Found problems: 5802
Consider the infinite grid of lattice points in $\mathbb{Z}^3$. Little D and Big Z play a game, where Little D first loses a shoe on an unmunched point in the grid. Then, Big Z munches a shoe-free plane perpendicular to one of the coordinate axes. They continue to alternate turns in this fashion, with Little D's goal to lose a shoe on each of $n$ consecutive lattice points on a line parallel to one of the coordinate axes. Determine all $n$ for which Little D can accomplish his goal.
[i]David Yang.[/i]
Find the largest prime that divides $1\cdot 2\cdot 3+2\cdot 3\cdot 4+\cdots +44\cdot 45\cdot 46$
Find all functions $f:\mathbb{Z}\to \mathbb{Z}$ satisfying the condition $f(n)-f(n+f(m))=m$ for all $m,n\in \mathbb{Z}$
Determine all integers $ n\geq 2$ having the following property: for any integers $a_1,a_2,\ldots, a_n$ whose sum is not divisible by $n$, there exists an index $1 \leq i \leq n$ such that none of the numbers $$a_i,a_i+a_{i+1},\ldots,a_i+a_{i+1}+\ldots+a_{i+n-1}$$ is divisible by $n$. Here, we let $a_i=a_{i-n}$ when $i >n$.
[i]Proposed by Warut Suksompong, Thailand[/i]
Let $a_0,b_0$ be positive integers, and define $a_{i+1}=a_i+\lfloor\sqrt{b_i}\rfloor$ and $b_{i+1}=b_i+\lfloor\sqrt{a_i}\rfloor$ for all $i\ge0$. Show that there exists a positive integer $n$ such that $a_n=b_n$.
[i]David Yang.[/i]
For positive integer $k$, let $p(k)$ denote the greatest odd divisor of $k$. Prove that for every positive integer $n$,
$$\frac{2n}{3} < \frac{p(1)}{1}+ \frac{p(2)}{2}+... +\frac{ p(n)}{n}<\frac{2(n + 1)}{3}$$
Consider the set $E = \{1,2,\ldots,2n\}$. Prove that an element $c \in E$ can belong to a subset $A \subset E$, having $n$ elements, and such that any two distinct elements in $A$ do not divide one each other, if and only if \[c > n \left( \frac{2}{3}\right )^{k+1},\] where $k$ is the exponent of $2$ in the factoring of $c$.
Suppose that a sequence $a_1,a_2,\ldots$ of positive real numbers satisfies \[a_{k+1}\geq\frac{ka_k}{a_k^2+(k-1)}\] for every positive integer $k$. Prove that $a_1+a_2+\ldots+a_n\geq n$ for every $n\geq2$.
Sir Alex plays the following game on a row of 9 cells. Initially, all cells are empty. In each move, Sir Alex is allowed to perform exactly one of the following two operations:
[list=1]
[*] Choose any number of the form $2^j$, where $j$ is a non-negative integer, and put it into an empty cell.
[*] Choose two (not necessarily adjacent) cells with the same number in them; denote that number by $2^j$. Replace the number in one of the cells with $2^{j+1}$ and erase the number in the other cell.
[/list]
At the end of the game, one cell contains $2^n$, where $n$ is a given positive integer, while the other cells are empty. Determine the maximum number of moves that Sir Alex could have made, in terms of $n$.
[i]Proposed by Warut Suksompong, Thailand[/i]
Let $a,b,c\ge0$. Show that $(a^2+2bc)^{2012}+(b^2+2ca)^{2012}+(c^2+2ab)^{2012}\le (a^2+b^2+c^2)^{2012}+2(ab+bc+ca)^{2012}$.
[i]Calvin Deng.[/i]
Let $n\ge 2$ be a given integer. Show that the number of strings of length $n$ consisting of $0'$s and $1'$s such that there are equal number of $00$ and $11$ blocks in each string is equal to
\[2\binom{n-2}{\left \lfloor \frac{n-2}{2}\right \rfloor}\]
Suppose that the function $f:\mathbb{R}\to\mathbb{R}$ satisfies
\[f(x^3 + y^3)=(x+y)(f(x)^2-f(x)f(y)+f(y)^2)\]
for all $x,y\in\mathbb{R}$.
Prove that $f(1996x)=1996f(x)$ for all $x\in\mathbb{R}$.
Let $\mathbb{Z}$ denote the set of all integers. Find all polynomials $P(x)$ with integer coefficients that satisfy the following property:
For any infinite sequence $a_1$, $a_2$, $\dotsc$ of integers in which each integer in $\mathbb{Z}$ appears exactly once, there exist indices $i < j$ and an integer $k$ such that $a_i +a_{i+1} +\dotsb +a_j = P(k)$.
