Found problems: 1187
Given an integer $n\ge 4$. $S=\{1,2,\ldots,n\}$. $A,B$ are two subsets of $S$ such that for every pair of $(a,b),a\in A,b\in B, ab+1$ is a perfect square. Prove that
\[\min \{|A|,|B|\}\le\log _2n.\]
Let be a nonnegative integer $ n $ such that $ \sqrt n $ is not integer. Show that the function
$$ f:\{ a+b\sqrt n | a,b\in\{ 0\}\cup\mathbb{N} , a^2-nb^2=1 \}\longrightarrow\{ 0\}\cup\mathbb{N} , f(x) =\lfloor x \rfloor $$
is injective and non-surjective.
The notation $\lfloor n \rfloor$ denotes the greatest integer less than or equal to $n$. Evaluate $\lfloor 2.1 \lfloor {-}4.3 \rfloor \rfloor$.
$\textbf{(A) }{-}11\qquad\textbf{(B) }{-}10\qquad\textbf{(C) }{-}9\qquad\textbf{(D) }{-}8\qquad\textbf{(E) }{-}4$
A sequence $a_1,a_2,\ldots ,a_n,\ldots$ of natural numbers is defined by the rule
\[a_{n+1}=a_n+b_n\ (n=1,2,\ldots)\]
where $b_n$ is the last digit of $a_n$. Prove that such a sequence contains infinitely many powers of $2$ if and only if $a_1$ is not divisible by $5$.
If $1<k_1<k_2<...<k_n$ and $a_1,a_2,...,a_n$ are integers such that for every integer $N,$ $k_i \mid N-a_i$ for some $1 \leq i \leq n,$ find the smallest possible value of $n.$
Find all integers $b$ such that there exists a positive real number $x$ with \[ \dfrac {1}{b} = \dfrac {1}{\lfloor 2x \rfloor} + \dfrac {1}{\lfloor 5x \rfloor} \] Here, $\lfloor y \rfloor$ denotes the greatest integer that is less than or equal to $y$.
Let $n\geq 0$ be an integer and let $p \equiv 7 \pmod 8$ be a prime number. Prove that
\[ \sum^{p-1}_{k=1} \left \{ \frac {k^{2^n}}p - \frac 12 \right\} = \frac {p-1}2 . \]
[i]Călin Popescu[/i]
Find all positive numbers $x$ such that $20\{x\}+0.5\lfloor x\rfloor = 2005$.
Let $a,b,c,d$ be positive irrational numbers with $a+b = 1$.
Show that $c+d = 1$ if and only if $[na]+[nb] = [nc]+[nd]$ for all positive integers $n$.
Let $\mathbb{N}_0$ and $\mathbb{Z}$ be the set of all non-negative integers and the set of all integers, respectively. Let $f:\mathbb{N}_0\rightarrow\mathbb{Z}$ be a function defined as
\[f(n)=-f\left(\left\lfloor\frac{n}{3}\right\rfloor \right)-3\left\{\frac{n}{3}\right\} \]
where $\lfloor x \rfloor$ is the greatest integer smaller than or equal to $x$ and $\{ x\}=x-\lfloor x \rfloor$. Find the smallest integer $n$ such that $f(n)=2010$.
Given positive integer $n$, find the largest real number $\lambda=\lambda(n)$, such that for any degree $n$ polynomial with complex coefficients $f(x)=a_n x^n+a_{n-1} x^{n-1}+\cdots+a_0$,
and any permutation $x_0,x_1,\cdots,x_n$ of $0,1,\cdots,n$, the following inequality holds $\sum_{k=0}^n|f(x_k)-f(x_{k+1})|\geq \lambda |a_n|$, where $x_{n+1}=x_0$.
Given any two real numbers $\alpha$ and $\beta , 0 \leq \alpha < \beta \leq 1$, prove that there exists a natural number $m$ such that
\[\alpha < \frac{\phi(m)}{m} < \beta.\]
Let $\lfloor m \rfloor$ be the largest integer smaller than $m$ . Assume $x,y \in \mathbb{R+}$ ,
For all positive integer $n$ , $\lfloor x \lfloor ny \rfloor \rfloor =n-1$ .
Prove : $xy=1$ , $y$ is an irrational number larger than $ 1 $ .
Find all primes $p \ge 3$ such that $p- \lfloor p/q \rfloor q$ is a square-free integer for any prime $q<p$.
The sequences $a_n$ and $b_n$ members are the last digits of $[\sqrt{10}^n]$ and $[\sqrt{2}^n]$ respectively (here $[ ...]$ denotes the whole part of a number). Are those sequences periodical?
Find the highest power of $2$ that divides exactly into $1996!=1\times2\times\cdots\times1996$.
A non-empty set $ S$ of positive integers is said to be [i]good[/i] if there is a coloring with $ 2008$ colors of all positive integers so that no number in $ S$ is the sum of two different positive integers (not necessarily in $ S$) of the same color. Find the largest value $ t$ can take so that the set $ S\equal{}\{a\plus{}1,a\plus{}2,a\plus{}3,\ldots,a\plus{}t\}$ is good, for any positive integer $ a$.
[hide="P.S."]I have the feeling that I've seen this problem before, so if I'm right, maybe someone can post some links...[/hide]
Determine all possible values of the expression$$x-\left [\frac{x}{2}\right ]-\left [\frac{x}{3}\right ]-\left [\frac{x} {6}\right ]$$by varying $x$ in the real numbers.
Clarification: The brackets indicate the integer part of the number they enclose.
Determine all positive integers $n$ for which $\dfrac{n^2+1}{[\sqrt{n}]^2+2}$ is an integer. Here $[r]$ denotes the greatest integer less than or equal to $r$.
Find all real numbers $a$ such that there exist $f:\mathbb{R} \to \mathbb{R}$ with$$f(x+f(y))=f(x)+a\lfloor y \rfloor $$for all $x,y\in \mathbb{R}$
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.
Find all positive real solutions to the equation
$x+\left\lfloor\frac x3\right\rfloor=\left\lfloor\frac{2x}3\right\rfloor+\left\lfloor\frac{3x}5\right\rfloor$
Find all pairs of positive integers $(a,b)$ satisfying the following conditions:
[list]
[*] $a$ divides $b^4+1$,
[*] $b$ divides $a^4+1$,
[*] $\lfloor\sqrt{a}\rfloor=\lfloor \sqrt{b}\rfloor$.
[/list]
[i]Yang Liu[/i]
In a checked $ 17\times 17$ table, $ n$ squares are colored in black. We call a line any of rows, columns, or any of two diagonals of the table. In one step, if at least $ 6$ of the squares in some line are black, then one can paint all the squares of this line in black.
Find the minimal value of $ n$ such that for some initial arrangement of $ n$ black squares one can paint all squares of the table in black in some steps.
$f(x) = [x] + [2x] + [3x] + [4x] + [5x] + [6x]$. What values does $f$ take?