Found problems: 5802
Is there a set $S$ of positive integers such that a number is in $S$ if and only if it is the sum of two distinct members of $S$ or a sum of two distinct positive integers not in $S$?
Given an undirected graph with $N$ vertices. For any set of $k$ vertices, where $1\le k\le N$, there are at most $2k-2$ edges, which join vertices of this set. Prove that the edges may be coloured in two colours so that each cycle contains edges of both colours. (Graph may contain multiple edges).
[i]I. Bogdanov, G. Chelnokov[/i]
Let $a > 1$ be a positive integer and $d > 1$ be a positive integer coprime to $a$. Let $x_1=1$, and for $k\geq 1$, define
$$x_{k+1} = \begin{cases}
x_k + d &\text{if } a \text{ does not divide } x_k \\
x_k/a & \text{if } a \text{ divides } x_k
\end{cases}$$
Find, in terms of $a$ and $d$, the greatest positive integer $n$ for which there exists an index $k$ such that $x_k$ is divisible by $a^n$.
The numbers $1$ to $1024$ are written one per square on a $32 \times 32$ board, so that the first row is $1, 2, ... , 32$, the second row is $33, 34, ... , 64$ and so on. Then the board is divided into four $16 \times 16$ boards and the position of these boards is moved round clockwise, so that
$AB$ goes to $DA$
$DC \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \, CB$
then each of the $16 \times 16 $ boards is divided into four equal $8 \times 8$ parts and each of these is moved around in the same way (within the $ 16 \times 16$ board). Then each of the $8 \times 8$ boards is divided into four $4 \times 4$ parts and these are moved around, then each $4 \times 4$ board is divided into $2 \times 2$ parts which are moved around, and finally the squares of each $2 \times 2$ part are moved around. What numbers end up on the main diagonal (from the top left to bottom right)?
Find all pairs of positive integers $\left(n;\;k\right)$ such that $n!=\left( n+1\right)^{k}-1$.
Let $n$ be a natural number and suppose that $ w_1, w_2, \ldots , w_n$ are $n$ weights . We call the set of $\{ w_1, w_2, \ldots , w_n\}$ to be a [i]Perfect Set [/i]if we can achieve all of the $1,2, \ldots, W$ weights with sums of $ w_1, w_2, \ldots , w_n$, where $W=\sum_{i=1}^n w_i $. Prove that if we delete the maximum weight of a Perfect Set, the other weights make again a Perfect Set.
Determine all functions $f:\mathbb{Z}\rightarrow\mathbb{Z}$ with the property that \[f(x-f(y))=f(f(x))-f(y)-1\] holds for all $x,y\in\mathbb{Z}$.
Let $n$ be a nonnegative integer. Determine the number of ways that one can choose $(n+1)^2$ sets $S_{i,j}\subseteq\{1,2,\ldots,2n\}$, for integers $i,j$ with $0\leq i,j\leq n$, such that:
[list]
[*] for all $0\leq i,j\leq n$, the set $S_{i,j}$ has $i+j$ elements; and
[*] $S_{i,j}\subseteq S_{k,l}$ whenever $0\leq i\leq k\leq n$ and $0\leq j\leq l\leq n$.
[/list]
[i]Proposed by Ricky Liu[/i]
Let $f(x)=x^3 +17$. Prove that for each natural number $n \ge 2$, there is a natural number $x$ for which $f(x)$ is divisible by $3^n$ but not $3^{n+1}$.
In any cell of an $n \times n$ table a number is written such that all the rows are distinct. Prove that we can remove a column such that the rows in the new table are still distinct.
Let $(a_i)_{i\in \mathbb{N}}$ and $(p_i)_{i\in \mathbb{N}}$ be two sequences of positive integers such that the following conditions hold:
$\bullet ~~a_1\ge 2$.
$\bullet~~ p_n$ is the smallest prime divisor of $a_n$ for every integer $n\ge 1$
$\bullet~~ a_{n+1}=a_n+\frac{a_n}{p_n}$ for every integer $n\ge 1$
Prove that there is a positive integer $N$ such that $a_{n+3}=3a_n$ for every integer $n>N$
For the NEMO, Kevin needs to compute the product
\[
9 \times 99 \times 999 \times \cdots \times 999999999.
\]
Kevin takes exactly $ab$ seconds to multiply an $a$-digit integer by a $b$-digit integer. Compute the minimum number of seconds necessary for Kevin to evaluate the expression together by performing eight such multiplications.
