Found problems: 5802
Let $a_0$, $a_1$, $a_2$, ... be an infinite sequence of real numbers satisfying the equation $a_n=\left|a_{n+1}-a_{n+2}\right|$ for all $n\geq 0$, where $a_0$ and $a_1$ are two different positive reals.
Can this sequence $a_0$, $a_1$, $a_2$, ... be bounded?
[i]Proposed by Mihai Bălună, Romania[/i]
If $a_1, a_2, \ldots$ is a sequence of real numbers such that for all $n$,
$$\sum_{k = 1}^n a_k \left( \frac{k}{n} \right)^2 = 1,$$
find the smallest $n$ such that $a_n < \frac{1}{2018}$.
Let $a(n)$ be the sequence defined by $a(1)=2$ and $a(n+1)=(a(n))^{n+1}-1$ for each integer $n\geq 1$. Suppose that $p>2$ is a prime and $k$ is a positive integer. Prove that some term of the sequence $a(n)$ is divisible by $p^k$.
[i]Proposed by John Berman[/i]
Given an odd $n\in\mathbb N$. In an $n\times n$ chessboard, you may place many $2\times2$ squares. How many grids, at most, are covered by exactly one square?
A sequence $\{a_n\}$ is defined recursively by $a_1=\frac{1}{2}, $ and for $n\ge 2,$ $0<a_n\leq a_{n-1}$ and
\[a_n^2(a_{n-1}+1)+a_{n-1}^2(a_n+1)-2a_na_{n-1}(a_na_{n-1}+a_n+1)=0.\]
$(1)$ Determine the general formula of the sequence $\{a_n\};$
$(2)$ Let $S_n=a_1+\cdots+a_n.$ Prove that for $n\ge 1,$ $\ln\left(\frac{n}{2}+1\right)<S_n<\ln(n+1).$
Some paper squares of $k$ distinct colors are placed on a rectangular table, with sides parallel to the sides of the table. Suppose that for any $k$ squares of distinct colors, some two of them can be nailed on the table with only one nail. Prove that there is a color such that all squares of that color can be nailed with $2k-2$ nails.
Consider a $100\times 100$ square unit lattice $\textbf{L}$ (hence $\textbf{L}$ has $10000$ points). Suppose $\mathcal{F}$ is a set of polygons such that all vertices of polygons in $\mathcal{F}$ lie in $\textbf{L}$ and every point in $\textbf{L}$ is the vertex of exactly one polygon in $\mathcal{F}.$ Find the maximum possible sum of the areas of the polygons in $\mathcal{F}.$
[i]Michael Ren and Ankan Bhattacharya, USA[/i]
Consider $2009$ cards, each having one gold side and one black side, lying on parallel on a long table. Initially all cards show their gold sides. Two player, standing by the same long side of the table, play a game with alternating moves. Each move consists of choosing a block of $50$ consecutive cards, the leftmost of which is showing gold, and turning them all over, so those which showed gold now show black and vice versa. The last player who can make a legal move wins.
(a) Does the game necessarily end?
(b) Does there exist a winning strategy for the starting player?
[i]Proposed by Michael Albert, Richard Guy, New Zealand[/i]
Does there exist a pair $(g,h)$ of functions $g,h:\mathbb{R}\rightarrow\mathbb{R}$ such that the only function $f:\mathbb{R}\rightarrow\mathbb{R}$ satisfying $f(g(x))=g(f(x))$ and $f(h(x))=h(f(x))$ for all $x\in\mathbb{R}$ is identity function $f(x)\equiv x$?
Michael and Steven are playing the card game War with a deck of $4$ cards numbered $1$ through $4.$ The deck is shuffled randomly and Michael gets a stack of $1$ card and Steven gets a stack of $3$ cards. In each round, the players reveal the top card from their stack, and the player whose card was higher collects both cards to the bottom of their stack in random order. A player wins when they get all four cards in their hand. The probability that Michael wins after exactly $5$ rounds is $\tfrac{m}{n}$ for coprime positive integers $m$ and $n.$ Find $m + n.$
For a nonempty set $\, S \,$ of integers, let $\, \sigma(S) \,$ be the sum of the elements of $\, S$. Suppose that $\, A = \{a_1, a_2, \ldots, a_{11} \} \,$ is a set of positive integers with $\, a_1 < a_2 < \cdots < a_{11} \,$ and that, for each positive integer $\, n\leq 1500, \,$ there is a subset $\, S \,$ of $\, A \,$ for which $\, \sigma(S) = n$. What is the smallest possible value of $\, a_{10}$?
Numbers $1,2,\ldots,50$ are written on the board. Anya does the following operation: removes the numbers $a$ and $b$ from the board and writes their sum - $a+b$, after which also notes down the number $ab(a+b)$. After $49$ of this operations only one number was left on the board. Anya summed up all the $49$ numbers in her notes and got $S$.
a) Prove that $S$ does not depend on the order of Anya's actions.
b) Calculate $S$.
