Found problems: 1187
1996 Argentina National Olympiad, 5
Determine all positive real numbers $x$ for which $$\left [x\right ]+\left [\sqrt{1996x}\right ]=1996$$ is verified
Clarification:The brackets indicate the integer part of the number they enclose.
2003 Argentina National Olympiad, 1
Find all positive numbers $x$ such that:$$\frac{1}{[x]}-\frac{1}{[2x]}=\frac{1}{6\{x\}}$$ where $[x]$ represents the integer part of $x$ and $\{x\}=x-[x]$.
1990 Baltic Way, 19
What is the largest possible number of subsets of the set $\{1, 2, \dots , 2n+1\}$ such that the intersection of any two subsets consists of one or several consecutive integers?
PEN I Problems, 6
Prove that for all positive integers $n$, \[\lfloor \sqrt{n}+\sqrt{n+1}+\sqrt{n+2}\rfloor =\lfloor \sqrt{9n+8}\rfloor.\]
2001 All-Russian Olympiad, 1
The integers from $1$ to $999999$ are partitioned into two groups: the first group consists of those integers for which the closest perfect square is odd, whereas the second group consists of those for which the closest perfect square is even. In which group is the sum of the elements greater?
2012 Waseda University Entrance Examination, 2
Consider a sequence $\{a_n\}_{n\geq 0}$ such that $a_{n+1}=a_n-\lfloor{\sqrt{a_n}}\rfloor\ (n\geq 0),\ a_0\geq 0$.
(1) If $a_0=24$, then find the smallest $n$ such that $a_n=0$.
(2) If $a_0=m^2\ (m=2,\ 3,\ \cdots)$, then for $j$ with $1\leq j\leq m$, express $a_{2j-1},\ a_{2j}$ in terms of $j,\ m$.
(3) Let $m\geq 2$ be integer and for integer $p$ with $1\leq p\leq m-1$, let $a\0=m^2-p$. Find $k$ such that $a_k=(m-p)^2$, then
find the smallest $n$ such that $a_n=0$.
1999 Korea Junior Math Olympiad, 3
Recall that $[x]$ denotes the largest integer not exceeding $x$ for real $x$. For integers $a, b$ in the interval $1 \leq a<b \leq 100$, find the number of ordered pairs $(a, b)$ satisfying the following equation.
$$[a+\frac{b}{a}]=[b+\frac{a}{b}]$$
2014 China Team Selection Test, 1
Prove that for any positive integers $k$ and $N$, \[\left(\frac{1}{N}\sum\limits_{n=1}^{N}(\omega (n))^k\right)^{\frac{1}{k}}\leq k+\sum\limits_{q\leq N}\frac{1}{q},\] where $\sum\limits_{q\leq N}\frac{1}{q}$ is the summation over of prime powers $q\leq N$ (including $q=1$).
Note: For integer $n>1$, $\omega (n)$ denotes number of distinct prime factors of $n$, and $\omega (1)=0$.
2005 USAMO, 6
For $m$ a positive integer, let $s(m)$ be the sum of the digits of $m$. For $n\ge 2$, let $f(n)$ be the minimal $k$ for which there exists a set $S$ of $n$ positive integers such that $s\left(\sum_{x\in X} x\right)=k$ for any nonempty subset $X\subset S$. Prove that there are constants $0<C_1<C_2$ with
\[C_1 \log_{10} n \le f(n) \le C_2 \log_{10} n.\]
2009 AMC 12/AHSME, 21
Ten women sit in $ 10$ seats in a line. All of the $ 10$ get up and then reseat themselves using all $ 10$ seats, each sitting in the seat she was in before or a seat next to the one she occupied before. In how many ways can the women be reseated?
$ \textbf{(A)}\ 89\qquad
\textbf{(B)}\ 90\qquad
\textbf{(C)}\ 120\qquad
\textbf{(D)}\ 2^{10}\qquad
\textbf{(E)}\ 2^2 3^8$
2002 Singapore Team Selection Test, 2
For each real number $x$, $\lfloor x \rfloor$ is the greatest integer less than or equal to $x$. For example $\lfloor 2.8 \rfloor = 2$. Let $r \ge 0$ be a real number such that for all integers $m, n, m|n$ implies $\lfloor mr \rfloor| \lfloor nr \rfloor$. Prove that $r$ is an integer.
