Found problems: 5802
In a football season, even number $n$ of teams plays a simple series, i.e. each team plays once against each other team. Show that ona can group the series into $n-1$ rounds such that in every round every team plays exactly one match.
Find all functions $ f: \mathbb{R} \rightarrow \mathbb{R}$ satisfying \[ f(x^3\plus{}y^3)\equal{}xf(x^2)\plus{}yf(y^2)\] for all real numbers $ x$ and $ y$.
[i]Hery Susanto, Malang[/i]
The following operation is allowed on a finite graph: Choose an arbitrary cycle of length 4 (if there is any), choose an arbitrary edge in that cycle, and delete it from the graph. For a fixed integer ${n\ge 4}$, find the least number of edges of a graph that can be obtained by repeated applications of this operation from the complete graph on $n$ vertices (where each pair of vertices are joined by an edge).
[i]Proposed by Norman Do, Australia[/i]
Let $f: \mathbb{N} \rightarrow \mathbb{N}$ be a function, and let $f^m$ be $f$ applied $m$ times. Suppose that for every $n \in \mathbb{N}$ there exists a $k \in \mathbb{N}$ such that $f^{2k}(n)=n+k$, and let $k_n$ be the smallest such $k$. Prove that the sequence $k_1,k_2,\ldots $ is unbounded.
[i]Proposed by Palmer Mebane, United States[/i]
The vertices of a connected graph cannot be coloured with less than $n+1$ colours (so that adjacent vertices have different colours).
Prove that $\dfrac{n(n-1)}{2}$ edges can be removed from the graph so that it remains connected.
[i]V. Dolnikov[/i]
[b]EDIT.[/b] It is confirmed by the official solution that the graph is tacitly assumed to be [b]finite[/b].
Let $x>1$ ,$n$ be positive integer. Prove that$$\sum_{k=1}^{n}\frac{\{kx \}}{[kx]}<\sum_{k=1}^{n}\frac{1}{2k-1}$$
Where $[kx ]$ be the integer part of $kx$ ,$\{kx \}$ be the decimal part of $kx$.
An equilateral triangle with side $20$ is divided by there series of parallel lines into $400$ equilateral triangles with side $1$. What maximum number of these small triangles can be crossed (internally) by one line?
Tuymaada 2017 Q7 Juniors
Find all polynomials $f(x)$ with real coefficients for which
\[f(x)f(2x^2) = f(2x^3 + x).\]
Positive reals $a,b,c$ satisfy $ab+bc+ca=abc$. Prove that:
$\frac{a^4+b^4}{ab(a^3+b^3)} + \frac{b^4+c^4}{bc(b^3+c^3)}+\frac{c^4+a^4}{ca(c^3+a^3)} \geq 1$
$N$ coins are placed on a table, $N - 1$ are genuine and have the same weight, and one is fake, with a different weight. Using a two pan balance, the goal is to determine with certainty the fake coin, and whether it is lighter or heavier than a genuine coin. Whenever one can deduce that one or more coins are genuine, they will be inmediately discarded and may no longer be used in subsequent weighings. Determine all $N$ for which the goal is achievable. (There are no limits regarding how many times one may use the balance).
Note: the only difference between genuine and fake coins is their weight; otherwise, they are identical.
Let $\left \{ c_i \right \}_{i=0}^{\infty}$ be a sequence of non-negative real numbers with $c_{2017}>0$. A sequence of polynomials is defined as
$$P_{-1}(x)=0 \ , \ P_0(x)=1 \ , \ P_{n+1}(x)=xP_n(x)+c_nP_{n-1}(x).$$
Prove that there doesn't exist any integer $n>2017$ and some real number $c$ such that
$$P_{2n}(x)=P_n(x^2+c).$$
[i]Proposed by Navid Safaei[/i]
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]
For every positive integer $n$ with prime factorization $n = \prod_{i = 1}^{k} p_i^{\alpha_i}$, define
\[\mho(n) = \sum_{i: \; p_i > 10^{100}} \alpha_i.\]
That is, $\mho(n)$ is the number of prime factors of $n$ greater than $10^{100}$, counted with multiplicity.
Find all strictly increasing functions $f: \mathbb{Z} \to \mathbb{Z}$ such that
\[\mho(f(a) - f(b)) \le \mho(a - b) \quad \text{for all integers } a \text{ and } b \text{ with } a > b.\]
[i]Proposed by Rodrigo Sanches Angelo, Brazil[/i]
Determine all positive integers $M$ such that the sequence $a_0, a_1, a_2, \cdots$ defined by \[ a_0 = M + \frac{1}{2} \qquad \textrm{and} \qquad a_{k+1} = a_k\lfloor a_k \rfloor \quad \textrm{for} \, k = 0, 1, 2, \cdots \] contains at least one integer term.