[i]Proposed by Evan Chen[/i]
Determine if there exist positive real numbers $x, \alpha$, so that for any non-empty finite set of positive integers $S$, the inequality
\[\left|x-\sum_{s\in S}\frac{1}{s}\right|>\frac{1}{\max(S)^\alpha}\]
holds, where $\max(S)$ is defined as the maximum element of the finite set $S$.
Call a sequence of positive integers $\{a_n\}$ good if for any distinct positive integers $m,n$, one has
$$\gcd(m,n) \mid a_m^2 + a_n^2 \text{ and } \gcd(a_m,a_n) \mid m^2 + n^2.$$
Call a positive integer $a$ to be $k$-good if there exists a good sequence such that $a_k = a$. Does there exists a $k$ such that there are exactly $2019$ $k$-good positive integers?
For any positive integer $n$ let $f(n)$ be the number of divisors of $n$ ending with $1$ or $9$ in base $10$ and let $g(n)$ be the number of divisors of $n$ ending with digit $3$ or $7$ in base $10$. Prove that $f(n)\geqslant g(n)$ for all nonnegative integers $n$.
[i](Swiss Mathematical Olympiad 2011, Final round, problem 9)[/i]
Let $\mathcal{L}$ be a finite collection of lines in the plane in general position (no two lines in $\mathcal{L}$ are parallel and no three are concurrent). Consider the open circular discs inscribed in the triangles enclosed by each triple of lines in $\mathcal{L}$. Determine the number of such discs intersected by no line in $\mathcal{L}$, in terms of $|\mathcal{L}|$.
[i]B. Aronov et al.[/i]
Find all the continuous functions $f : \mathbb{R} \mapsto\mathbb{R}$ such that $\forall x,y \in \mathbb{R}$,
$(1+f(x)f(y))f(x+y)=f(x)+f(y)$.
Let $\,S\,$ be a finite set of points in three-dimensional space. Let $\,S_{x},\,S_{y},\,S_{z}\,$ be the sets consisting of the orthogonal projections of the points of $\,S\,$ onto the $yz$-plane, $zx$-plane, $xy$-plane, respectively. Prove that \[ \vert S\vert^{2}\leq \vert S_{x} \vert \cdot \vert S_{y} \vert \cdot \vert S_{z} \vert, \] where $\vert A \vert$ denotes the number of elements in the finite set $A$.
[hide="Note"] Note: The orthogonal projection of a point onto a plane is the foot of the perpendicular from that point to the plane. [/hide]
There are $n$ coins lying in a circle. Each coin has two sides, $+$ and $-$. A $flop$ means to flip every coin that has two different neighbors simultaneously, while leaving the others alone. For instance, $++-+$, after one $flop$, becomes $+---$.
For $n$ coins, let us define $M$ to be a $perfect$ $number$ if for any initial arrangement of the coins, the arrangement of the coins after $m$ $flops$ is exactly the same as the initial one.
(a) When $n=1024$, find a perfect number $M$.
(b) Find all $n$ for which a perfect number $M$ exist.
The infinite sequence $a_0,a _1, a_2, \dots$ of (not necessarily distinct) integers has the following properties: $0\le a_i \le i$ for all integers $i\ge 0$, and \[\binom{k}{a_0} + \binom{k}{a_1} + \dots + \binom{k}{a_k} = 2^k\] for all integers $k\ge 0$. Prove that all integers $N\ge 0$ occur in the sequence (that is, for all $N\ge 0$, there exists $i\ge 0$ with $a_i=N$).
Suppose a function $f : \mathbb{Z}^+ \rightarrow \mathbb{Z}^+$ satisfies $f(f(n)) + f(n+1) = n+2$ for all positive integer $n$. Prove that $f(f(n)+n) = n+1$ for all positive integer $n$.
Find all pairs $(m,n)$ of nonnegative integers for which \[m^2 + 2 \cdot 3^n = m\left(2^{n+1} - 1\right).\]
[i]Proposed by Angelo Di Pasquale, Australia[/i]
Chim Tu has a large rectangular table. On it, there are finitely many pieces of paper with nonoverlapping interiors, each one in the shape of a convex polygon. At each step, Chim Tu is allowed to slide one piece of paper in a straight line such that its interior does not touch any other piece of paper during the slide. Can Chim Tu always slide all the pieces of paper off the table in finitely many steps?
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]
There are $4n$ pebbles of weights $1, 2, 3, \dots, 4n.$ Each pebble is coloured in one of $n$ colours and there are four pebbles of each colour. Show that we can arrange the pebbles into two piles so that the following two conditions are both satisfied:
[list]
[*]The total weights of both piles are the same.
[*] Each pile contains two pebbles of each colour.
[/list]
[i]Proposed by Milan Haiman, Hungary and Carl Schildkraut, USA[/i]