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

2006 AIME Problems, 13
Tags: floor function
How many integers $ N$ less than 1000 can be written as the sum of $ j$ consecutive positive odd integers from exactly 5 values of $ j\ge 1$?

2010 Hanoi Open Mathematics Competitions, 6
Tags: floor function , algebra
Find the greatest integer less than $(2 +\sqrt3)^5$ . (A): $721$ (B): $722$ (C): $723$ (D): $724$ (E) None of the above.

1992 Vietnam Team Selection Test, 1
Tags: floor function , number theory
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 $n$.

PEN I Problems, 1
Tags: floor function
Let $ \alpha$ be the positive root of the equation $ x^{2} \equal{} 1991x \plus{} 1$. For natural numbers $ m$ and $ n$ define \[ m*n \equal{} mn \plus{} \lfloor\alpha m \rfloor \lfloor \alpha n\rfloor. \] Prove that for all natural numbers $ p$, $ q$, and $ r$, \[ (p*q)*r \equal{} p*(q*r). \]

2005 China Team Selection Test, 3
Tags: floor function , euler , 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 \[\sum_{j=0}^{n} a_j x_j^m \equiv 0 \pmod m, \qquad \text{for all}\quad m \in \{ r+1, r+2, \cdots, 2r+1 \}.\]

2020 HMNT (HMMO), 1
Tags: floor function , algebra
For how many positive integers $n \le 1000$ does the equation in real numbers $x^{\lfloor x \rfloor } = n$ have a positive solution for $x$?

2019 Peru IMO TST, 4
Tags: sequence , floor function , number theory
Let $k\geq 0$ an integer. The sequence $a_0,\ a_1,\ a_2, \ a_3, \ldots$ is defined as follows: [LIST] [*] $a_0=k$ [/*] [*] For $n\geq 1$, we have that $a_n$ is the smallest integer greater than $a_{n-1}$ so that $a_n+a_{n-1}$ is a perfect square. [/*] [/LIST] Prove that there are exactly $\left \lfloor{\sqrt{2k}} \right \rfloor$ positive integers that cannot be written as the difference of two elements of such a sequence. [i]Note.[/i] If $x$ is a real number, $\left \lfloor{x} \right \rfloor$ denotes the greatest integer smaller or equal than $x$.

1991 Baltic Way, 6
Tags: floor function
Solve the equation $[x] \cdot \{x\} = 1991x$. (Here $[x]$ denotes the greatest integer less than or equal to $x$, and $\{x\}=x-[x]$.)

2007 Putnam, 5
Tags: derivative , calculus , polynomial , linear algebra , floor function , algebra
Let $ k$ be a positive integer. Prove that there exist polynomials $ P_0(n),P_1(n),\dots,P_{k\minus{}1}(n)$ (which may depend on $ k$) such that for any integer $ n,$ \[ \left\lfloor\frac{n}{k}\right\rfloor^k\equal{}P_0(n)\plus{}P_1(n)\left\lfloor\frac{n}{k}\right\rfloor\plus{} \cdots\plus{}P_{k\minus{}1}(n)\left\lfloor\frac{n}{k}\right\rfloor^{k\minus{}1}.\] ($ \lfloor a\rfloor$ means the largest integer $ \le a.$)

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

2011 Switzerland - Final Round, 9
Tags: strong induction , ceiling function , induction , floor function , 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]

2011 AIME Problems, 15
Tags: quadratic , polynomial , algebra , probability , floor function
Let $P(x)=x^2-3x-9$. A real number $x$ is chosen at random from the interval $5\leq x \leq 15$. The probability that $\lfloor \sqrt{P(x)} \rfloor = \sqrt{P(\lfloor x \rfloor )}$ is equal to $\dfrac{\sqrt{a}+\sqrt{b}+\sqrt{c}-d}{e}$, where $a,b,c,d$ and $e$ are positive integers and none of $a,b,$ or $c$ is divisible by the square of a prime. Find $a+b+c+d+e$.

