Found problems: 1187
The sum of several non-negative numbers is not greater than 200, while the sum of their squares is not less than 2500. Prove that among them there are four numbers whose sum is not less than 50.
[i]Proposed by A. Khabrov[/i]
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.
Find the folllowing limit.
\[ \lim_{n\to\infty} \frac{(2n\plus{}1)\int_0^1 x^{n\minus{}1}\sin \left(\frac{\pi}{2}x\right)dx}{(n\plus{}1)^2\int_0^1 x^{n\minus{}1}\cos \left(\frac{\pi}{2}x\right)dx}\ \ (n\equal{}1,\ 2,\ \cdots).\]
Let $f(x, y) = \left\lfloor \frac{5x}{2y} \right\rfloor + \left\lceil \frac{5y}{2x} \right\rceil$. Suppose $x, y$ are chosen independently uniformly at random from the interval $(0, 1]$. Let $p$ be the probability that $f(x, y) < 6$. If $p$ can be expressed in the form $m/n$ for relatively prime positive integers $m$ and $n$, compute $m + n$.
(Note: $\lfloor x\rfloor $ is defined as the greatest integer less than or equal to $x$ and $\lceil x \rceil$ is defined as the least integer greater than or equal to$ x$.)
Prove for all natural $n$ that $\left. {{{40}^n} \cdot n!} \right|(5n)!$
Let $m>1$ be an integer. Determine the number of positive integer solutions of the equation $\left\lfloor\frac xm\right\rfloor=\left\lfloor\frac x{m-1}\right\rfloor$.
Show that for all primes $p$, \[\sum^{p-1}_{k=1}\left \lfloor \frac{k^{3}}{p}\right \rfloor =\frac{(p+1)(p-1)(p-2)}{4}.\]
Suppose $x$ is a random real number between $1$ and $4$, and $y$ is a random real number between $1$ and $9$. If the expected value of \[ \left\lceil \log_2 x \right\rceil - \left\lfloor \log_3 y \right\rfloor \] can be expressed as $\frac mn$ where $m$ and $n$ are relatively prime positive integers, compute $100m + n$.
[i]Proposed by Lewis Chen[/i]
Let $n$ be a natural number and $f(n) = 2n - 1995 \lfloor \frac{n}{1000} \rfloor$($\lfloor$ $\rfloor$ denotes the floor function).
1. Show that if for some integer $r$: $f(f(f...f(n)...))=1995$ (where the function $f$ is applied $r$ times), then $n$ is multiple of $1995$.
2. Show that if $n$ is multiple of 1995, then there exists r such that:$f(f(f...f(n)...))=1995$ (where the function $f$ is applied $r$ times). Determine $r$ if $n=1995.500=997500$
Prove that for all $n\in\mathbb{Z}^+$, we have \[ \sum\limits_{p=1}^n\sum\limits_{q=1}^p\left\lfloor -\frac{1+\sqrt{8q+(2p-1)^2}}{2}\right\rfloor =-\frac{n(n+1)(n+2)}{3} \]
Let $a=\lfloor \sqrt{n} \rfloor$ for given positive integer $n.$
Express the summation $\sum_{k=1}^{n}\lfloor \sqrt{k} \rfloor$ in terms of $n$ and $a.$
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 $n$.
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].
Define $ f: \mathbb{R}\to\mathbb{R}$ by
\[ f(x)\equal{}\begin{cases}x&\text{if }x\le e\\ xf(\ln x)&\text{if }x>e\end{cases}\]
Does $ \displaystyle\sum_{n\equal{}1}^{\infty}\frac1{f(n)}$ converge?
Find the number of different numbers of the form $\left\lfloor\frac{i^2}{2015} \right\rfloor$, with $i = 1,2, ..., 2015$.
For a positive integer $n$, two payers $A$ and $B$ play the following game: Given a pile of $s$ stones, the players take turn alternatively with $A$ going first. On each turn the player is allowed to take either one stone, or a prime number of stones, or a positive multiple of $n$ stones. The winner is the one who takes the last stone. Assuming both $A$ and $B$ play perfectly, for how many values of $s$ the player $A$ cannot win?
For each $\alpha\in \mathbb{R}$ define $f_{\alpha}(x)=\lfloor{\alpha x}\rfloor$. Let $n\in \mathbb{N}$. Show there exists a real $\alpha$ such that for $1\le \ell \le n$ :
\[ f_{\alpha}^{\ell}(n^2)=n^2-\ell=f_{\alpha^{\ell}}(n^2).\]
Here $f^{\ell}_{\alpha}(x)=(f_{\alpha}\circ f_{\alpha}\circ \cdots \circ f_{\alpha})(x)$ where the composition is carried out $\ell$ times.
Let $\lfloor x\rfloor$ denote the greatest integer less than or equal to $x$. Let $m$ be a positive integer, $m\geq 3$. For every integer $i$ with $1\leq i\leq m$, let \[S_{m,i}=\left\{\left\lfloor\dfrac{2^m-1}{2^{i-1}}n-2^{m-i}+1\right\rfloor\,:\,n=1,2,3,\ldots\right\}.\] For example, for $m=3$,
\begin{align*}S_{3,1}&=\{\lfloor 7n-3\rfloor\,:\,n=1,2,3,\ldots\}
\\&=\{4,11,18,\ldots\},
\\S_{3,2}&=\left\{\left\lfloor\dfrac72n-1\right\rfloor\,:\,n=1,2,3,\ldots\right\}
\\&=\{2,6,9,\ldots\},
\\S_{3,3}&=\left\{\left\lfloor\dfrac74n\right\rfloor\,:\,n=1,2,3,\ldots\right\}
\\&=\{1,3,5,\ldots\}.\end{align*}
Prove that for all $m\geq 3$, each positive integer occurs in exactly one of the sets $S_{m,i}$.
Determine all pairs $(a, b)$ of real numbers such that $a\lfloor bn\rfloor =b\lfloor an\rfloor$ for all positive integer $n$.
Let $ S$ be the set of the $ 2005$ smallest multiples of $ 4$, and let $ T$ be the set of the $ 2005$ smallest positive multiples of $ 6$. How many elements are common to $ S$ and $ T$?
$ \textbf{(A)}\ 166\qquad
\textbf{(B)}\ 333\qquad
\textbf{(C)}\ 500\qquad
\textbf{(D)}\ 668\qquad
\textbf{(E)}\ 1001$
Find all reals $x$ satisfying $0 \le x \le 5$ and
$\lfloor x^2-2x \rfloor = \lfloor x \rfloor ^2 - 2 \lfloor x \rfloor$.
From the interval $(2^{2n},2^{3n})$ are selected $2^{2n-1}+1$ odd numbers. Prove that there are two among the selected numbers, none of which divides the square of the other.
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 $BC$.
Prove or disprove that there exists a positive real number $u$ such that $\lfloor u^n \rfloor -n$ is an even integer for all positive integer $n$.
Suppose $x$ is a positive real number such that $\{x\}, [x]$ and $x$ are in a geometric progression. Find the least positive integer $n$ such that $x^n > 100$. (Here $[x]$ denotes the integer part of $x$ and $\{x\} = x - [x]$.)