Olympiad problems

PEN G Problems, 20

You are given three lists A, B, and C. List A contains the numbers of the form $10^{k}$ in base 10, with $k$ any integer greater than or equal to 1. Lists B and C contain the same numbers translated into base 2 and 5 respectively: \[\begin{array}{lll}A & B & C \\ 10 & 1010 & 20 \\ 100 & 1100100 & 400 \\ 1000 & 1111101000 & 13000 \\ \vdots & \vdots & \vdots \end{array}.\] Prove that for every integer $n > 1$, there is exactly one number in exactly one of the lists B or C that has exactly $n$ digits.

2018 VJIMC, 2

Tags: integer part , floor function , inequalities , algebra

Let $n$ be a positive integer and let $a_1\le a_2 \le \dots \le a_n$ be real numbers such that \[a_1+2a_2+\dots+na_n=0.\] Prove that \[a_1[x]+a_2[2x]+\dots+a_n[nx] \ge 0\] for every real number $x$. (Here $[t]$ denotes the integer satisfying $[t] \le t<[t]+1$.)

1999 All-Russian Olympiad, 6

Tags: floor function , inequalities , number theory , algebra , induction

Prove that for all natural numbers $n$, \[ \sum_{k=1}^{n^2} \left\{ \sqrt{k} \right\} \le \frac{n^2-1}{2}. \] Here, $\{x\}$ denotes the fractional part of $x$.

2012 Online Math Open Problems, 33

Tags: floor function , probability

You are playing a game in which you have $3$ envelopes, each containing a uniformly random amount of money between $0$ and $1000$ dollars. (That is, for any real $0 \leq a < b \leq 1000$, the probability that the amount of money in a given envelope is between $a$ and $b$ is $\frac{b-a}{1000}$.) At any step, you take an envelope and look at its contents. You may choose either to keep the envelope, at which point you finish, or discard it and repeat the process with one less envelope. If you play to optimize your expected winnings, your expected winnings will be $E$. What is $\lfloor E\rfloor,$ the greatest integer less than or equal to $E$? [i]Author: Alex Zhu[/i]

2015 Romania Team Selection Test, 4

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

2020 AMC 12/AHSME, 21

Tags: floor function

How many positive integers $n$ satisfy$$\dfrac{n+1000}{70} = \lfloor \sqrt{n} \rfloor?$$(Recall that $\lfloor x\rfloor$ is the greatest integer not exceeding $x$.) $\textbf{(A) } 2 \qquad\textbf{(B) } 4 \qquad\textbf{(C) } 6 \qquad\textbf{(D) } 30 \qquad\textbf{(E) } 32$

2014 USA TSTST, 4

Tags: modular arithmetic , linear algebra , floor function , vector , pigeonhole principle , polynomial , algebra

Let $P(x)$ and $Q(x)$ be arbitrary polynomials with real coefficients, and let $d$ be the degree of $P(x)$. Assume that $P(x)$ is not the zero polynomial. Prove that there exist polynomials $A(x)$ and $B(x)$ such that: (i) both $A$ and $B$ have degree at most $d/2$ (ii) at most one of $A$ and $B$ is the zero polynomial. (iii) $\frac{A(x)+Q(x)B(x)}{P(x)}$ is a polynomial with real coefficients. That is, there is some polynomial $C(x)$ with real coefficients such that $A(x)+Q(x)B(x)=P(x)C(x)$.

2008 ITest, 7

Tags: quadratic , floor function , inequalities

Find the number of integers $n$ for which $n^2+10n<2008$.

2006 Taiwan National Olympiad, 2

Tags: function , algebra , floor function

Find all reals $x$ satisfying $0 \le x \le 5$ and $\lfloor x^2-2x \rfloor = \lfloor x \rfloor ^2 - 2 \lfloor x \rfloor$.

PEN A Problems, 83

Tags: divisibility theory , floor function

Find all $n \in \mathbb{N}$ such that $ \lfloor \sqrt{n}\rfloor$ divides $n$.

2022 SAFEST Olympiad, 1

Tags: summation , number theory , algebra , floor function

Which positive integers $n$ make the equation \[\sum_{i=1}^n \sum_{j=1}^n \left\lfloor \frac{ij}{n+1} \right\rfloor=\frac{n^2(n-1)}{4}\] true?

1992 China National Olympiad, 2

Tags: floor function , combinatorics

Find the maximum possible number of edges of a simple graph with $8$ vertices and without any quadrilateral. (a simple graph is an undirected graph that has no loops (edges connected at both ends to the same vertex) and no more than one edge between any two different vertices.)

