Found problems: 1187
At a conference there are $n$ mathematicians. Each of them knows exactly $k$ fellow mathematicians. Find the smallest value of $k$ such that there are at least three mathematicians that are acquainted each with the other two.
[color=#BF0000]Rewording of the last line for clarification:[/color]
Find the smallest value of $k$ such that there (always) exists $3$ mathematicians $X,Y,Z$ such that $X$ and $Y$ know each other, $X$ and $Z$ know each other and $Y$ and $Z$ know each other.
Find the minimum positive noninteger root of $ \sin x=\sin \lfloor x \rfloor $.
F. Petrov
Each term of an infinite sequence of natural numbers is obtained from the previous term by adding to it one of its nonzero digits. Prove that this sequence contains an even number.
For fixed positive integers $s, t$, define $a_n$ as the following.
$a_1 = s, a_2 = t$, and $\forall n \ge 1$, $a_{n+2} = \lfloor \sqrt{a_n+(n+2)a_{n+1}+2008} \rfloor$.
Prove that the solution set of $a_n \not= n$, $n \in \mathbb{N}$ is finite.
Find the number of positive integers $n$ less than $1000$ for which there exists a positive real number $x$ such that $n = x \lfloor x \rfloor$.
[b]Note[/b]: $\lfloor x \rfloor$ is the greatest integer less than or equal to $x$.
Prove that for all positive integers $n$, \[\lfloor \sqrt[3]{n}+\sqrt[3]{n+1}\rfloor =\lfloor \sqrt[3]{8n+3}\rfloor.\]
Find all real numbers $x$ such that
$\lfloor x^2-2x \rfloor+2\lfloor x \rfloor=\lfloor x \rfloor^2$.
(For a real number $x$, $\lfloor x \rfloor$ denote the greatest integer less than or equal to $x$.)
A set of 10 distinct integers $S$ is chosen. Let $M$ be the number of nonempty subsets of $S$ whose elements have an even sum. What is the minimum possible value of $M$?
[hide="Clarifications"]
[list]
[*] $S$ is the ``set of 10 distinct integers'' from the first sentence.[/list][/hide]
[i]Ray Li[/i]
Let $ a, b, c$ be positive integers satisfying the conditions $ b > 2a$ and $ c > 2b.$ Show that there exists a real number $ \lambda$ with the property that all the three numbers $ \lambda a, \lambda b, \lambda c$ have their fractional parts lying in the interval $ \left(\frac {1}{3}, \frac {2}{3} \right].$
Find the sum of the decimal digits of \[ \left\lfloor \frac{51525354555657\dots979899}{50} \right\rfloor. \] Here $\left\lfloor x \right\rfloor$ is the greatest integer not exceeding $x$.
[i]Proposed by Evan Chen[/i]
Let $n$ be an integer greater than $1$. Define
\[x_1 = n, y_1 = 1, x_{i+1} =\left[ \frac{x_i+y_i}{2}\right] , y_{i+1} = \left[ \frac{n}{x_{i+1}}\right], \qquad \text{for }i = 1, 2, \ldots\ ,\]
where $[z]$ denotes the largest integer less than or equal to $z$. Prove that
\[ \min \{x_1, x_2, \ldots, x_n \} =[ \sqrt n ]\]
Let $\sigma(x)$ denote the sum of divisors of natural number $x$, including $1$ and $x$. For every $n\in \mathbb{N}$ define $f(n)$ as number of natural numbers $m, m\leq n$, for which $\sigma(m)$ is odd number. Prove that there are infinitely many natural numbers $n$, such that $f(n)|n$.
Let $S$ be the set of $n$-uples $\left( x_{1}, x_{2}, \ldots, x_{n}\right)$ such that $x_{i}\in \{ 0, 1 \}$ for all $i \in \overline{1,n}$, where $n \geq 3$. Let $M(n)$ be the smallest integer with the property that any subset of $S$ with at least $M(n)$ elements contains at least three $n$-uples \[\left( x_{1}, \ldots, x_{n}\right), \, \left( y_{1}, \ldots, y_{n}\right), \, \left( z_{1}, \ldots, z_{n}\right) \] such that
\[\sum_{i=1}^{n}\left( x_{i}-y_{i}\right)^{2}= \sum_{i=1}^{n}\left( y_{i}-z_{i}\right)^{2}= \sum_{i=1}^{n}\left( z_{i}-x_{i}\right)^{2}. \]
(a) Prove that $M(n) \leq \left\lfloor \frac{2^{n+1}}{n}\right\rfloor+1$.
