Found problems: 5802
Determine if there are positive integers $a, b$ such that all terms of the sequence defined by
\[ x_{1}= 2010,x_{2}= 2011\\ x_{n+2}= x_{n}+ x_{n+1}+a\sqrt{x_{n}x_{n+1}+b}\quad (n\ge 1) \] are integers.
Let $k$ be an odd number that is greater than or equal to $3$. Prove that there exists a $k^{th}$-degree integer-valued polynomial with non-integer-coefficients that has the following properties:
(1) $f(0)=0$ and $f(1)=1$; and.
(2) There exist infinitely many positive integers $n$ so that if the following equation: \[ n= f(x_1)+\cdots+f(x_s), \] has integer solutions $x_1, x_2, \dots, x_s$, then $s \geq 2^k-1$.
Determine whether or not there exist 15 integers $m_1,\ldots,m_{15}$
such that~
$$\displaystyle \sum_{k=1}^{15}\,m_k\cdot\arctan(k) = \arctan(16). \eqno(1)$$
(Proposed by Gerhard Woeginger, Eindhoven University of Technology)
Write down some numbers $a_1,a_2,\ldots, a_n$ from left to right on a line. Step 1, we write $a_1+a_2$ between $a_1,a_2$; $a_2+a_3$ between $a_2,a_3$, …, $a_{n-1}+a_n$ between $a_{n-1},a_n$, and then we have new sequence $b=(a_1, a_1+a_2,a_2,a_2+a_3,a_3, \ldots, a_{n-1}, a_{n-1}+a_n, a_n)$. Step 2, we do the same thing with sequence b to have the new sequence c again…. And so on. If we do 2013 steps, count the number of the number 2013 appear on the line if
a) $n=2$, $a_1=1, a_2=1000$
b) $n=1000$, $a_i=i, i=1,2\ldots, 1000$
Sorry for my bad English
[color=#008000]Moderator says: alternate phrasing here: https://www.artofproblemsolving.com/Forum/viewtopic.php?f=42&t=516134[/color]
The numbers $1, 2, 3, \dots, 1024$ are written on a blackboard. They are divided into pairs. Then each pair is wiped off the board and non-negative difference of its numbers is written on the board instead. $512$ numbers obtained in this way are divided into pairs and so on. One number remains on the blackboard after ten such operations. Determine all its possible values.
[i]Proposed by A. Golovanov[/i]
Find, with proof, all positive integers $n$ with the following property: There are only finitely many positive multiples of $n$ which have exactly $n$ positive divisors
For a positive integer $K$, define a sequence, $\{a_n\}_n$, as following $a_1=K$, \[ a_{n+1} = \{ \begin{array} {cc} a_n-1 , & \mbox{ if } a_n \mbox{ is even} \\ \frac{a_n-1}2 , & \mbox{ if } a_n \mbox{ is odd} \end{array}, \] for all $n\geq 1$.
Find the smallest value of $K$, which makes $a_{2005}$ the first term equal to 0.
Define a sequence $(a_n)_{n \geq 1}$ by $a_1 =1$ and $a_2 =2$ and $a_{n+2} = 2 a_{n+1} - a_n + 2$ for $n \geq 1$. prove that for any $m$ , $a_m a_{m+1}$ is also a term in this sequence.
For a prime $ p$ and a given integer $ n$ let $ \nu_p(n)$ denote the exponent of $ p$ in the prime factorisation of $ n!$. Given $ d \in \mathbb{N}$ and $ \{p_1,p_2,\ldots,p_k\}$ a set of $ k$ primes, show that there are infinitely many positive integers $ n$ such that $ d\mid \nu_{p_i}(n)$ for all $ 1 \leq i \leq k$.
[i]Author: Tejaswi Navilarekkallu, India[/i]
Prove that $\sum \frac{1}{i_1i_2 \ldots i_k} = n$ is taken over all non-empty subsets $\left\{i_1,i_2, \ldots, i_k\right\}$ of $\left\{1,2,\ldots,n\right\}$. (The $k$ is not fixed, so we are summing over all the $2^n-1$ possible nonempty subsets.)
Given integer $n\geq 2$ and real numbers $x_1,x_2,\cdots, x_n$ in the interval $[0,1]$. Prove that there exist real numbers $a_0,a_1,\cdots,a_n$ satisfying the following conditions:
(1) $a_0+a_n=0$;
(2) $|a_i|\leq 1$, for $i=0,1,\cdots,n$;
(3) $|a_i-a_{i-1}|=x_i$, for $i=1,2,\cdots,n$.
Euler marks $n$ different points in the Euclidean plane. For each pair of marked points, Gauss writes down the number $\lfloor \log_2 d \rfloor$ where $d$ is the distance between the two points. Prove that Gauss writes down less than $2n$ distinct values.
[i]Note:[/i] For any $d>0$, $\lfloor \log_2 d\rfloor$ is the unique integer $k$ such that $2^k\le d<2^{k+1}$.
