Found problems: 5802
The sequences $ (a_n),(b_n)$ are defined by $ a_1\equal{}1,b_1\equal{}2$ and \[a_{n \plus{} 1} \equal{} \frac {1 \plus{} a_n \plus{} a_nb_n}{b_n}, \quad b_{n \plus{} 1} \equal{} \frac {1 \plus{} b_n \plus{} a_nb_n}{a_n}.\]
Show that $ a_{2008} < 5$.
Given positive integer $m,n$, color the points of the regular $(2m+2n)$-gon in black and white, $2m$ in black and $2n$ in white.
The [i]coloring distance[/i] $d(B,C) $ of two black points $B,C$ is defined as the smaller number of white points in the two paths linking the two black points.
The [i]coloring distance[/i] $d(W,X) $ of two white points $W,X$ is defined as the smaller number of black points in the two paths linking the two white points.
We define the matching of black points $\mathcal{B}$ : label the $2m$ black points with $A_1,\cdots,A_m,B_1,\cdots,B_m$ satisfying no $A_iB_i$ intersects inside the gon.
We define the matching of white points $\mathcal{W}$ : label the $2n$ white points with $C_1,\cdots,C_n,D_1,\cdots,D_n$ satisfying no $C_iD_i$ intersects inside the gon.
We define $P(\mathcal{B})=\sum^m_{i=1}d(A_i,B_i), P(\mathcal{W} )=\sum^n_{j=1}d(C_j,D_j) $.
Prove that: $\max_{\mathcal{B}}P(\mathcal{B})=\max_{\mathcal{W}}P(\mathcal{W})$
Let $(a_n)_{n\geq 1}$ be a sequence of positive real numbers with the property that
$$(a_{n+1})^2 + a_na_{n+2} \leq a_n + a_{n+2}$$
for all positive integers $n$. Show that $a_{2022}\leq 1$.
Let $c>0$ be a given positive real and $\mathbb{R}_{>0}$ be the set of all positive reals. Find all functions $f \colon \mathbb{R}_{>0} \to \mathbb{R}_{>0}$ such that \[f((c+1)x+f(y))=f(x+2y)+2cx \quad \textrm{for all } x,y \in \mathbb{R}_{>0}.\]
We choose random a unitary polynomial of degree $n$ and coefficients in the set $1,2,...,n!$. Prove that the probability for this polynomial to be special is between $0.71$ and $0.75$, where a polynomial $g$ is called special if for every $k>1$ in the sequence $f(1), f(2), f(3),...$ there are infinitely many numbers relatively prime with $k$.
Find all the positive integers $x,y,z$ satisfiing : $x^{2}+y^{2}+z^{2}=2xyz$
Let $ABC$ be an equilateral triangle. From the vertex $A$ we draw a ray towards the interior of the triangle such that the ray reaches one of the sides of the triangle. When the ray reaches a side, it then bounces off following the law of reflection, that is, if it arrives with a directed angle $\alpha$, it leaves with a directed angle $180^{\circ}-\alpha$. After $n$ bounces, the ray returns to $A$ without ever landing on any of the other two vertices. Find all possible values of $n$.
Let $n\ge k$ be positive integers, and let $\mathcal{F}$ be a family of finite sets with the following properties:
(i) $\mathcal{F}$ contains at least $\binom{n}{k}+1$ distinct sets containing exactly $k$ elements;
(ii) for any two sets $A, B\in \mathcal{F}$, their union $A\cup B$ also belongs to $\mathcal{F}$.
Prove that $\mathcal{F}$ contains at least three sets with at least $n$ elements.
(Proposed by Fedor Petrov, St. Petersburg State University)
A bounded sequence $ (x_n)_{n\ge 1}$ of real numbers satisfies $ x_n \plus{} x_{n \plus{} 1} \ge 2x_{n \plus{} 2}$ for all $ n \ge 1$. Prove that this sequence has a finite limit.
Given a positive integer $n>1$. Denote $T$ a set that contains all ordered sets $(x;y;z)$ such that $x,y,z$ are all distinct positive integers and $1\leq x,y,z\leq 2n$. Also, a set $A$ containing ordered sets $(u;v)$ is called [i]"connected"[/i] with $T$ if for every $(x;y;z)\in T$ then $\{(x;y),(x;z),(y;z)\} \cap A \neq \varnothing$.
a) Find the number of elements of set $T$.
b) Prove that there exists a set "connected" with $T$ that has exactly $2n(n-1)$ elements.
c) Prove that every set "connected" with $T$ has at least $2n(n-1)$ elements.
We are given a 5x5 square grid, divided to 1x1 tiles. Two tiles are called [b]linked[/b] if they lie in the same row or column, and the distance between their centers is 2 or 3. For example, in the picture the gray tiles are the ones linked to the red tile.
[img]https://i.imgur.com/JVTQ9wB.png[/img]
Sammy wants to mark as many tiles in the grid as possible, such that no two of them are linked. What is the maximal number of tiles he can mark?
Let $S$ be a set of $a+b+3$ points on a sphere, where $a$, $b$ are nonnegative integers and no four points of $S$ are coplanar. Determine how many planes pass through three points of $S$ and separate the remaining points into $a$ points on one side of the plane and $b$ points on the other side.
