Found problems: 5802
Find all functions $f: (0, \infty) \to (0, \infty)$ such that
\begin{align*}
f(y(f(x))^3 + x) = x^3f(y) + f(x)
\end{align*}
for all $x, y>0$.
[i]Proposed by Jason Prodromidis, Greece[/i]
Determine all $f:\mathbb{Z}^+ \rightarrow \mathbb{Z}^+$ such that $f(m)\geq m$ and $f(m+n) \mid f(m)+f(n)$ for all $m,n\in \mathbb{Z}^+$
Prove that there exist at least $100!$ ways to write $100!$ as sum of elements of set {$1!,2!,3!...99!$}
(each number in sum can be two or more times)
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 $a^n$.
Show that for all positive integers $n\geq 2$ the last digit of the number $2^{2^n}+1$ is $7$ .
For $ n\geq2$ let $ a_1, a_2, \ldots a_n$ be positive real numbers such that
\[ (a_1 \plus{} a_2 \plus{} \cdots \plus{} a_n)\left(\frac {1}{a_1} \plus{} \frac {1}{a_2} \plus{} \cdots \plus{} \frac {1}{a_n}\right) \leq \left(n \plus{} \frac {1}{2}\right)^2.
\]
Prove that $ \max(a_1, a_2, \ldots, a_n)\leq 4\min(a_1, a_2, \ldots, a_n)$.
Determine all infinite sets $A$ of positive integers with the following propety:
If $a,b \in A$ and $a \ge b$ then $\left\lfloor \frac{a}{b} \right\rfloor \in A$
In the city built are $2019$ metro stations. Some pairs of stations are connected. tunnels, and from any station through the tunnels you can reach any other. The mayor ordered to organize several metro lines: each line should include several different stations connected in series by tunnels (several lines can pass through the same tunnel), and in each station must lie at least on one line. To save money no more than $k$ lines should be made. It turned out that the order of the mayor is not feasible. What is the largest $k$ it could to happen?
Determine all functions $f: \mathbb{R}^+ \to \mathbb{R}^+$ satisfying
\[f\bigl(x + y^2 f(y)\bigr) = f\bigl(1 + yf(x)\bigr)f(x)\]
for any positive reals $x$, $y$, where $\mathbb{R}^+$ is the collection of all positive real numbers.
[i]Proposed by Ming Hsiao.[/i]
Let $n\geq 2$ be an integer and let $a_1, a_2, \ldots, a_n$ be positive real numbers with sum $1$. Prove that $$\sum_{k=1}^n \frac{a_k}{1-a_k}(a_1+a_2+\cdots+a_{k-1})^2 < \frac{1}{3}.$$
$1)$. Prove a graph with $2n$ vertices and $n+2$ edges has an independent set of size $n$ (there are $n$ vertices such that no two of them are adjacent ).
$2)$.Find the number of graphs with $2n$ vertices and $n+3$ edges , such that among any $n$ vertices there is an edge connecting two of them
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]
Find all functions $f : \mathbb{Z} \rightarrow \mathbb{Z}$ such that for all integers $m,n$,
\[f(m - n + f(n)) = f(m) + f(n).\]
Let $a_0,a_1,a_2,\dots $ be a sequence of real numbers such that $a_0=0, a_1=1,$ and for every $n\geq 2$ there exists $1 \leq k \leq n$ satisfying \[ a_n=\frac{a_{n-1}+\dots + a_{n-k}}{k}. \]Find the maximum possible value of $a_{2018}-a_{2017}$.
In a sports tournament of $n$ players, each pair of players plays against each other exactly one match and there are no draws.Show that the players can be arranged in an order $P_1,P_2, .... , P_n$ such that $P_i$ defeats $P_{i+1}$ for all $1 \le i \le n-1$.
Suppose $A\subset \{(a_1,a_2,\dots,a_n)\mid a_i\in \mathbb{R},i=1,2\dots,n\}$. For any $\alpha=(a_1,a_2,\dots,a_n)\in A$ and $\beta=(b_1,b_2,\dots,b_n)\in A$, we define
\[ \gamma(\alpha,\beta)=(|a_1-b_1|,|a_2-b_2|,\dots,|a_n-b_n|), \] \[ D(A)=\{\gamma(\alpha,\beta)\mid\alpha,\beta\in A\}. \] Please show that $|D(A)|\geq |A|$.
