Found problems: 1187
Let $\alpha\geq 1$ be a real number. Define the set
$$A(\alpha)=\{\lfloor \alpha\rfloor,\lfloor 2\alpha\rfloor, \lfloor 3\alpha\rfloor,\dots\}$$
Suppose that all the positive integers that [b]does not belong[/b] to the $A(\alpha)$ are exactly the positive integers that have the same remainder $r$ in the division by $2021$ with $0\leq r<2021$. Determine all the possible values of $\alpha$.
$P$ is a monic polynomial of odd degree greater than one such that there exists a function $f : \mathbb{R} \rightarrow \mathbb{N}$ such that for each $x \in \mathbb{R}$ ,\[f(P(x))=P(f(x))\]
(a) Prove that there are a finite number of natural numbers in range of $f$.
(b) Prove that if $f$ is not constant then the equation $P(x)-x=0$ has at least two real solutions.
(c) For each natural $n>1$ prove that there exists a function $f : \mathbb{R} \rightarrow \mathbb{N}$ and a monic polynomial of odd degree greater than one $P$ such that for each $x \in \mathbb{R}$ ,\[f(P(x))=P(f(x))\] and range of $f$ contains exactly $n$ different numbers.
Time allowed for this problem was 105 minutes.
Let $m, k$, and $c$ be positive integers with $k > c$, and let $\lambda$ be a positive, non-integer real root of the equation $\lambda^{m+1} - k \lambda^m - c = 0$. Let $f : Z^+ \to Z$ be defined by $f(n) = \lfloor \lambda n \rfloor$ for all $n \in Z^+$. Show that $f^{m+1}(n) \equiv cn - 1$ (mod $k$) for all $n \in Z^+$. (Here, $Z^+$ denotes the set of positive integers, $ \lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$, and $f^{m+1}(n) = f(f(... f(n)...))$ where $f$ appears $m + 1$ times.)
Let $p_1, p_2, p_3, \ldots$ be the prime numbers listed in increasing order, and let $x_0$ be a real number between 0 and 1. For positive integer $k$, define
\[ x_k = \begin{cases} 0 & \mbox{if} \; x_{k-1} = 0, \\[.1in] {\displaystyle \left\{ \frac{p_k}{x_{k-1}} \right\}} & \mbox{if} \; x_{k-1} \neq 0, \end{cases} \]
where $\{x\}$ denotes the fractional part of $x$. (The fractional part of $x$ is given by $x - \lfloor x \rfloor$ where $\lfloor x \rfloor$ is the greatest integer less than or equal to $x$.) Find, with proof, all $x_0$ satisfying $0 < x_0 < 1$ for which the sequence $x_0, x_1, x_2, \ldots$ eventually becomes 0.
Let $p$ be an odd prime number. Show that the smallest positive quadratic nonresidue of $p$ is smaller than $\sqrt{p}+1$.
A real number $a$ satisfies the equality $\frac{1}{a} = a - [a]$. Prove that $a$ is irrational.
For each positive integer $n$, define $V_n=\lfloor 2^n\sqrt{2020}\rfloor+\lfloor 2^n\sqrt{2021}\rfloor$. Prove that, in the sequence $V_1,V_2,\ldots,$ there are infinitely many odd integers, as well as infinitely many even integers.
[i]Remark.[/i] $\lfloor x\rfloor$ is the largest integer that does not exceed the real number $x$.
Find the smallest possible value of the expression \[\left\lfloor\frac{a+b+c}{d}\right\rfloor+\left\lfloor\frac{b+c+d}{a}\right\rfloor+\left\lfloor\frac{c+d+a}{b}\right\rfloor+\left\lfloor\frac{d+a+b}{c}\right\rfloor\]
in which $a,~ b,~ c$, and $d$ vary over the set of positive integers.
(Here $\lfloor x\rfloor$ denotes the biggest integer which is smaller than or equal to $x$.)
You are given three lists $A$, $B$, and $C$. List $A$ contains the numbers of the form $10^k$ in base $10$, with $k$ any integer greater than or equal to $1$. Lists $B$ and $C$ contain the same numbers translated into base $2$ and $5$ respectively:
$$\begin{array}{lll} A & B & C \\ 10 & 1010 & 20 \\ 100 & 1100100 & 400 \\ 1000 & 1111101000 & 13000 \\ \vdots & \vdots & \vdots \end{array}$$
Prove that for every integer $n > 1$, there is exactly one number in exactly one of the lists $B$ or $C$ that has exactly $n$ digits.
