This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 1187

2010 Tuymaada Olympiad, 4
Tags: floor function , algebra , polynomial , limit , number theory
Prove that for any positive real number $\alpha$, the number $\lfloor\alpha n^2\rfloor$ is even for infinitely many positive integers $n$.

2020 Taiwan TST Round 3, 2
Tags: number theory , floor function
Let $H = \{ \lfloor i\sqrt{2}\rfloor : i \in \mathbb Z_{>0}\} = \{1,2,4,5,7,\dots \}$ and let $n$ be a positive integer. Prove that there exists a constant $C$ such that, if $A\subseteq \{1,2,\dots, n\}$ satisfies $|A| \ge C\sqrt{n}$, then there exist $a,b\in A$ such that $a-b\in H$. (Here $\mathbb Z_{>0}$ is the set of positive integers, and $\lfloor z\rfloor$ denotes the greatest integer less than or equal to $z$.)

2008 AMC 12/AHSME, 23
Tags: floor function , 3d geometry , pyramid , symmetry , number theory , logarithm , geometry
The sum of the base-$ 10$ logarithms of the divisors of $ 10^n$ is $ 792$. What is $ n$? $ \textbf{(A)}\ 11\qquad \textbf{(B)}\ 12\qquad \textbf{(C)}\ 13\qquad \textbf{(D)}\ 14\qquad \textbf{(E)}\ 15$

MOAA Team Rounds, 2022.13
Tags: floor function , algebra
Determine the number of distinct positive real solutions to $$\lfloor x \rfloor ^{\{x\}} = \frac{1}{2022}x^2$$ . Note: $\lfloor x \rfloor$ is known as the floor function, which returns the greatest integer less than or equal to $x$. Furthermore, $\{x\}$ is defined as $x - \lfloor x \rfloor$.

2005 Singapore MO Open, 1
Tags: floor function , number theory
An integer is square-free if it is not divisible by $a^2$ for any integer $a>1$. Let $S$ be the set of positive square-free integers. Determine, with justification, the value of\[\sum_{k\epsilon S}\left[\sqrt{\frac{10^{10}}{k}}\right]\]where $[x]$ denote the greatest integer less than or equal to $x$

2010 China Team Selection Test, 3
Tags: floor function , inequalities , algorithm , combinatorics
Let $n_1,n_2, \cdots, n_{26}$ be pairwise distinct positive integers satisfying (1) for each $n_i$, its digits belong to the set $\{1,2\}$; (2) for each $i,j$, $n_i$ can't be obtained from $n_j$ by adding some digits on the right. Find the smallest possible value of $\sum_{i=1}^{26} S(n_i)$, where $S(m)$ denotes the sum of all digits of a positive integer $m$.

2005 IMAR Test, 3
Tags: ratio , ceiling function , floor function , algebra
A flea moves in the positive direction on the real Ox axis, starting from the origin. He can only jump over distances equal with $\sqrt 2$ or $\sqrt{2005}$. Prove that there exists $n_0$ such that the flea can reach any interval $[n,n+1]$ with $n\geq n_0$.

2013 National Olympiad First Round, 35
Tags: floor function , algebra
What is the least positive integer $n$ such that $\overbrace{f(f(\dots f}^{21 \text{ times}}(n)))=2013$ where $f(x)=x+1+\lfloor \sqrt x \rfloor$? ($\lfloor a \rfloor$ denotes the greatest integer not exceeding the real number $a$.) $ \textbf{(A)}\ 1214 \qquad\textbf{(B)}\ 1202 \qquad\textbf{(C)}\ 1186 \qquad\textbf{(D)}\ 1178 \qquad\textbf{(E)}\ \text{None of above} $

2007 Bundeswettbewerb Mathematik, 4
Tags: floor function , function , algebra
Let $a$ be a positive integer. How many non-negative integer solutions x does the equation $\lfloor \frac{x}{a}\rfloor = \lfloor \frac{x}{a+1}\rfloor$ have? $\lfloor ~ \rfloor$ ---> [url=http://en.wikipedia.org/wiki/Floor_function]Floor Function[/url].

2014-2015 SDML (High School), 15
Tags: quadratic , floor function
Find the sum of all $\left\lfloor x\right\rfloor$ such that $x^2-15\left\lfloor x\right\rfloor+36=0$. $\text{(A) }15\qquad\text{(B) }26\qquad\text{(C) }45\qquad\text{(D) }49\qquad\text{(E) }75$

2003 Vietnam Team Selection Test, 3
Tags: floor function , calculus , integration , induction , algebra
Let $f(0, 0) = 5^{2003}, f(0, n) = 0$ for every integer $n \neq 0$ and \[\begin{array}{c}\ f(m, n) = f(m-1, n) - 2 \cdot \Bigg\lfloor \frac{f(m-1, n)}{2}\Bigg\rfloor + \Bigg\lfloor\frac{f(m-1, n-1)}{2}\Bigg\rfloor + \Bigg\lfloor\frac{f(m-1, n+1)}{2}\Bigg\rfloor \end{array}\] for every natural number $m > 0$ and for every integer $n$. Prove that there exists a positive integer $M$ such that $f(M, n) = 1$ for all integers $n$ such that $|n| \leq \frac{(5^{2003}-1)}{2}$ and $f(M, n) = 0$ for all integers n such that $|n| > \frac{5^{2003}-1}{2}.$

