n \mod 2012$.

2005 China Team Selection Test, 3

Tags: euler , floor function , number theory

Let $a_1,a_2 \dots a_n$ and $x_1, x_2 \dots x_n$ be integers and $r\geq 2$ be an integer. It is known that \[\sum_{j=0}^{n} a_j x_j^k =0 \qquad \text{for} \quad k=1,2, \dots r.\] Prove that \[\sum_{j=0}^{n} a_j x_j^m \equiv 0 \pmod m, \qquad \text{for all}\quad m \in \{ r+1, r+2, \cdots, 2r+1 \}.\]

2019 Silk Road, 4

Tags: combinatorics , sum , floor function , even , algebra , 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 $.)

2020 BMT Fall, 10

Tags: floor function , number theory

How many integers $100 \le x \le 999$ have the property that, among the six digits in $\lfloor 280 + \frac{x}{100} \rfloor$ and $x$, exactly two are identical?

2000 AIME Problems, 6

Tags: geometry , trapezoid , ratio , quadratic , floor function

One base of a trapezoid is 100 units longer than the other base. The segment that joins the midpoints of the legs divides the trapezoid into two regions whose areas are in the ratio $2: 3.$ Let $x$ be the length of the segment joining the legs of the trapezoid that is parallel to the bases and that divides the trapezoid into two regions of equal area. Find the greatest integer that does not exceed $x^2/100.$

1976 IMO Longlists, 44

Tags: floor function , geometry

A circle of radius $1$ rolls around a circle of radius $\sqrt{2}$. Initially, the tangent point is colored red. Afterwards, the red points map from one circle to another by contact. How many red points will be on the bigger circle when the center of the smaller one has made $n$ circuits around the bigger one?

2023 Romania National Olympiad, 3

Tags: floor function , algebra

We say that a natural number $n$ is interesting if it can be written in the form \[ n = \left\lfloor \frac{1}{a} \right\rfloor + \left\lfloor \frac{1}{b} \right\rfloor + \left\lfloor \frac{1}{c} \right\rfloor, \] where $a,b,c$ are positive real numbers such that $a + b + c = 1.$ Determine all interesting numbers. ( $\lfloor x \rfloor$ denotes the greatest integer not greater than $x$.)

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]

2015 Romania Team Selection Tests, 4

Tags: floor function , density , kronecker , algebra , 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 $\{\left \lfloor{a^2n}\right \rfloor-\left \lfloor{a\left \lfloor{an}\right \rfloor}\right \rfloor : n \in \mathbb{N}_{>0}\}=\{1 , 2 , \ldots ,\left \lfloor{a}\right \rfloor\}$.

1984 Iran MO (2nd round), 1

Tags: function , floor function , domain , limit , algebra

Let $f$ and $g$ be two functions such that \[f(x)=\frac{1}{\lfloor | x | \rfloor}, \quad g(x)=\frac{1}{|\lfloor x \rfloor |}.\] Find the domains of $f$ and $g$ and then prove that \[\lim_{x \to -1^+} f(x)= \lim_{x \to 1^- } g(x).\]

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?

2004 China Team Selection Test, 2

Tags: floor function , number theory , logarithm

Let u be a fixed positive integer. Prove that the equation $n! = u^{\alpha} - u^{\beta}$ has a finite number of solutions $(n, \alpha, \beta).$

MathLinks Contest 2nd, 6.1

Tags: algebra , floor function , function

Determine the parity of the positive integer $N$, where $$N = \lfloor \frac{2002!}{2001 \cdot2003} \rfloor.$$

PEN M Problems, 2

Tags: floor function , induction , recursive sequences

An integer sequence $\{a_{n}\}_{n \ge 1}$ is defined by \[a_{1}=1, \; a_{n+1}=a_{n}+\lfloor \sqrt{a_{n}}\rfloor.\] Show that $a_{n}$ is a square if and only if $n=2^{k}+k-2$ for some $k \in \mathbb{N}$.

2012 Hanoi Open Mathematics Competitions, 4

Tags: floor function , radical , algebra

What is the largest integer less than or equal to $4x^3 - 3x$, where $x=\frac{\sqrt[3]{2+\sqrt3}+\sqrt[3]{2-\sqrt3}}{2}$ ? (A) $1$, (B) $2$, (C) $3$, (D) $4$, (E) None of the above.

2006 AIME Problems, 14

Tags: floor function , 3d geometry , pyramid , symmetry , trigonometry , tetrahedron , geometry

A tripod has three legs each of length 5 feet. When the tripod is set up, the angle between any pair of legs is equal to the angle between any other pair, and the top of the tripod is 4 feet from the ground. In setting up the tripod, the lower 1 foot of one leg breaks off. Let $h$ be the height in feet of the top of the tripod from the ground when the broken tripod is set up. Then $h$ can be written in the form $\frac m{\sqrt{n}},$ where $m$ and $n$ are positive integers and $n$ is not divisible by the square of any prime. Find $\lfloor m+\sqrt{n}\rfloor.$ (The notation $\lfloor x\rfloor$ denotes the greatest integer that is less than or equal to $x$.)

1955 Moscow Mathematical Olympiad, 311

Tags: number theory , floor function

Find all numbers $a$ such that (1) all numbers $[a], [2a], . . . , [Na]$ are distinct and (2) all numbers $\left[ \frac{1}{a}\right], \left[ \frac{2}{a}\right], ..., \left[ \frac{M}{a}\right]$ are distinct.

1998 Romania Team Selection Test, 1

Tags: floor function , combinatorics

Let $n\ge 2$ be an integer. Show that there exists a subset $A\in \{1,2,\ldots ,n\}$ such that: i) The number of elements of $A$ is at most $2\lfloor\sqrt{n}\rfloor+1$ ii) $\{ |x-y| \mid x,y\in A, x\not= y\} = \{ 1,2,\ldots n-1 \}$ [i]Radu Todor[/i]

2021 Harvard-MIT Mathematics Tournament., 8

Tags: combinatorics , floor function , number theory

For each positive real number $\alpha$, define $$\lfloor \alpha \mathbb{N}\rfloor :=\{\lfloor \alpha m \rfloor\; |\; m\in \mathbb{N}\}.$$ Let $n$ be a positive integer. A set $S\subseteq \{1,2,\ldots,n\}$ has the property that: for each real $\beta >0$, $$ \text{if}\; S\subseteq \lfloor \beta \mathbb{N} \rfloor, \text{then}\; \{1,2,\ldots,n\} \subseteq \lfloor \beta\mathbb{N}\rfloor.$$ Determine, with proof, the smallest positive size of $S$.

2022/2023 Tournament of Towns, P2

Tags: floor function , number theory

А positive integer $n{}$ is given. For every $x{}$ consider the sum \[Q(x)=\sum_{k=1}^{10^n}\left\lfloor\frac{x}{k}\right\rfloor.\]Find the difference $Q(10^n)-Q(10^n-1)$. [i]Alexey Tolpygo[/i]

2014 Junior Balkan Team Selection Tests - Romania, 4

Tags: floor function , coloring , combinatorics

Let $n \ge 6$ be an integer. We have at our disposal $n$ colors. We color each of the unit squares of an $n \times n$ board with one of the $n$ colors. a) Prove that, for any such coloring, there exists a path of a chess knight from the bottom-left to the upper-right corner, that does not use all the colors. b) Prove that, if we reduce the number of colors to $\lfloor 2n/3 \rfloor + 2$, then the statement from a) is true for infinitely many values of $n$ and it is false also for infinitely many values of $n$