Denote by $\{x\}$ the fractional part of a real number $x$, that is $\{x\} = x - \rfloor x \lfloor $ where $\rfloor x \lfloor $ is the maximum integer not greater than$ x$ . Prove that
a) For every integer $n$, we have $\{n\sqrt{17}\}> \frac{1}{2\sqrt{17} n}$
b) The value $\frac{1}{2\sqrt{17} }$ is the largest constant $c$ such that the inequality $\{n\sqrt{17}\}> c n $ holds for all positive integers $n$
Is there a positive integer $n$ such that the fractional part of
\[
\left(3+\sqrt{5}\right)^n >0.99 ?
\]
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.
Solve in positive real numbers: $n+ \lfloor \sqrt{n} \rfloor+\lfloor \sqrt[3]{n} \rfloor=2014$
For a positive integer $n \ge 2 $, define set $ T = \{ (i,j) | 1 \le i < j \le n , i | j \} $. For nonnegative real numbers $ x_1 , x_2 , \cdots , x_n $ with $ x_1 + x_2 + \cdots + x_n = 1 $, find the maximum value of
\[ \sum_{(i,j) \in T} x_i x_j \]
in terms of $n$.
Let $m$ and $n$ be arbitrary non-negative integers. Prove that \[\frac{(2m)!(2n)!}{m! n!(m+n)!}\] is an integer.
How many natural number $n$ less than $2015$ that is divisible by $\lfloor\sqrt[3]{n}\rfloor$ ?
For any positive integer n, evaluate $$\sum_{i=0}^{\lfloor \frac{n+1}{2} \rfloor}
{n-i+1 \choose i}$$
, where $\lfloor n \rfloor$ is the greatest integer less than or equal to $n$ .
Prove that for any positive integer $n\geq 2$ we have that \[\sum_{k=2}^n \lfloor \sqrt[k]{n}\rfloor=\sum_{k=2}^n\lfloor\log_{k}n\rfloor.\]
Find the number of positive integers $n$ satisfying $\phi(n) | n$ such that
\[\sum_{m=1}^{\infty} \left( \left[ \frac nm \right] - \left[\frac{n-1}{m} \right] \right) = 1992\]
What is the largest number among them? As usual, $\phi(n)$ is the number of positive integers less than or equal to $n$ and relatively prime to $n.$
If $n>4$ is a composite number, then $2n$ divides $(n-1)!$.
A transformation of the first quadrant of the coordinate plane maps each point $(x,y)$ to the point $(\sqrt{x},\sqrt{y}).$ The vertices of quadrilateral $ABCD$ are $A=(900,300), B=(1800,600), C=(600,1800),$ and $D=(300,900).$ Let $k$ be the area of the region enclosed by the image of quadrilateral $ABCD.$ Find the greatest integer that does not exceed $k.$
Let $ \alpha$ be the positive root of the equation $ x^{2} \equal{} 1991x \plus{} 1$. For natural numbers $ m$ and $ n$ define
\[ m*n \equal{} mn \plus{} \lfloor\alpha m \rfloor \lfloor \alpha n\rfloor.
\]
Prove that for all natural numbers $ p$, $ q$, and $ r$,
\[ (p*q)*r \equal{} p*(q*r).
\]
Find the minimum number of colors necessary to color the integers from $1$ to $2007$ such that if distinct integers $a$, $b$, and $c$ are the same color, then $a \nmid b$ or $b \nmid c$.
Let $n \ge 2$ be an integer. Find all real numbers $a$ such that there exist real numbers $x_1,x_2,\dots,x_n$ satisfying
\[x_1(1-x_2)=x_2(1-x_3)=\dots=x_n(1-x_1)=a.\]
[i]Proposed by Walther Janous and Gerhard Kirchner, Innsbruck.[/i]
Let $S$ be the set of all squarefree numbers and $n$ be a natural number. Prove that
$$\sum_{k\in S}\left\lfloor\sqrt{\frac nk}\right\rfloor=n.$$