2012 BMT Spring, 5
Let $p > 1$ be relatively prime to $10$. Let $n$ be any positive number and$ d$ be the last digit of $n$. Define $f(n) = \lfloor \frac{n}{10} \rfloor + d \cdot m$. Then, we can call $m$ a [i]divisibility multiplier[/i] for $p$, if $f(n)$ is divisible by $p$ if and only if $n$ is divisible by $p$. Find a divisibility multiplier for $2013$.
2004 Italy TST, 3
Find all functions $f:\mathbb{N}\rightarrow \mathbb{N}$ such that for all $m,n\in\mathbb{N}$,
\[(2^m+1)f(n)f(2^mn)=2^mf(n)^2+f(2^mn)^2+(2^m-1)^2n. \]
2005 Rioplatense Mathematical Olympiad, Level 3, 1
Find all numbers $n$ that can be expressed in the form $n=k+2\lfloor\sqrt{k}\rfloor+2$ for some nonnegative integer $k$.
2008 ITest, 32
A right triangle has perimeter $2008$, and the area of a circle inscribed in the triangle is $100\pi^3$. Let $A$ be the area of the triangle. Compute $\lfloor A\rfloor$.
2012 Online Math Open Problems, 22
Find the largest prime number $p$ such that when $2012!$ is written in base $p$, it has at least $p$ trailing zeroes.
[i]Author: Alex Zhu[/i]
PEN E Problems, 38
Prove that if $c > \dfrac{8}{3}$, then there exists a real number $\theta$ such that $\lfloor\theta^{c^n}\rfloor$ is prime for every positive integer $n$.
1999 Canada National Olympiad, 1
Find all real solutions to the equation $4x^2 - 40 \lfloor x \rfloor + 51 = 0$.
2007 Balkan MO, 4
For a given positive integer $n >2$, let $C_{1},C_{2},C_{3}$ be the boundaries of three convex $n-$ gons in the plane , such that
$C_{1}\cap C_{2}, C_{2}\cap C_{3},C_{1}\cap C_{3}$ are finite. Find the maximum number of points of the sets $C_{1}\cap C_{2}\cap C_{3}$.
2019 Federal Competition For Advanced Students, P1, 4
Find all pairs $(a, b)$ of real numbers such that $a \cdot \lfloor b \cdot n\rfloor = b \cdot \lfloor a \cdot n \rfloor$ applies to all positive integers$ n$.
(For a real number $x, \lfloor x\rfloor$ denotes the largest integer that is less than or equal to $x$.)
2014 Purple Comet Problems, 12
The first number in the following sequence is $1$. It is followed by two $1$'s and two $2$'s. This is followed by three $1$'s, three $2$'s, and three $3$'s. The sequence continues in this fashion.
\[1,1,1,2,2,1,1,1,2,2,2,3,3,3,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,\dots.\]
Find the $2014$th number in this sequence.
2011 AIME Problems, 15
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 $a+b+c+d+e$.
1971 IMO Longlists, 53
Denote by $x_n(p)$ the multiplicity of the prime $p$ in the canonical representation of the number $n!$ as a product of primes. Prove that $\frac{x_n(p)}{n}<\frac{1}{p-1}$ and $\lim_{n \to \infty}\frac{x_n(p)}{n}=\frac{1}{p-1}$.
PEN P Problems, 21
Let $A$ be the set of positive integers of the form $a^2 +2b^2$, where $a$ and $b$ are integers and $b \neq 0$. Show that if $p$ is a prime number and $p^2 \in A$, then $p \in A$.
2019 Hong Kong TST, 4
Let $ABC$ be an acute-angled triangle such that $\angle{ACB} = 45^{\circ}$. Let $G$ be the point of intersection of the three medians and let $O$ be the circumcentre. Suppose $OG=1$ and $OG \parallel BC$. Determine the length of the segment $BC$.