Found problems: 5802
Determine all functions $f$ defined on the set of all positive integers and taking non-negative integer values, satisfying the three conditions:
[list]
[*] $(i)$ $f(n) \neq 0$ for at least one $n$;
[*] $(ii)$ $f(x y)=f(x)+f(y)$ for every positive integers $x$ and $y$;
[*] $(iii)$ there are infinitely many positive integers $n$ such that $f(k)=f(n-k)$ for all $k<n$.
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Find all positive integers $n$ such that for any integer $k$ there exists an integer $a$ for which $a^3+a-k$ is divisible by $n$.
[i]Warut Suksompong, Thailand[/i]
Let $n \geq 2$ be an integer. Carl has $n$ books arranged on a bookshelf. Each book has a height and a width. No two books have the same height, and no two books have the same width. Initially, the books are arranged in increasing order of height from left to right. In a move, Carl picks any two adjacent books where the left book is wider and shorter than the right book, and swaps their locations. Carl does this repeatedly until no further moves are possible. Prove that regardless of how Carl makes his moves, he must stop after a finite number of moves, and when he does stop, the books are sorted in increasing order of width from left to right.
[i]Proposed by Milan Haiman[/i]
For a positive integer $n$ let $S(n)$ be the sum of digits in the decimal representation of $n$. Any positive integer obtained by removing several (at least one) digits from the right-hand end of the decimal representation of $n$ is called a [i]stump[/i] of $n$. Let $T(n)$ be the sum of all stumps of $n$. Prove that $n=S(n)+9T(n)$.
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 $f$ be a function defined on the set of positive rational numbers with the property that $f(a\cdot b)=f(a)+f(b)$ for all positive rational numbers $a$ and $b$. Suppose that $f$ also has the property that $f(p)=p$ for every prime number $p$. For which of the following numbers $x$ is $f(x)<0?$
$\textbf{(A) } \frac{17}{32} \qquad \textbf{(B) } \frac{11}{16} \qquad \textbf{(C) } \frac{7}{9} \qquad \textbf{(D) } \frac{7}{6} \qquad \textbf{(E) } \frac{25}{11}$
Let a sequence of real numbers $a_0, a_1,a_2, \cdots$ satisfies the condition:
$$\sum_{n=0}^ma_n\cdot(-1)^n\cdot{m\choose n}=0$$
for all sufficiently large values of $m$. Show that there exists a polynomial $P$ such that $a_n=P(n)$ for all $n\geq 0$
Let $m,n\geq 2$ be integers. Let $f(x_1,\dots, x_n)$ be a polynomial with real coefficients such that $$f(x_1,\dots, x_n)=\left\lfloor \frac{x_1+\dots + x_n}{m} \right\rfloor\text{ for every } x_1,\dots, x_n\in \{0,1,\dots, m-1\}.$$ Prove that the total degree of $f$ is at least $n$.
We are given an infinite row of cells extending infinitely in both directions. Some cells contain one or more stones. The total number of stones is finite. At each move, the player performs one of the following three operations:
[b]1. [/b]Take three stones from some cell, and add one stone to the cells located one cell to the left and one cell to the right, each skipping one cell in between.
[b]2. [/b]Take two stones from some cell, and add one stone to the cell one cell to the left, skipping one cell and one stone to the adjacent cell to the right.
[b]3.[/b] Take one stone from each of two adjacent cells, and add one stone to the cell to the right of these two cells.
The process ends when no moves are possible. Prove that the process always terminates and the final distribution of stones does not depend on the choices of moves made by the player.
[img]https://i.imgur.com/IjcIDOa.png[/img]
[i]Proposed by Luka Tsulaia, Georgia[/i]
The partition of $2n$ positive integers into $n$ pairs is called [i]square-free[/i] if the product of numbers in each pair is not a perfect square.Prove that if for $2n$ distinct positive integers, there exists one square-free partition, then there exists at least $n!$ square-free partitions.
Let $n$ be a positive integer, and $x$ be a positive real number. Prove that $$\sum_{k=1}^{n} \left( x \left[\frac{k}{x}\right] - (x+1)\left[\frac{k}{x+1}\right]\right) \leq n,$$ where $[x]$ denotes the largest integer not exceeding $x$.