2010 Laurențiu Panaitopol, Tulcea, 1
Tags: equation , algebra , trigonometric equation , floor , monotone functions , floor function
Solve in the real numbers the equation $ \arcsin x=\lfloor 2x \rfloor . $ [i]Petre Guțescu[/i]

2014 China Western Mathematical Olympiad, 4
Tags: algorithm , hard combinatorics , combinatorics , floor function , algebra
Given a positive integer $n$, let $a_1,a_2,..,a_n$ be a sequence of nonnegative integers. A sequence of one or more consecutive terms of $a_1,a_2,..,a_n$ is called $dragon$ if their aritmetic mean is larger than 1. If a sequence is a $dragon$, then its first term is the $head$ and the last term is the $tail$. Suppose $a_1,a_2,..,a_n$ is the $head$ or/and $tail$ of some $dragon$ sequence; determine the minimum value of $a_1+a_2+\cdots +a_n$ in terms of $n$.

2010 AMC 12/AHSME, 22
Tags: graphing lines , slope , analytic geometry , function , floor function , inequalities
What is the minimum value of $ f(x) \equal{} |x \minus{} 1| \plus{} |2x \minus{} 1| \plus{} |3x \minus{} 1| \plus{} \cdots \plus{} |119x \minus{} 1|$? $ \textbf{(A)}\ 49 \qquad \textbf{(B)}\ 50 \qquad \textbf{(C)}\ 51 \qquad \textbf{(D)}\ 52 \qquad \textbf{(E)}\ 53$

1985 Tournament Of Towns, (086) 2
Tags: fractional part , integer part , floor function
The integer part $I (A)$ of a number $A$ is the greatest integer which is not greater than $A$ , while the fractional part $F(A)$ is defined as $A - I(A)$ . (a) Give an example of a positive number $A$ such that $F(A) + F( 1/A) = 1$ . (b) Can such an $A$ be a rational number? (I. Varge, Romania)

2014 Contests, 2
Tags: number theory , floor function
Let $n$ be a natural number. Prove that, \[ \left\lfloor \frac{n}{1} \right\rfloor+ \left\lfloor \frac{n}{2} \right\rfloor + \cdots + \left\lfloor \frac{n}{n} \right\rfloor + \left\lfloor \sqrt{n} \right\rfloor \] is even.

2018 Moldova Team Selection Test, 12
Tags: divisibility theory , modular arithmetic , floor function
Let $p>3$ is a prime number and $k=\lfloor\frac{2p}{3}\rfloor$. Prove that \[{p \choose 1}+{p \choose 2}+\cdots+{p \choose k}\] is divisible by $p^{2}$.

2020 BMT Fall, 10
Tags: number theory , floor function
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?

2001 Austrian-Polish Competition, 6
Tags: sequence , integer , floor function , algebra , number theory
Let $k$ be a fixed positive integer. Consider the sequence definited by \[a_{0}=1 \;\; , a_{n+1}=a_{n}+\left\lfloor\root k \of{a_{n}}\right\rfloor \;\; , n=0,1,\cdots\] where $\lfloor x\rfloor$ denotes the greatest integer less than or equal to $x$. For each $k$ find the set $A_{k}$ containing all integer values of the sequence $(\sqrt[k]{a_{n}})_{n\geq 0}$.

1985 ITAMO, 10
Tags: integration , calculus , floor function , function
How many of the first 1000 positive integers can be expressed in the form \[ \lfloor 2x \rfloor + \lfloor 4x \rfloor + \lfloor 6x \rfloor + \lfloor 8x \rfloor, \] where $x$ is a real number, and $\lfloor z \rfloor$ denotes the greatest integer less than or equal to $z$?

2007 Pre-Preparation Course Examination, 1
Tags: absolute value , number theory , limit , inequalities , floor function
Let $a\geq 2$ be a natural number. Prove that $\sum_{n=0}^\infty\frac1{a^{n^{2}}}$ is irrational.

2014 Bosnia Herzegovina Team Selection Test, 2
Tags: combinatorics , floor function
It is given regular $n$-sided polygon, $n \geq 6$. How many triangles they are inside the polygon such that all of their sides are formed by diagonals of polygon and their vertices are vertices of polygon?

2021 Thailand TST, 2
Tags: number theory , floor function
Prove that, for all positive integers $m$ and $n$, we have $$\left\lfloor m\sqrt{2} \right\rfloor\cdot\left\lfloor n\sqrt{7} \right\rfloor<\left\lfloor mn\sqrt{14} \right\rfloor.$$

PEN A Problems, 13
Tags: logarithm , divisibility theory , inequalities , floor function
Show that for all prime numbers $p$, \[Q(p)=\prod^{p-1}_{k=1}k^{2k-p-1}\] is an integer.