Found problems: 1340
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.
A total of $n$ people compete in a mathematical match which contains $15$ problems where $n>12$. For each problem, $1$ point is given for a right answer and $0$ is given for a wrong answer. Analysing each possible situation, we find that if the sum of points every group of $12$ people get is no less than $36$, then there are at least $3$ people that got the right answer of a certain problem, among the $n$ people. Find the least possible $n$.
Show that there is a natural number $n$ such that $n!$ when written in decimal notation ends exactly in 1993 zeros.
How many four-digit multiples of $8$ are greater than $2008$?
Find the number of ending zeros of $2014!$ in base 9. Give your answer in base 9.
The sequence of numbers $a_1, a_2, a_3, ...$ is defined recursively by $a_1 = 1, a_{n + 1} = \lfloor \sqrt{a_1+a_2+...+a_n} \rfloor $ for $n \ge 1$. Find all numbers that appear more than twice at this sequence.
Find all functions $f:\mathbb{N}\rightarrow \mathbb{N}$ such that for all $m,n\in\mathbb{N}$,
\[(2^m+1)f(n)f(2^mn)=2^mf(n)^2+f(2^mn)^2+(2^m-1)^2n. \]
The sequence $ \{a_n \} $ is defined as follows: $ a_0 = 1 $ and $ {a_n} = \sum \limits_ {k = 1} ^ {[\sqrt n]} {{a_ {n - {k ^ 2 }}}} $ for $ n \ge 1. $
Prove that among $ a_1, a_2, \ldots, a_ {10 ^ 6} $ there are at least $500$ even numbers.
(Here, $ [x] $ is the largest integer not exceeding $ x $.)
In chess tournament participates $n$ participants ($n >1$). In tournament each of participants plays with each other exactly $1$ game. For each game participant have $1$ point if he wins game, $0,5$ point if game is drow and $0$ points if he lose game.
If after ending of tournament participant have at least $ 75 %
$ of maximum possible points he called $winner$ $of$ $tournament$.
Find maximum possible numbers of $winners$ $of$ $tournament$.
Find the number of positive integers $n \le 2014$ such that there exists integer $x$ that satisfies the condition that $\frac{x+n}{x-n}$ is an odd perfect square.
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 $x$.
There are $n$ people at a party. Prove that there are two people such that, of the remaining $n-2$ people, there are at least $\left\lfloor\frac{n}{2}\right\rfloor-1$ of them, each of whom either knows both or else knows neither of the two. Assume that knowing is a symmetric relation, and that $\lfloor x\rfloor$ denotes the greatest integer less than or equal to $x$.
Let $\mathcal P$ denote the set of planes in three-dimensional space with positive $x$, $y$, and $z$ intercepts summing to one. A point $(x,y,z)$ with $\min \{x,y,z\} > 0$ lies on exactly one plane in $\mathcal P$. What is the maximum possible integer value of $\left(\frac{1}{4} x^2 + 2y^2 + 16z^2\right)^{-1}$?
[i]Proposed by Sammy Luo[/i]
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.
Find the number of positive integers $n \le 2014$ such that there exists integer $x$ that satisfies the condition that $\frac{x+n}{x-n}$ is an odd perfect square.
Given that \[ \frac 1{2!17!}+\frac 1{3!16!}+\frac 1{4!15!}+\frac 1{5!14!}+\frac 1{6!13!}+\frac 1{7!12!}+\frac 1{8!11!}+\frac 1{9!10!}=\frac N{1!18!} \] find the greatest integer that is less than $\frac N{100}.$
If $a,b,x,y$ are real numbers then find all solutions of $a,b$ such that $\left \lfloor ax+by \right \rfloor + \left \lfloor bx+ay \right \rfloor = (a+b) \left \lfloor x+y \right \rfloor$
Consider a random string $s$ of $10^{2020}$ base-ten digits (there can be leading zeroes). We say a substring $s' $ (which has no leading zeroes) is self-locating if $s' $ appears in $s$ at index $s' $ where the string is indexed at $ 1$. For example the substring $11$ in the string “$122352242411$” is selflocating since the $11$th digit is $ 1$ and the $12$th digit is $ 1$. Let the expected number of self-locating substrings in s be $G$. Compute $\lfloor G \rfloor$.
Let $a_1,a_2,a_3,...$ be a sequence of positive real numbers such that:
(i) For all positive integers $m,n$, we have $a_{mn}=a_ma_n$
(ii) There exists a positive real number $B$ such that for all positive integers $m,n$ with $m<n$, we have $a_m < Ba_n$
Find all possible values of $\log_{2015}(a_{2015}) - \log_{2014}(a_{2014})$
Find the positive integer $n$ for which \[ \lfloor \log_2{1}\rfloor+\lfloor\log_2{2}\rfloor+\lfloor\log_2{3}\rfloor+\cdots+\lfloor\log_2{n}\rfloor=1994. \] (For real $x$, $\lfloor x\rfloor$ is the greatest integer $\le x.$)
Let be a sequence of functions $ a_n:\mathbb{R}\longrightarrow\mathbb{Z} $ defined as $ a_n(x)=\sum_{i=1}^n (-1)^i\lfloor xi\rfloor . $
[b]a)[/b] Find the real numbers $ y $ such that $ \left( a_n(y) \right)_{n\ge 1} $ converges to $ 1. $
[b]b)[/b] Find the real numbers $ z $ such that $ \left( a_n(z) \right)_{n\ge 1} $ converges.
For each real number $ x$< let $ \lfloor x \rfloor$ be the integer satisfying $ \lfloor x \rfloor \le x < \lfloor x \rfloor \plus{}1$ and let $ \{x\}\equal{}x\minus{}\lfloor x \rfloor$. Let $ c$ be a real number such that \[ \{n\sqrt{3}\}>\dfrac{c}{n\sqrt{3}}\] for all positive integers $ n$. Prove that $ c \le 1$.
The sequence $S_0,S_1,S_2,\ldots$ is defined by[list][*]$S_n=1$ for $0\le n\le 2011$, and
[*]$S_{n+2012}=S_{n+2011}+S_n$ for $n\ge 0$.[/list]Prove that $S_{2011a}-S_a$ is a multiple of $2011$ for all nonnegative integers $a$.
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]
The $n^{th}$ term of a sequence is $t_n$. For $n \ge 1$, $t_n$ is given by the relation:
$$t_n= n^3+\frac12 n^2+ \frac13 n + \frac14$$
The $n^{th}$ term of a second sequence $T_n$, where $T_n$ represents the smallest integer greater than $t_n$. Calculate: $$(T_1+T_2+...+T_{1014}) -(t_1+t_2+...+t_{1014}) $$