2021 Harvard-MIT Mathematics Tournament., 8

Tags: number theory , floor function , combinatorics

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 $S$.

2010 Contests, 2

Tags: floor function

Calculate $\displaystyle{\sum_{n=1}^\infty\left(\lfloor\sqrt[n]{2010}\rfloor-1\right)}$ where $\lfloor x\rfloor$ is the largest integer less than or equal to $x$.

2003 Estonia National Olympiad, 4

Tags: integer , divide , infinity , divisible , number theory , floor function

Prove that there exist infinitely many positive integers $n$ such that $\sqrt{n}$ is not an integer and $n$ is divisible by $[\sqrt{n}] $.

2024 India National Olympiad, 6

Tags: floor function

For each positive integer $n \ge 3$, define $A_n$ and $B_n$ as \[A_n = \sqrt{n^2 + 1} + \sqrt{n^2 + 3} + \cdots + \sqrt{n^2+2n-1}\] \[B_n = \sqrt{n^2 + 2} + \sqrt{n^2 + 4} + \cdots + \sqrt{n^2 + 2n}.\] Determine all positive integers $n\ge 3$ for which $\lfloor A_n \rfloor = \lfloor B_n \rfloor$. Note. For any real number $x$, $\lfloor x\rfloor$ denotes the largest integer $N\le x$. [i]Anant Mudgal and Navilarekallu Tejaswi[/i]

2016 NIMO Summer Contest, 9

Tags: algebra , function , floor function

Compute the number of real numbers $t$ such that \[t = 50 \sin(t - \lfloor t \rfloor).\] Here $\lfloor \cdot\rfloor$ denotes the greatest integer function. [i]Proposed by David Altizio[/i]

1989 IMO Longlists, 75

Tags: floor function

Solve in the set of real numbers the equation \[ 3x^3 \minus{} [x] \equal{} 3,\] where $ [x]$ denotes the integer part of $ x.$

2013 South East Mathematical Olympiad, 5

Tags: floor function , calculus , algebra

$f(x)=\sum\limits_{i=1}^{2013}\left[\dfrac{x}{i!}\right]$. A integer $n$ is called [i]good[/i] if $f(x)=n$ has real root. How many good numbers are in $\{1,3,5,\dotsc,2013\}$?

2012 Math Prize for Girls Olympiad, 2

Tags: induction , geometry , floor function , rectangle

Let $m$ and $n$ be integers greater than 1. Prove that $\left\lfloor \dfrac{mn}{6} \right\rfloor$ non-overlapping 2-by-3 rectangles can be placed in an $m$-by-$n$ rectangle. Note: $\lfloor x \rfloor$ means the greatest integer that is less than or equal to $x$.

1992 AIME Problems, 15

Tags: factorial , geometry , floor function

Define a positive integer $ n$ to be a factorial tail if there is some positive integer $ m$ such that the decimal representation of $ m!$ ends with exactly $ n$ zeroes. How many positive integers less than $ 1992$ are not factorial tails?

2007 Iran MO (3rd Round), 2

Tags: floor function , number theory

We call the mapping $ \Delta:\mathbb Z\backslash\{0\}\longrightarrow\mathbb N$, a degree mapping if and only if for each $ a,b\in\mathbb Z$ such that $ b\neq0$ and $ b\not|a$ there exist integers $ r,s$ such that $ a \equal{} br\plus{}s$, and $ \Delta(s) <\Delta(b)$. a) Prove that the following mapping is a degree mapping: \[ \delta(n)\equal{}\mbox{Number of digits in the binary representation of }n\] b) Prove that there exist a degree mapping $ \Delta_{0}$ such that for each degree mapping $ \Delta$ and for each $ n\neq0$, $ \Delta_{0}(n)\leq\Delta(n)$. c) Prove that $ \delta \equal{}\Delta_{0}$ [img]http://i16.tinypic.com/4qntmd0.png[/img]

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 $a_n \equiv n \mod 2012$.

2014 APMO, 4

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

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]

2014 Harvard-MIT Mathematics Tournament, 6

Tags: floor function

We have a calculator with two buttons that displays and integer $x$. Pressing the first button replaces $x$ by $\lfloor \frac{x}{2} \rfloor$, and pressing the second button replaces $x$ by $4x+1$. Initially, the calculator displays $0$. How many integers less than or equal to $2014$ can be achieved through a sequence of arbitrary button presses? (It is permitted for the number displayed to exceed 2014 during the sequence. Here, $\lfloor y \rfloor$ denotes the greatest integer less than or equal to the real number $y$).