Olympiad problems

2014 Saudi Arabia IMO TST, 2

Determine all functions $f:[0,\infty)\rightarrow\mathbb{R}$ such that $f(0)=0$ and \[f(x)=1+5f\left(\left\lfloor{\frac{x}{2}\right\rfloor}\right)-6f\left(\left\lfloor{\frac{x}{4}\right\rfloor}\right)\] for all $x>0$.

2014 EGMO, 5

Tags: floor function , reachability , combinatorics

Let $n$ be a positive integer. We have $n$ boxes where each box contains a non-negative number of pebbles. In each move we are allowed to take two pebbles from a box we choose, throw away one of the pebbles and put the other pebble in another box we choose. An initial configuration of pebbles is called [i]solvable[/i] if it is possible to reach a configuration with no empty box, in a finite (possibly zero) number of moves. Determine all initial configurations of pebbles which are not solvable, but become solvable when an additional pebble is added to a box, no matter which box is chosen.

2023 Purple Comet Problems, 18

Tags: algebra , floor function

For real number $x$, let $\lfloor x\rfloor$ denote the greatest integer less than or equal to $x$, and let $\{x\}$ denote the fractional part of $x$, that is $\{x\} = x -\lfloor x\rfloor$. The sum of the solutions to the equation $2\lfloor x\rfloor^2 + 3\{x\}^2 = \frac74 x \lfloor x\rfloor$ can be written as $\frac{p}{q} $, where $p$ and $q$ are prime numbers. Find $10p + q$.

2023 USA EGMO Team Selection Test, 5

Tags: floor function , algebra , function

Let $\lfloor \bullet \rfloor$ denote the floor function. For nonnegative integers $a$ and $b$, their [i]bitwise xor[/i], denoted $a \oplus b$, is the unique nonnegative integer such that $$ \left \lfloor \frac{a}{2^k} \right \rfloor+ \left\lfloor\frac{b}{2^k} \right\rfloor - \left\lfloor \frac{a\oplus b}{2^k}\right\rfloor$$ is even for every $k \ge 0$. Find all positive integers $a$ such that for any integers $x>y\ge 0$, we have \[ x\oplus ax \neq y \oplus ay. \] [i]Carl Schildkraut[/i]

2006 JBMO ShortLists, 6

Tags: factorial , floor function , number theory

Prove that for every composite number $ n>4$, numbers $ kn$ divides $ (n\minus{}1)!$ for every integer $ k$ such that $ 1\le k\le \lfloor \sqrt{n\minus{}1} \rfloor$.

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 $a_{2024}$.

2006 AMC 12/AHSME, 20

Tags: probability , floor function , function , calculus , integration , inequalities , logarithm

Let $ x$ be chosen at random from the interval $ (0,1)$. What is the probability that \[ \lfloor\log_{10}4x\rfloor \minus{} \lfloor\log_{10}x\rfloor \equal{} 0? \]Here $ \lfloor x\rfloor$ denotes the greatest integer that is less than or equal to $ x$. $ \textbf{(A) } \frac 18 \qquad \textbf{(B) } \frac 3{20} \qquad \textbf{(C) } \frac 16 \qquad \textbf{(D) } \frac 15 \qquad \textbf{(E) } \frac 14$

2011 Math Prize For Girls Problems, 12

Tags: floor function , logarithm

If $x$ is a real number, let $\lfloor x \rfloor$ be the greatest integer that is less than or equal to $x$. If $n$ is a positive integer, let $S(n)$ be defined by \[ S(n) = \left\lfloor \frac{n}{10^{\lfloor \log n \rfloor}} \right\rfloor + 10 \left( n - 10^{\lfloor \log n \rfloor} \cdot \left\lfloor \frac{n}{10^{\lfloor \log n \rfloor}} \right\rfloor \right) \, . \] (All the logarithms are base 10.) How many integers $n$ from 1 to 2011 (inclusive) satisfy $S(S(n)) = n$?