Let $\leftarrow$ denote the left arrow key on a standard keyboard. If one opens a text editor and types the keys "ab$\leftarrow$ cd $\leftarrow \leftarrow$ e $\leftarrow \leftarrow$ f", the result is "faecdb". We say that a string $B$ is [i]reachable[/i] from a string $A$ if it is possible to insert some amount of $\leftarrow$'s in $A$, such that typing the resulting characters produces $B$. So, our example shows that "faecdb" is reachable from "abcdef".
Prove that for any two strings $A$ and $B$, $A$ is reachable from $B$ if and only if $B$ is reachable from $A$.
Find all positive integers $n$ for which all positive divisors of $n$ can be put into the cells of a rectangular table under the following constraints:
[list]
[*]each cell contains a distinct divisor;
[*]the sums of all rows are equal; and
[*]the sums of all columns are equal.
[/list]
Find all functions $ f: \mathbb{N} \to \mathbb{N} $ such that $$ f\left(x+yf\left(x\right)\right) = x+f\left(y\right)f\left(x\right) $$ holds for all $ x,y \in \mathbb{N} $
Find all (weakly) increasing $f\colon \mathbb{R}\to \mathbb{R}$ for which
\[f(f(x)+y)=f(f(y)+x)\]
holds for all $x, y\in \mathbb{R}$.
There are $101$ positive integers $a_1, a_2, \ldots, a_{101}$ such that for every index $i$, with $1 \leqslant i \leqslant 101$, $a_i+1$ is a multiple of $a_{i+1}$. Determine the greatest possible value of the largest of the $101$ numbers.
A rectangle $\mathcal{R}$ with odd integer side lengths is divided into small rectangles with integer side lengths. Prove that there is at least one among the small rectangles whose distances from the four sides of $\mathcal{R}$ are either all odd or all even.
[i]Proposed by Jeck Lim, Singapore[/i]
Celeste has an unlimited amount of each type of $n$ types of candy, numerated type 1, type 2, ... type n. Initially she takes $m>0$ candy pieces and places them in a row on a table. Then, she chooses one of the following operations (if available) and executes it:
$1.$ She eats a candy of type $k$, and in its position in the row she places one candy type $k-1$ followed by one candy type $k+1$ (we consider type $n+1$ to be type 1, and type 0 to be type $n$).
$2.$ She chooses two consecutive candies which are the same type, and eats them.
Find all positive integers $n$ for which Celeste can leave the table empty for any value of $m$ and any configuration of candies on the table.
$\textit{Proposed by Federico Bach and Santiago Rodriguez, Colombia}$
Let $N$ be the number of polynomials $P(x_1, x_2, \dots, x_{2016})$ of degree at most $2015$ with coefficients in the set $\{0, 1, 2 \}$ such that $P(a_1,a_2,\cdots ,a_{2016}) \equiv 1 \pmod{3}$ for all $(a_1,a_2,\cdots ,a_{2016}) \in \{0, 1\}^{2016}.$
Compute the remainder when $v_3(N)$ is divided by $2011$, where $v_3(N)$ denotes the largest integer $k$ such that $3^k | N.$
[i]Proposed by Yang Liu[/i]
In a country, mathematicians chose an $\alpha> 2$ and issued coins in denominations of 1 ruble, as well as $\alpha ^k$ rubles for each positive integer k. $\alpha$ was chosen so that the value of each coins, except the smallest, was irrational. Is it possible that any natural number of rubles can be formed with at most 6 of each denomination of coins?
Prove that any uncountable subset of the Euclidean $ n$-space contains an countable subset with the property that the distances between different pairs of points are different (that is, for any points $ P_1 \not\equal{} P_2$ and $ Q_1\not\equal{} Q_2$ of this subset, $ \overline{P_1P_2}\equal{}\overline{Q_1Q_2}$ implies either $ P_1\equal{}Q_1$ and $ P_2\equal{}Q_2$, or $ P_1\equal{}Q_2$ and $ P_2\equal{}Q_1$). Show that a similar statement is not valid if the Euclidean $ n$-space is replaced with a (separable) Hilbert space.
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 $f(n)=2010$.
Let $a_1, a_2, \ldots, a_n$ be $n$ positive integers that are not divisible by each other, i.e. for any $i \neq j$, $a_i$ is not divisible by $a_j$. Show that
\[ a_1+a_2+\cdots+a_n \ge 1.1n^2-2n. \]
[i]Note:[/i] A proof of the inequality when $n$ is sufficient large will be awarded points depending on your results.
Let $n$ and $k$ be positive integers. Prove that for $a_1, \dots, a_n \in [1,2^k]$ one has
\[ \sum_{i = 1}^n \frac{a_i}{\sqrt{a_1^2 + \dots + a_i^2}} \le 4 \sqrt{kn}. \]