Found problems: 5802
Assume that a face of a convex polyhedron $ P$ has a common edge with every other face. Show that there exists a simple closed polygon that consists of edges of $ P$ and passes through all vertices.
[i]L .Lovasz[/i]
Let $\{a_n\}_{n\geq 1}$ be a sequence of real numbers which satisfies the following relation:
\[a_{n+1}=10^n a_n^2\]
(a) Prove that if $a_1$ is small enough, then $\displaystyle\lim_{n\to\infty} a_n =0$.
(b) Find all possible values of $a_1\in \mathbb{R}$, $a_1\geq 0$, such that $\displaystyle\lim_{n\to\infty} a_n =0$.
Let $a$ and $b$ be two positive integers. Prove that the integer
\[a^2+\left\lceil\frac{4a^2}b\right\rceil\]
is not a square. (Here $\lceil z\rceil$ denotes the least integer greater than or equal to $z$.)
[i]Russia[/i]
Define \[\begin{cases}d(n, 0)=d(n, n)=1&(n \ge 0),\\ md(n, m)=md(n-1, m)+(2n-m)d(n-1,m-1)&(0<m<n).\end{cases}\] Prove that $d(n, m)$ are integers for all $m, n \in \mathbb{N}$.
For some positive integer $n,$ Elmo writes down the equation
\[x_1+x_2+\dots+x_n=x_1+x_2+\dots+x_n.\]
Elmo inserts at least one $f$ to the left side of the equation and adds parentheses to create a valid functional equation. For example, if $n=3,$ Elmo could have created the equation
\[f(x_1+f(f(x_2)+x_3))=x_1+x_2+x_3.\]
Cookie Monster comes up with a function $f: \mathbb{Q}\to\mathbb{Q}$ which is a solution to Elmo's functional equation. (In other words, Elmo's equation is satisfied for all choices of $x_1,\dots,x_n\in\mathbb{Q})$. Is it possible that there is no integer $k$ (possibly depending on $f$) such that $f^k(x)=x$ for all $x$?
[i]Srinivas Arun[/i]
Let $ ABC$ be an equilateral triangle with side length equal to $ N \in \mathbb{N}.$ Consider the set $ S$ of all points $ M$ inside the triangle $ ABC$ satisfying
\[ \overrightarrow{AM} \equal{} \frac{1}{N} \cdot \left(n \cdot \overrightarrow{AB} \plus{} m \cdot \overrightarrow{AC} \right)\]
with $ m, n$ integers, $ 0 \leq n \leq N,$ $ 0 \leq m \leq N$ and $ n \plus{} m \leq N.$
Every point of S is colored in one of the three colors blue, white, red such that
[b](i) [/b]no point of $ S \cap [AB]$ is coloured blue
[b](ii)[/b] no point of $ S \cap [AC]$ is coloured white
[b](iii)[/b] no point of $ S \cap [BC]$ is coloured red
Prove that there exists an equilateral triangle the following properties:
[b](1)[/b] the three vertices of the triangle are points of $ S$ and coloured blue, white and red, respectively.
[b](2)[/b] the length of the sides of the triangle is equal to 1.
[i]Variant:[/i] Same problem but with a regular tetrahedron and four different colors used.
[i]Biribol[/i] is a game played between two teams of 4 people each (teams are not fixed). Find all the possible values of $ n$ for which it is possible to arrange a tournament with $ n$ players in such a way that every couple of people plays a match in opposite teams exactly once.
Let $A$ be the set of all binary sequences of length $n$ and denote $o =(0, 0, \ldots , 0) \in A$. Define the addition on $A$ as $(a_1, \ldots , a_n)+(b_1, \ldots , b_n) =(c_1, \ldots , c_n)$, where $c_i = 0$ when $a_i = b_i$ and $c_i = 1$ otherwise. Suppose that $f\colon A \to A$ is a function such that $f(0) = 0$, and for each $a, b \in A$, the sequences $f(a)$ and $f(b)$ differ in exactly as many places as $a$ and $b$ do. Prove that if $a$ , $b$, $c \in A$ satisfy $a+ b + c = 0$, then $f(a)+ f(b) + f(c) = 0$.
For every positive integer $k>1$ prove that there exist a real number $x$ so that for every positive integer $n<1398$:
$$\left\{x^n\right\}<\left\{x^{n-1}\right\} \Longleftrightarrow k\mid n.$$
[i]Proposed by Mohammad Amin Sharifi[/i]
Let $n$ and $k$ be positive integers. Two infinite sequences $\{s_i\}_{i\geq 1}$ and $\{t_i\}_{i\geq 1}$ are [i]equivalent[/i] if, for all positive integers $i$ and $j$, $s_i = s_j$ if and only if $t_i = t_j$. A sequence $\{r_i\}_{i\geq 1}$ has [i]equi-period[/i] $k$ if $r_1, r_2, \ldots $ and $r_{k+1}, r_{k+2}, \ldots$ are equivalent.
Suppose $M$ infinite sequences with equi-period $k$ whose terms are in the set $\{1, \ldots, n\}$ can be chosen such that no two chosen sequences are equivalent to each other. Determine the largest possible value of $M$ in terms of $n$ and $k$.
Determine all positive integers $n$ for which there exists an integer $m$ such that ${2^{n}-1}$ is a divisor of ${m^{2}+9}$.