Prove or disprove the following hypotheses.
a) For all $k \geq 2,$ each sequence of $k$ consecutive positive integers contains a number that is not divisible by any prime number less than $k.$
b) For all $k\geq 2,$ each sequence of $k$ consecutive positive integers contains a number that is relatively prime to all other members of the sequence.
There are several dominoes on a board such that each domino occupies two adjacent cells and none of the dominoes are adjacent by side or vertex. The bottom left and top right cells of the board are free. A token starts at the bottom left cell and can move to a cell adjacent by side: one step to the right or upwards at each turn. Is it always possible to move from the bottom left to the top right cell without passing through dominoes if the size of the board is a) $100 \times 101$ cells and b) $100 \times 100$ cells?
[i]Nikolay Chernyatiev[/i]
[b]a)[/b] Solve in $ \mathbb{R} $ the equation $ 2^x=x+1. $
[b]b)[/b] If a function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ has the property that
$$ (f\circ f)(x)=2^x-1,\quad\forall x\in\mathbb{R} , $$
then $ f(0)+f(1)=1. $
For which $n\ge 3$ does there exist positive integers $a_1<a_2<\cdots <a_n$, such that: $$a_n=a_1+...+a_{n-1}, \hspace{0.5cm} \frac{1}{a_1}=\frac{1}{a_2}+...+\frac{1}{a_n}$$ are both true?
[i]Proposed by Ivan Chan Kai Chin[/i]
The sequence $a_i$ is defined as $a_1 = 2, a_2 = 3$, and
$a_{n+1} = 2a_{n-1}$ or $a_{n+1} = 3a_n - 2a_{n-1}$ for all integers $n \ge 2$.
Prove that no term in $a_i$ is in the range $[1612, 2012]$.
Find all functions $f$ from the reals into the reals such that \[ f(ab) = f(a+b) \] for all irrational $a, b$.
Let $a_1, a_2, a_3, . . .$ be a sequence of positive real numbers that satisfies $a_1 = 1$ and $a^2_{n+1} + a_{n+1} = a_n$ for every natural number $n$. Prove that $a_n \ge \frac{1}{n}$ for every natural number $n$.
Determine all pairs $(a, b)$ of real numbers such that $a\lfloor bn\rfloor =b\lfloor an\rfloor$ for all positive integer $n$.
Let $n \in \mathbb N$ and $A_n$ set of all permutations $(a_1, \ldots, a_n)$ of the set $\{1, 2, \ldots , n\}$ for which
\[k|2(a_1 + \cdots+ a_k), \text{ for all } 1 \leq k \leq n.\]
Find the number of elements of the set $A_n$.
[i]Proposed by Vidan Govedarica, Serbia[/i]
Determine all positive integers $n$ for which there exists an integer $m$ such that $2^{n}-1$ divides $m^{2}+9$.
For all integers $x$ and $y$, let $a_{x, y}$ be a real number. Suppose that $a_{0, 0} = 0$. Suppose that only a finite number of the $a_{x, y}$ are nonzero. Prove that
\[
\sum_{x = -\infty}^\infty \sum_{y = -\infty}^{\infty} a_{x,y} ( a_{x, 2x + y} + a_{x + 2y, y} )
\le \sqrt{3} \sum_{x = -\infty}^\infty \sum_{y = -\infty}^{\infty} a_{x, y}^2 \, .
\]
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.
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Determine the largest integer $N$ for which there exists a table $T$ of integers with $N$ rows and $100$ columns that has the following properties:
$\text{(i)}$ Every row contains the numbers $1$, $2$, $\ldots$, $100$ in some order.
$\text{(ii)}$ For any two distinct rows $r$ and $s$, there is a column $c$ such that $|T(r,c) - T(s, c)|\geq 2$. (Here $T(r,c)$ is the entry in row $r$ and column $c$.)
Let $s_1, s_2, s_3, \dots$ be an infinite, nonconstant sequence of rational numbers, meaning it is not the case that $s_1 = s_2 = s_3 = \dots.$ Suppose that $t_1, t_2, t_3, \dots$ is also an infinite, nonconstant sequence of rational numbers with the property that $(s_i - s_j)(t_i - t_j)$ is an integer for all $i$ and $j$. Prove that there exists a rational number $r$ such that $(s_i - s_j)r$ and $(t_i - t_j)/r$ are integers for all $i$ and $j$.