[i]Proposed by Pranjal Srivastava[/i]
Prove that:
a) the sequence $a_n=\frac{1}{n+1}+\frac{1}{n+2}+\ldots+\frac{1}{n+n},\ n\ge 1$ is monotonic.
b) there is a sequence $(a_n)_{n\ge 1}\in \{0,1\}$ such that:
\[\lim_{n\to \infty} \left(\frac{a_1}{n+1}+\frac{a_2}{n+2}+\ldots +\frac{a_n}{n+n}\right)=\frac{1}{2}\]
[i]Radu Gologan[/i]
Find the smallest positive integer $n$ or show no such $n$ exists, with the following property: there are infinitely many distinct $n$-tuples of positive rational numbers $(a_1, a_2, \ldots, a_n)$ such that both
$$a_1+a_2+\dots +a_n \quad \text{and} \quad \frac{1}{a_1} + \frac{1}{a_2} + \dots + \frac{1}{a_n}$$
are integers.
Consider two odd natural numbers $a$ and $b$ where $a$ is a divisor of $b^2+2$ and $b$ is a divisor of $a^2+2.$ Prove that $a$ and $b$ are the terms of the series of natural numbers $\langle v_n\rangle$ defined by
\[v_1 = v_2 = 1; v_n = 4v_ {n-1}-v_{n-2} \ \ \text{for} \ n\geq 3.\]
In a regular 100-gon, 41 vertices are colored black and the remaining 59 vertices are colored white. Prove that there exist 24 convex quadrilaterals $Q_{1}, \ldots, Q_{24}$ whose corners are vertices of the 100-gon, so that
[list]
[*] the quadrilaterals $Q_{1}, \ldots, Q_{24}$ are pairwise disjoint, and
[*] every quadrilateral $Q_{i}$ has three corners of one color and one corner of the other color.
[/list]
Given that $ a_1\equal{}1$, $ a_2\equal{}5$, $ \displaystyle a_{n\plus{}1} \equal{} \frac{a_n \cdot a_{n\minus{}1}}{\sqrt{a_n^2 \plus{} a_{n\minus{}1}^2 \plus{} 1}}$. Find a expression of the general term of $ \{ a_n \}$.
Let $ \mathbb{Z}$ be the set of all integers. Define the set $ \mathbb{H}$ as follows:
(1). $ \dfrac{1}{2} \in \mathbb{H}$,
(2). if $ x \in \mathbb{H}$, then $ \dfrac{1}{1\plus{}x} \in \mathbb{H}$ and also $ \dfrac{x}{1\plus{}x} \in \mathbb{H}$.
Prove that there exists a bijective function $ f: \mathbb{Z} \rightarrow \mathbb{H}$.
Let $f(u)$ be a continuous function and, for any real number $u$, let $[u]$ denote the greatest integer less than or equal to $u$. Show that for any $x>1$,
\[\int_{1}^{x} [u]([u]+1)f(u)du = 2\sum_{i=1}^{[x]} i \int_{i}^{x} f(u)du\]
A configuration of $4027$ points in the plane is called Colombian if it consists of $2013$ red points and $2014$ blue points, and no three of the points of the configuration are collinear. By drawing some lines, the plane is divided into several regions. An arrangement of lines is good for a Colombian configuration if the following two conditions are satisfied:
i) No line passes through any point of the configuration.
ii) No region contains points of both colors.
Find the least value of $k$ such that for any Colombian configuration of $4027$ points, there is a good arrangement of $k$ lines.
Proposed by [i]Ivan Guo[/i] from [i]Australia.[/i]
[b](a)[/b] Prove that every positive integer $n$ can be written uniquely in the form \[n=\sum_{j=1}^{2k+1}(-1)^{j-1}2^{m_j},\] where $k\geq 0$ and $0\le m_1<m_2\cdots <m_{2k+1}$ are integers.
This number $k$ is called [i]weight[/i] of $n$.
[b](b)[/b] Find (in closed form) the difference between the number of positive integers at most $2^{2017}$ with even weight and the number of positive integers at most $2^{2017}$ with odd weight.
A $2010\times 2010$ board is divided into corner-shaped figures of three cells. Prove that it is possible to mark one cell in each figure such that each row and each column will have the same number of marked cells.
[i]I. Bogdanov & O. Podlipsky[/i]
Determine all bounded functions $f: R \to R$, such that $f (f (x) + y) = f (x) + f (y)$, for all real $x, y$.
Determine all positive integers $n$ such that the following statement holds: If a convex polygon with with $2n$ sides $A_1 A_2 \ldots A_{2n}$ is inscribed in a circle and $n-1$ of its $n$ pairs of opposite sides are parallel, which means if the pairs of opposite sides
\[(A_1 A_2, A_{n+1} A_{n+2}), (A_2 A_3, A_{n+2} A_{n+3}), \ldots , (A_{n-1} A_n, A_{2n-1} A_{2n})\]
are parallel, then the sides \[ A_n A_{n+1}, A_{2n} A_1\] are parallel as well.
Some unit squares of $ 2007\times 2007 $ square board are colored. Let $ (i,j) $ be a unit square belonging to the $ith$ line and $jth$ column and $ S_{i,j} $ be the set of all colored unit squares $(x,y)$ satisfying $ x\leq i, y\leq j $. At the first step in each colored unit square $(i,j)$ we write the number of colored unit squares in $ S_{i,j} $ . In each step, in each colored unit square $(i,j)$ we write the sum of all numbers written in $ S_{i,j} $ in the previous step. Prove that after finite number of steps, all numbers in the colored unit squares will be odd.