Find the smallest positive integer $n$ or show no such $n$ exists, with the following property: there are infinitely many distinct $n$-tuples of positive rational numbers $(a_1, a_2, \ldots, a_n)$ such that both
$$a_1+a_2+\dots +a_n \quad \text{and} \quad \frac{1}{a_1} + \frac{1}{a_2} + \dots + \frac{1}{a_n}$$
are integers.
A token is placed at each vertex of a regular $2n$-gon. A [i]move[/i] consists in choosing an edge of the $2n$-gon and swapping the two tokens placed at the endpoints of that edge. After a finite number of moves have been performed, it turns out that every two tokens have been swapped exactly once. Prove that some edge has never been chosen.
Let $P_1,P_2,\dots,P_n$ be $n$ distinct points over a line in the plane ($n\geq2$). Consider all the circumferences with diameters $P_iP_j$ ($1\leq{i,j}\leq{n}$) and they are painted with $k$ given colors. Lets call this configuration a ($n,k$)-cloud.
For each positive integer $k$, find all the positive integers $n$ such that every possible ($n,k$)-cloud has two mutually exterior tangent circumferences of the same color.
Given are the real line and the two unique marked points $0$ and $1$. We can perform as many times as we want the following operation: we take two already marked points $a$ and $b$ and mark the reflection of $a$ over $b$. Let $f(n)$ be the minimum number of operations needed to mark on the real line the number $n$ (which is the number at a distance $\left| n\right|$ from $0$ and it is on the right of $0$ if $n>0$ and on the left of $0$ if $n<0$). For example, $f(0)=f(1)=0$ and $f(-1)=f(2)=1$. Find $f(n)$.
In the country some mathematicians know each other and any division of them into two sets contain 2 friends from different sets.It is known that if you put any set of four or more mathematicians at a round table so that any two neighbours know each other , then at the table there are two friends not sitting next to each other.We denote by $c_i $ the number of sets of $i$ pairwise familiar mathematicians(by saying "familiar" it means know each other).Prove that
$c_1-c_2+c_3-c_4+...=1$
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.
[/list]
Find all positive integers $m, n$ satisfying $n!+2^{n-1}=2^m$.
Let $n\geq2$ be an integer. An $n$-tuple $(a_1,a_2,\dots,a_n)$ of not necessarily different positive integers is [i]expensive[/i] if there exists a positive integer $k$ such that $$(a_1+a_2)(a_2+a_3)\dots(a_{n-1}+a_n)(a_n+a_1)=2^{2k-1}.$$
a) Find all integers $n\geq2$ for which there exists an expensive $n$-tuple.
b) Prove that for every odd positive integer $m$ there exists an integer $n\geq2$ such that $m$ belongs to an expensive $n$-tuple.
[i]There are exactly $n$ factors in the product on the left hand side.[/i]
Let $N$ be a positive integer. Consider a $N \times N$ array of square unit cells. Two corner cells that lie on the same longest diagonal are colored black, and the rest of the array is white. A [i]move[/i] consists of choosing a row or a column and changing the color of every cell in the chosen row or column.
What is the minimal number of additional cells that one has to color black such that, after a finite number of moves, a completely black board can be reached?
Let $\Gamma$ consist of all polynomials in $x$ with integer coefficients. For $f$ and $g$ in $\Gamma$ and $m$ a positive integer, let $f \equiv g \pmod{m}$ mean that every coefficient of $f-g$ is an integral multiple of $m$. Let $n$ and $p$ be positive integers with $p$ prime. Given that $f,g,h,r$ and $s$ are in $\Gamma$ with $rf+sg\equiv 1 \pmod{p}$ and $fg \equiv h \pmod{p}$, prove that there exist $F$ and $G$ in $\Gamma$ with $F \equiv f \pmod{p}$, $G \equiv g \pmod{p}$, and $FG \equiv h \pmod{p^n}$.
Let \(d(n)\) denote the number of positive divisors of \(n\). The sequence \(a_0\), \(a_1\), \(a_2\), \(\ldots\) is defined as follows: \(a_0=1\), and for all integers \(n\ge1\), \[a_n=d(a_{n-1})+d(d(a_{n-2}))+\cdots+ {\underbrace{d(d(\ldots d(a_0)\ldots))}_{n\text{ times}}}.\] Show that for all integers \(n\ge1\), we have \(a_n\le3n\).
[i]Proposed by Karthik Vedula[/i]
For which real numbers $a$ does the sequence $(u_n )$ defined by the initial condition $u_0 =a$ and the recursion $u_{n+1} =2u_n - n^2$ have $u_n >0$ for all $n \geq 0?$
Let be a natural number $ n, $ two $ \text{n-tuplets} $ of real numbers $ a:=\left( a_1,a_2,\ldots, a_n \right) , b:=\left( b_1,b_2,\ldots, b_n \right) , $ and the function $ f:\mathbb{R}\longrightarrow\mathbb{R}, f(x)=\sum_{i=1}^na_i\cos \left( b_ix \right) $. Prove that if the numbers of $ b $ are all positive and pairwise distinct,
[b]a)[/b] then, $ f\ge 0 $ implies that the numbers of $ a $ are all equal.
[b]b)[/b] if the numbers of $ a $ are all nonzero and $ f $ is periodic, then the ratio of any two numbers of $ b $ is rational.
[i]Marin Tolosi[/i]
Proof that
$$ \sum_{m=1}^n5^{\omega (m)} \le \sum_{k=1}^n\lfloor \frac{n}{k} \rfloor \tau (k)^2 \le \sum_{m=1}^n5^{\Omega (m)} .$$