Let $ n$ be positive integer, $ A,B\subseteq[0,n]$ are sets of integers satisfying $ \mid A\mid \plus{} \mid B\mid\ge n \plus{} 2.$ Prove that there exist $ a\in A, b\in B$ such that $ a \plus{} b$ is a power of $ 2.$
Let $A$ be an $n\times n$ matrix of real numbers for some $n\ge 1.$ For each positive integer $k,$ let $A^{[k]}$ be the matrix obtained by raising each entry to the $k$th power. Show that if $A^k=A^{[k]}$ for $k=1,2,\cdots,n+1,$ then $A^k=A^{[k]}$ for all $k\ge 1.$
Let $f : [0, 1] \to [0, 1]$ satisfy $f(0) = 0, f(1) = 1$ and
\[f(x + y) - f(x) = f(x) - f(x - y)\]
for all $x, y \geq 0$ with $x - y, x + y \in [0, 1].$ Prove that $f(x) = x$ for all $x \in [0, 1].$
Let $m$ and $n$ be positive integers with $m\geq n$. There are $m$ cupcakes of different flavors arranged around a circle and $n$ people who like cupcakes. Each person assigns a nonnegative real number score to each cupcake, depending on how much they like the cupcake. Suppose that for each person $P$, it is possible to partition the circle of $m$ cupcakes into $n$ groups of consecutive cupcakes so that the sum of $P$'s scores of the cupcakes in each group is at least $1$. Prove that it is possible to distribute the $m$ cupcakes to the $n$ people so that each person $P$ receives cupcakes of total score at least $1$ with respect to $P$.
Does there exist a positive integer $ n$ such that $ n$ has exactly 2000 prime divisors and $ n$ divides $ 2^n \plus{} 1$?
All contestants at one contest are sitting in $n$ columns and are forming a "good" configuration. (We define one configuration as "good" when we don't have 2 friends sitting in the same column). It's impossible for all the students to sit in $n-1$ columns in a "good" configuration. Prove that we can always choose contestants $M_1,M_2,...,M_n$ such that $M_i$ is sitting in the $i-th$ column, for each $i=1,2,...,n$ and $M_i$ is friend of $M_{i+1}$ for each $i=1,2,...,n-1$.
Let $n$ be an integer greater than $1$ and let $X$ be an $n$-element set. A non-empty collection of subsets $A_1, ..., A_k$ of $X$ is tight if the union $A_1 \cup \cdots \cup A_k$ is a proper subset of $X$ and no element of $X$ lies in exactly one of the $A_i$s. Find the largest cardinality of a collection of proper non-empty subsets of $X$, no non-empty subcollection of which is tight.
[i]Note[/i]. A subset $A$ of $X$ is proper if $A\neq X$. The sets in a collection are assumed to be distinct. The whole collection is assumed to be a subcollection.
Let $x_1, x_2, \ldots, x_n$ be positive reals; for any positive integer $k$, let $S_k=x_1^k+x_2^k+\ldots+x_n^k$.
(a) Given that $S_1<S_2$, show that $S_1, S_2, S_3, \ldots$ is strictly increasing.
(b) Prove that there exists a positive integer $n$ and positive reals $x_1, x_2, \ldots, x_n$, such that $S_1>S_2$ and $S_1, S_2, S_3, \ldots$ is not strictly decreasing.
Vishal starts with $n$ copies of the number $1$ written on the board. Every minute, he takes two numbers $a, b$ and replaces them with either $a+b$ or $\min(a^2, b^2)$. After $n-1$ there is $1$ number on the board. Let the maximal possible value of this number be $f(n)$. Prove $2^{n/3}<f(n)\leq 3^{n/3}$.
The [i]Collatz's function[i] is a mapping $f:\mathbb{Z}_+\to\mathbb{Z}_+$ satisfying \[
f(x)=\begin{cases}
3x+1,& \mbox{as }x\mbox{ is odd}\\
x/2, & \mbox{as }x\mbox{ is even.}\\
\end{cases}
\] In addition, let us define the notation $f^1=f$ and inductively $f^{k+1}=f\circ f^k,$ or to say in another words, $f^k(x)=\underbrace{f(\ldots (f}_{k\text{ times}}(x)\ldots ).$
Prove that there is an $x\in\mathbb{Z}_+$ satisfying \[f^{40}(x)> 2012x.\]
Let $\mathbb N$ be the set of all positive integers. A subset $A$ of $\mathbb N$ is [i]sum-free[/i] if, whenever $x$ and $y$ are (not necessarily distinct) members of $A$, their sum $x+y$ does not belong to $A$. Determine all surjective functions $f:\mathbb N\to\mathbb N$ such that, for each sum-free subset $A$ of $\mathbb N$, the image $\{f(a):a\in A\}$ is also sum-free.
[i]Note: a function $f:\mathbb N\to\mathbb N$ is surjective if, for every positive integer $n$, there exists a positive integer $m$ such that $f(m)=n$.[/i]
Initially, one of the two boxes on the table is empty and the other contains $29$ different colored marbles. By starting with the full box and performing moves in order, in each move, one or more marbles are selected from that box and transferred to the other box. At most, how many moves can be made without selecting the same set of marbles more than once?
(1) Let $a,b,c$ be positive real numbers satisfying $(a^2+b^2+c^2)^2>2(a^4+b^4+c^4)$. Prove that $a,b,c$ can be the lengths of three sides of a triangle respectively.
(2) Let $a_1,a_2,\dots ,a_n$ be $n$ ($n>3$) positive real numbers satisfying $(a_1^2+a_2^2+\dots +a_n^2)^2>(n-1)(a_1^4+ a_2^4+\dots +a_n^4)$. Prove that any three of $a_1,a_2,\dots ,a_n$ can be the lengths of three sides of a triangle respectively.