2003 USA Team Selection Test, 1
Tags: pigeonhole principle , floor function , algebra
For a pair of integers $a$ and $b$, with $0 < a < b < 1000$, set $S\subseteq \{ 1, 2, \dots , 2003\}$ is called a [i]skipping set[/i] for $(a, b)$ if for any pair of elements $s_1, s_2 \in S$, $|s_1 - s_2|\not\in \{ a, b\}$. Let $f(a, b)$ be the maximum size of a skipping set for $(a, b)$. Determine the maximum and minimum values of $f$.

2016 Germany 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.

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]

2002 China Girls Math Olympiad, 5
Tags: floor function , inequalities
There are $ n \geq 2$ permutations $ P_1, P_2, \ldots, P_n$ each being an arbitrary permutation of $ \{1,\ldots,n\}.$ Prove that \[ \sum^{n\minus{}1}_{i\equal{}1} \frac{1}{P_i \plus{} P_{i\plus{}1}} > \frac{n\minus{}1}{n\plus{}2}.\]

1982 IMO Longlists, 9
Tags: function , floor function , number theory
Given any two real numbers $\alpha$ and $\beta , 0 \leq \alpha < \beta \leq 1$, prove that there exists a natural number $m$ such that \[\alpha < \frac{\phi(m)}{m} < \beta.\]

2010 China Team Selection Test, 3
Tags: floor function , inequalities , algorithm , combinatorics
Let $n_1,n_2, \cdots, n_{26}$ be pairwise distinct positive integers satisfying (1) for each $n_i$, its digits belong to the set $\{1,2\}$; (2) for each $i,j$, $n_i$ can't be obtained from $n_j$ by adding some digits on the right. Find the smallest possible value of $\sum_{i=1}^{26} S(n_i)$, where $S(m)$ denotes the sum of all digits of a positive integer $m$.

2024 Czech-Polish-Slovak Junior Match, 4
Tags: divisibility , floor function , number theory
How many positive integers $n<2024$ are divisible by $\lfloor \sqrt{n}\rfloor-1$?

2012 Princeton University Math Competition, A8
Tags: floor function , algebra
If $n$ is an integer such that $n \ge 2^k$ and $n < 2^{k+1}$, where $k = 1000$, compute the following: $$n - \left( \lfloor \frac{n -2^0}{2^1} \rfloor + \lfloor \frac{n -2^1}{2^2} \rfloor + ...+ \lfloor \frac{n -2^{k-1}}{2^k} \rfloor \right)$$

1988 IMO Longlists, 2
Tags: floor function , algebra
Let $\left[\sqrt{(n+1)^2 + n^2} \right], n = 1,2, \ldots,$ where $[x]$ denotes the integer part of $x.$ Prove that [b]i.)[/b] there are infinitely many positive integers $m$ such that $a_{m+1} - a_m > 1;$ [b]ii.)[/b] there are infinitely many positive integers $m$ such that $a_{m+1} - a_m = 1.$

2007 Mathematics for Its Sake, 1
Tags: floor function , real analysis , sequence , parity , number theory
Prove that the parity of each term of the sequence $ \left( \left\lfloor \left( \lfloor \sqrt q \rfloor +\sqrt{q} \right)^n \right\rfloor \right)_{n\ge 1} $ is opposite to the parity of its index, where $ q $ is a squarefree natural number.

2008 Postal Coaching, 1
Tags: floor function , number theory , natural , algebra , irrational , fraction
For each positive $ x \in \mathbb{R}$, define $ E(x)=\{[nx]: n\in \mathbb{N}\}$ Find all irrational $ \alpha >1$ with the following property: If a positive real $ \beta$ satisfies $ E(\beta) \subset E(\alpha)$. then $ \frac{\beta}{\alpha}$ is a natural number.

1984 Iran MO (2nd round), 2
Tags: function , trigonometry , floor function , limit , algebra
Consider the function \[f(x)= \sin \biggl( \frac{\pi}{2} \lfloor x \rfloor \biggr).\] Find the period of $f$ and sketch diagram of $f$ in one period. Also prove that $\lim_{x \to 1} f(x)$ does not exist.

2011 Switzerland - Final Round, 9
Tags: ceiling function , floor function , induction , strong induction , number theory
For any positive integer $n$ let $f(n)$ be the number of divisors of $n$ ending with $1$ or $9$ in base $10$ and let $g(n)$ be the number of divisors of $n$ ending with digit $3$ or $7$ in base $10$. Prove that $f(n)\geqslant g(n)$ for all nonnegative integers $n$. [i](Swiss Mathematical Olympiad 2011, Final round, problem 9)[/i]

2007 Regional Competition For Advanced Students, 2
Tags: floor function , 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.$