(b) Compute $M(3)$ and $M(4)$.
For a real number $x$, let $\lfloor x\rfloor$ stand for the largest integer that is less than or equal to $x$. Prove that
\[ \left\lfloor{(n-1)!\over n(n+1)}\right\rfloor \]
is even for every positive integer $n$.
Indians find those sequences of non-negative real numbers $x_0, x_1,...$ [i]mystical [/i]t hat satisfy $x_0 < 2021$, $x_{i+1} = \lfloor x_i \rfloor \{x_i\}$ for every $i \ge 0$, furthermore the sequence contains an integer different from $0$. How many sequences are mystical according to the Indians?
Given a positive integer $n,$ what is the largest $k$ such that the numbers $1,2,\dots,n$ can be put into $k$ boxes so that the sum of the numbers in each box is the same?
[When $n=8,$ the example $\{1,2,3,6\},\{4,8\},\{5,7\}$ shows that the largest $k$ is [i]at least[/i] 3.]
A sequence $\{an\}$ of positive integers is defined by
\[a_n=\left[ n +\sqrt n + \frac 12 \right] , \qquad \forall n \in \mathbb N\]
Determine the positive integers that occur in the sequence.
Let $n$ be a positive integer. Prove that at least one of the integers $[2^n \cdot \sqrt2]$, $[2^{n+1} \cdot \sqrt2]$, $...$, $[2^{2n} \cdot \sqrt2]$ is even, where $[a]$ denotes the integer part of $a$.
Let $k \ge 1$ be an integer and $a_1, a_2, ... , a_k, b1, b_2, ..., b_k$ rational numbers with the property that for any irrational numbers $x_i >1$, $i = 1, 2, ..., k$, there exist the positive integers $n_1, n_2, ... , n_k, m_1, m_2, ..., m_k$ such that $$a_1\lfloor x^{n_1}_1\rfloor + a_2 \lfloor x^{n_2}_2\rfloor + ...+ a_k\lfloor x^{n_k}_k\rfloor=b_1\lfloor x^{m_1}_1\rfloor +2_1\lfloor x^{m_2}_2\rfloor+...+b_k\lfloor x^{m_k}_k\rfloor $$
Prove that $a_i = b_i$ for all $i = 1, 2, ... , k$.
Let $ n \ge 2009$ be an integer and define the set:
\[ S \equal{} \{2^x|7 \le x \le n, x \in \mathbb{N}\}.
\]
Let $ A$ be a subset of $ S$ and the sum of last three digits of each element of $ A$ is $ 8$. Let $ n(X)$ be the number of elements of $ X$. Prove that
\[ \frac {28}{2009} < \frac {n(A)}{n(S)} < \frac {82}{2009}.
\]
For every positive integer $n$, define the number of non-empty subsets $\mathcal N\subseteq \{1,\ldots ,n\}$ such that $\gcd(n\in\mathcal N)=1$. Show that $f(n)$ is a perfect square if and only if $n=1$.
For each integer $ n > 1,$ let $ F(n)$ be the number of solutions of the equation $ \sin x \equal{} \sin nx$ on the interval $ [0,\pi].$ What is $ \sum_{n \equal{} 2}^{2007}F(n)?$
$ \textbf{(A)}\ 2,014,524 \qquad \textbf{(B)}\ 2,015,028 \qquad \textbf{(C)}\ 2,015,033 \qquad \textbf{(D)}\ 2,016,532 \qquad \textbf{(E)}\ 2,017,033$
For real number $x$, let $\lfloor x\rfloor$ denote the greatest integer less than or equal to $x$, and let $\{x\}$ denote the fractional part of $x$, that is $\{x\} = x -\lfloor x\rfloor$. The sum of the solutions to the equation $2\lfloor x\rfloor^2 + 3\{x\}^2 = \frac74 x \lfloor x\rfloor$ can be written as $\frac{p}{q} $, where $p$ and $q$ are prime numbers. Find $10p + q$.
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?
Show that $\sum^{\infty}_{n=1}\frac{(-1)^{n-1}\sin(\log n)}{n^\alpha}$ converges iff $\alpha>0$.