The integer number $n > 1$ is given and a set $S \subset \{0, 1, 2, \ldots, n-1\}$ with $|S| > \frac{3}{4} n$. Prove that there exist integer numbers $a, b, c$ such that the remainders after the division by $n$ of the numbers:
\[a, b, c, a+b, b+c, c+a, a+b+c\]
belong to $S$.
In how many ways can the set of integers $\{1,2,\ldots ,1995\}$ be partitioned into three non-empty sets so that none of these sets contains any pair of consecutive integers?
Let $n$ be a positive integer. There are $2018n+1$ cities in the Kingdom of Sellke Arabia. King Mark wants to build two-way roads that connect certain pairs of cities such that for each city $C$ and integer $1\le i\le 2018,$ there are exactly $n$ cities that are a distance $i$ away from $C.$ (The [i]distance[/i] between two cities is the least number of roads on any path between the two cities.)
For which $n$ is it possible for Mark to achieve this?
[i]Proposed by Michael Ren[/i]
Let $n$ be a natural number. Prove that \[ \left\lfloor \frac{n+2^0}{2^1} \right\rfloor + \left\lfloor \frac{n+2^1}{2^2} \right\rfloor +\cdots +\left\lfloor \frac{n+2^{n-1}}{2^n}\right\rfloor =n. \]
[hide="Remark"]For any real number $x$, the number $\lfloor x \rfloor$ represents the largest integer smaller or equal with $x$.[/hide]
In the plane, there are $n \geqslant 6$ pairwise disjoint disks $D_{1}, D_{2}, \ldots, D_{n}$ with radii $R_{1} \geqslant R_{2} \geqslant \ldots \geqslant R_{n}$. For every $i=1,2, \ldots, n$, a point $P_{i}$ is chosen in disk $D_{i}$. Let $O$ be an arbitrary point in the plane. Prove that \[O P_{1}+O P_{2}+\ldots+O P_{n} \geqslant R_{6}+R_{7}+\ldots+R_{n}.\]
(A disk is assumed to contain its boundary.)
Find all functions $f : \mathbb{Z}\rightarrow \mathbb{Z}$ satisfying
\[f^{a^{2} + b^{2}}(a+b) = af(a) +bf(b)\]
for all integers $a$ and $b$
Consider the sequence $(x_n)_{n\ge 1}$ where $x_1=1,x_2=2011$ and $x_{n+2}=4022x_{n+1}-x_n$ for all $n\in\mathbb{N}$. Prove that $\frac{x_{2012}+1}{2012}$ is a perfect square.
suppose that $A$ is the set of all Closed intervals $[a,b] \subset \mathbb{R}$. Find all functions $f:\mathbb{R} \rightarrow A$ such that
$\bullet$ $x \in f(y) \Leftrightarrow y \in f(x)$
$\bullet$ $|x-y|>2 \Leftrightarrow f(x) \cap f(y)=\varnothing$
$\bullet$ For all real numbers $0\leq r\leq 1$, $f(r)=[r^2-1,r^2+1]$
Proposed by Matin Yousefi
Edges of a complete graph with $2m$ vertices are properly colored with $2m-1$ colors. It turned out that for any two colors all the edges colored in one of these two colors can be described as union of several $4$-cycles. Prove that $m$ is a power of $2$.
$(GBR 1)$ The polynomial $P(x) = a_0x^k + a_1x^{k-1} + \cdots + a_k$, where $a_0,\cdots, a_k$ are integers, is said to be divisible by an integer $m$ if $P(x)$ is a multiple of $m$ for every integral value of $x$. Show that if $P(x)$ is divisible by $m$, then $a_0 \cdot k!$ is a multiple of $m$. Also prove that if $a, k,m$ are positive integers such that $ak!$ is a multiple of $m$, then a polynomial $P(x)$ with leading term $ax^k$can be found that is divisible by $m.$
Let $n \geq 5$ be an integer. Consider $n$ squares with side lengths $1, 2, \dots , n$, respectively. The squares are arranged in the plane with their sides parallel to the $x$ and $y$ axes. Suppose that no two squares touch, except possibly at their vertices. Show that it is possible to arrange these squares in a way such that every square touches exactly two other squares.
Let $n{}$ be a positive integer, and let $\mathcal{C}$ be a collection of subsets of $\{1,2,\ldots,2^n\}$ satisfying both of the following conditions:[list=1]
[*]Every $(2^n-1)$-element subset of $\{1,2,\ldots,2^n\}$ is a member of $\mathcal{C}$, and
[*]Every non-empty member $C$ of $\mathcal{C}$ contains an element $c$ such that $C\setminus\{c\}$ is again a member of $\mathcal{C}$.
[/list]Determine the smallest size $\mathcal{C}$ may have.
[i]Serbia, Pavle Martinovic ́[/i]