Prove that every positive rational number can be expressed uniquely as a finite sum of the form $$a_1+\frac{a_2}{2!}+\frac{a_3}{3!}+\dots+\frac{a_n}{n!},$$ where $a_n$ are integers such that $0 \leq a_n \leq n-1$ for all $n > 1$.
Positive integer $k$ and $k$ colors are given. We will say that a set of $2k$ points on a plane is $colorful$, if it contains exactly 2 points of each color and if lines connecting every two points of the same color are pairwise distinct. Find, in terms of $k$ the least integer $n\geq 2$ such that: in every set of $nk$ points of a plane, no three of which are collinear, consisting of $n$ points of every color there exists a $colorful$ subset.
a) For each $n \ge 2$, find the maximum constant $c_{n}$ such that
$\frac 1{a_{1}+1}+\frac 1{a_{2}+1}+\ldots+\frac 1{a_{n}+1}\ge c_{n}$
for all positive reals $a_{1},a_{2},\ldots,a_{n}$ such that $a_{1}a_{2}\cdots a_{n}= 1$.
b) For each $n \ge 2$, find the maximum constant $d_{n}$ such that
$\frac 1{2a_{1}+1}+\frac 1{2a_{2}+1}+\ldots+\frac 1{2a_{n}+1}\ge d_{n}$
for all positive reals $a_{1},a_{2},\ldots,a_{n}$ such that $a_{1}a_{2}\cdots a_{n}= 1$.
For each positive integer $n$, the Bank of Cape Town issues coins of denomination $\frac1n$. Given a finite collection of such coins (of not necessarily different denominations) with total value at most most $99+\frac12$, prove that it is possible to split this collection into $100$ or fewer groups, such that each group has total value at most $1$.
Find all functions $f:\mathbb Z\rightarrow \mathbb Z$ such that, for all integers $a,b,c$ that satisfy $a+b+c=0$, the following equality holds:
\[f(a)^2+f(b)^2+f(c)^2=2f(a)f(b)+2f(b)f(c)+2f(c)f(a).\]
(Here $\mathbb{Z}$ denotes the set of integers.)
[i]Proposed by Liam Baker, South Africa[/i]
We define a sequence $a_n$ so that $a_0=1$ and
\[a_{n+1} = \begin{cases} \displaystyle \frac{a_n}2 & \textrm { if } a_n \equiv 0 \pmod 2, \\ a_n + d & \textrm{ otherwise. } \end{cases} \]
for all postive integers $n$.
Find all positive integers $d$ such that there is some positive integer $i$ for which $a_i=1$.
Some of the vertices of a convex $n$-gon are connected by segments, such that any two of them have no common interior point. Prove that, for any $n$ points in general position, there exists a one-to-one correspondence between the points and the vertices of the $n$ gon, such that any two segments between the points, corresponding to the respective segments from the $n$ gon, have no common interior point.
We denote by $\mathbb{R}^\plus{}$ the set of all positive real numbers.
Find all functions $f: \mathbb R^ \plus{} \rightarrow\mathbb R^ \plus{}$ which have the property:
\[f(x)f(y)\equal{}2f(x\plus{}yf(x))\]
for all positive real numbers $x$ and $y$.
[i]Proposed by Nikolai Nikolov, Bulgaria[/i]
Given an endless supply of white, blue and red cubes. In a circle arrange any $N$ of them. The robot, standing in any place of the circle, goes clockwise and, until one cube remains, constantly repeats this operation: destroys the two closest cubes in front of him and puts a new one behind him a cube of the same color if the destroyed ones are the same, and the third color if the destroyed two are different colors.
We will call the arrangement of the cubes [i]good [/i] if the color of the cube remaining at the very end does not depends on where the robot started. We call $N$ [i]successful [/i] if for any choice of $N$ cubes all their arrangements are good. Find all successful $N$.
I. Bogdanov
Given a positive integer $n$, let $D$ is the set of positive divisors of $n$, and let $f: D \to \mathbb{Z}$ be a function. Prove that the following are equivalent:
(a) For any positive divisor $m$ of $n$,
\[ n ~\Big|~ \sum_{d|m} f(d) \binom{n/d}{m/d}. \]
(b) For any positive divisor $k$ of $n$,
\[ k ~\Big|~ \sum_{d|k} f(d). \]
Let $(a_n)$ be a decreasing sequence of positive numbers with limit $0$ such that
$$b_n = a_n -2 a_{n+1}+a_{n+2} \geq 0$$
for all $n.$ Prove that
$$\sum_{n=1}^{\infty} n b_n =a_1.$$