2011 Spain Mathematical Olympiad, 3

Tags: algebra , polynomial , induction , floor function , modular arithmetic , number theory

The sequence $S_0,S_1,S_2,\ldots$ is defined by[list][*]$S_n=1$ for $0\le n\le 2011$, and [*]$S_{n+2012}=S_{n+2011}+S_n$ for $n\ge 0$.[/list]Prove that $S_{2011a}-S_a$ is a multiple of $2011$ for all nonnegative integers $a$.

2005 Czech-Polish-Slovak Match, 4

Tags: floor function , algebra , polynomial , combinatorics

We distribute $n\ge1$ labelled balls among nine persons $A,B,C, \dots , I$. How many ways are there to do this so that $A$ gets the same number of balls as $B,C,D$ and $E$ together?

2011 NIMO Summer Contest, 3

Tags: floor function

Define $\lfloor x \rfloor$ as the largest integer less than or equal to $x$. Define $\{x \} = x - \lfloor x \rfloor$. For example, $\{ 3 \} = 3-3 = 0$, $\{ \pi \} = \pi - 3$, and $\{ - \pi \} = 4-\pi$. If $\{n\} + \{ 3n\} = 1.4$, then find the sum of all possible values of $100\{n\}$. [i]Proposed by Isabella Grabski [/i]

Indonesia MO Shortlist - geometry, g9

Tags: floor function , geometry , equal area

Given triangle $ABC$. Let $A_1B_1$, $A_2B_2$,$ ...$, $A_{2008}B_{2008}$ be $2008$ lines parallel to $AB$ which divide triangle $ABC$ into $2009$ equal areas. Calculate the value of $$ \left\lfloor \frac{A_1B_1}{2A_2B_2} + \frac{A_1B_1}{2A_3B_3} + ... + \frac{A_1B_1}{2A_{2008}B_{2008}} \right\rfloor$$

1980 All Soviet Union Mathematical Olympiad, 301

Tags: floor function , algebra , equation

Prove that there is an infinite number of such numbers $B$ that the equation $\lfloor x^3/2\rfloor + \lfloor y^3/2 \rfloor = B$ has at least $1980$ integer solutions $(x,y)$. ($\lfloor z\rfloor$ denotes the greatest integer not exceeding $z$.)

2013 Romanian Masters In Mathematics, 1

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

For a positive integer $a$, define a sequence of integers $x_1,x_2,\ldots$ by letting $x_1=a$ and $x_{n+1}=2x_n+1$ for $n\geq 1$. Let $y_n=2^{x_n}-1$. Determine the largest possible $k$ such that, for some positive integer $a$, the numbers $y_1,\ldots,y_k$ are all prime.

2021 BMT, 3

Tags: floor function , algebra

Let $x$ be a solution to the equation $\lfloor x \lfloor x + 2\rfloor + 2\rfloor = 10$. Compute the smallest $C$ such that for any solution $x$, $x < C$. Here, $\lfloor m \rfloor$ is defined as the greatest integer less than or equal to $m$. For example, $\lfloor 3\rfloor = 3$ and $\lfloor -4.25\rfloor = -5$.

2007 Princeton University Math Competition, 2

Tags: floor function

Suppose that $A$ is a set of positive integers less than $N$ and that no two distinct elements of $A$ sum to a perfect square. That is, if $a_1, a_2 \in A$ and $a_1 \neq a_2$ then $|a_1+a_2|$ is not a square of an integer. Prove that the maximum number of elements in $A$ is at least $\left\lfloor\frac{11}{32}N\right\rfloor$ .

2002 Vietnam Team Selection Test, 1

Tags: floor function , combinatorics

Let $n\geq 2$ be an integer and consider an array composed of $n$ rows and $2n$ columns. Half of the elements in the array are colored in red. Prove that for each integer $k$, $1<k\leq \dsp \left\lfloor \frac n2\right\rfloor+1$, there exist $k$ rows such that the array of size $k\times 2n$ formed with these $k$ rows has at least \[ \frac { k! (n-2k+2) } {(n-k+1)(n-k+2)\cdots (n-1)} \] columns which contain only red cells.