Let $ n \ge 2$ be a fixed integer and let $ a_{i,j} (1 \le i < j \le n)$ be some positive integers. For a sequence $ x_1, ... , x_n$ of reals, let $ K(x_1, .... , x_n)$ be the product of all expressions $ (x_i \minus{} x_j)^{a_{i,j}}$ where $ 1 \le i < j \le n$. Prove that if the inequality $ K(x_1, .... , x_n) \ge 0$ holds independently of the choice of the sequence $ x_1, ... , x_n$ then all integers $ a_{i,j}$ are even.
Let $(a_n)_{n \geq 1}$ be a sequence of real numbers. We define a sequence of real functions $(f_n)_{n \geq 0}$ such that for all $x \in \mathbb{R}$, the following holds:
\[
f_0(x) = 1 \quad \text{and} \quad f_n(x) = \int_{a_n}^{x} f_{n-1}(t) \, dt \quad \text{for } n \geq 1.
\]
Find all possible sequences $(a_n)_{n \geq 1}$ such that $f_n(0) = 0$ for all $n \geq 2$.\\
[b]Note:[/b] It is not necessarily true that $f_1(0) = 0$.
We have a $m\times n$ table and $m\geq{4}$ and we call a $1\times 1$ square a room. When we put an alligator coin in a room, it menaces all the rooms in his column and his adjacent rooms in his row. What's the minimum number of alligator coins required, such that each room is menaced at least by one alligator coin? (Notice that all alligator coins are vertical.)
Let $n$ be a "Good Number" if sum of all divisors of $n$ is less than $2n$ for $n\in \mathbb{Z}.$
Does there exist an infinite set $M$ that satisfies the following?
For all $a,b\in M,$ $a+b$ is good number. ($a=b$ is allowed.)
A calculator has a key which replaces the displayed entry with its square, and another key which replaces the displayed entry with its reciprocal. Let $y$ be the final result if one starts with an entry $x \neq 0$ and alternately squares and reciprocates $n$ times each. Assuming the calculator is completely accurate (e.g., no roundoff or overflow), then $y$ equals
A. $x^{((-2)^n)}$
B. $x^{2n}$
C. $x^{-2n}$
D. $x^{-(2^n)}$
E. $x^{((-1)^n 2n)}$
Define a [i]beautiful number[/i] to be an integer of the form $a^n$, where $a\in\{3,4,5,6\}$ and $n$ is a positive integer.
Prove that each integer greater than $2$ can be expressed as the sum of pairwise distinct beautiful numbers.
[i]Proposed by Matthew Babbitt[/i]
Let $m$ and $n$ be positive integers. Mr. Fat has a set $S$ containing every rectangular tile with integer side lengths and area of a power of $2$. Mr. Fat also has a rectangle $R$ with dimensions $2^m \times 2^n$ and a $1 \times 1$ square removed from one of the corners. Mr. Fat wants to choose $m + n$ rectangles from $S$, with respective areas $2^0, 2^1, \ldots, 2^{m + n - 1}$, and then tile $R$ with the chosen rectangles. Prove that this can be done in at most $(m + n)!$ ways.
[i]Palmer Mebane.[/i]
Given integer $a_1\geq 2$. For integer $n\geq 2$, define $a_n$ to be the smallest positive integer which is not coprime to $a_{n-1}$ and not equal to $a_1,a_2,\cdots, a_{n-1}$. Prove that every positive integer except 1 appears in this sequence $\{a_n\}$.
Let $n$ be a positive integer. We start with $n$ piles of pebbles, each initially containing a single pebble. One can perform moves of the following form: choose two piles, take an equal number of pebbles from each pile and form a new pile out of these pebbles. Find (in terms of $n$) the smallest number of nonempty piles that one can obtain by performing a finite sequence of moves of this form.