2013 USA TSTST, 3

Tags: analytic geometry , graphing lines , slope , geometric transformation , reflection , floor function

Divide the plane into an infinite square grid by drawing all the lines $x=m$ and $y=n$ for $m,n \in \mathbb Z$. Next, if a square's upper-right corner has both coordinates even, color it black; otherwise, color it white (in this way, exactly $1/4$ of the squares are black and no two black squares are adjacent). Let $r$ and $s$ be odd integers, and let $(x,y)$ be a point in the interior of any white square such that $rx-sy$ is irrational. Shoot a laser out of this point with slope $r/s$; lasers pass through white squares and reflect off black squares. Prove that the path of this laser will form a closed loop.

2012 IberoAmerican, 3

Tags: floor function , number theory , logarithm

Show that, for every positive integer $n$, there exist $n$ consecutive positive integers such that none is divisible by the sum of its digits. (Alternative Formulation: Call a number good if it's not divisible by the sum of its digits. Show that for every positive integer $n$ there are $n$ consecutive good numbers.)

2014 Contests, 4

Tags: pigeonhole principle , floor function , number theory , algebra , combinatorics

Let $n$ and $b$ be positive integers. We say $n$ is $b$-discerning if there exists a set consisting of $n$ different positive integers less than $b$ that has no two different subsets $U$ and $V$ such that the sum of all elements in $U$ equals the sum of all elements in $V$. (a) Prove that $8$ is $100$-discerning. (b) Prove that $9$ is not $100$-discerning. [i]Senior Problems Committee of the Australian Mathematical Olympiad Committee[/i]

2023 BMT, 10

Tags: floor function , number theory

Let $a$ denote the positive real root of the polynomial $x^2 -3x-2$. Compute the remainder when $\lfloor a^{1000}\rfloor $ is divided by the prime number $997$. Here, $\lfloor r\rfloor$ denotes the greatest integer less than $r$.

2004 Uzbekistan National Olympiad, 1

Tags: floor function , function , algebra

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

2018 Mathematical Talent Reward Programme, MCQ: P 2

Tags: algebra , floor function , function

$\lim _{x \rightarrow 0^{+}} \frac{[x]}{\tan x}$ where $[x]$ is the greatest integer function [list=1] [*] -1 [*] 0 [*] 1 [*] Does not exists [/list]

2015 Saudi Arabia GMO TST, 1

Tags: floor function , sequence , recurrence relation , algebra

Let be given the sequence $(x_n)$ defined by $x_1 = 1$ and $x_{n+1} = 3x_n + \lfloor x_n \sqrt5 \rfloor$ for all $n = 1,2,3,...,$ where $\lfloor x \rfloor$ denotes the greatest integer that does not exceed $x$. Prove that for any positive integer $n$ we have $$x_nx_{n+2} - x^2_{n+1} = 4^{n-1}$$ Trần Nam Dũng

2014 AMC 12/AHSME, 21

Tags: function , floor function , algebra , domain , logarithm

For every real number $x$, let $\lfloor x\rfloor$ denote the greatest integer not exceeding $x$, and let \[f(x)=\lfloor x\rfloor(2014^{x-\lfloor x\rfloor}-1).\] The set of all numbers $x$ such that $1\leq x<2014$ and $f(x)\leq 1$ is a union of disjoint intervals. What is the sum of the lengths of those intervals? $\textbf{(A) }1\qquad \textbf{(B) }\dfrac{\log 2015}{\log 2014}\qquad \textbf{(C) }\dfrac{\log 2014}{\log 2013}\qquad \textbf{(D) }\dfrac{2014}{2013}\qquad \textbf{(E) }2014^{\frac1{2014}}\qquad$