Found problems: 5802
A dissection of a convex polygon into finitely many triangles by segments is called a [i]trilateration[/i] if no three vertices of the created triangles lie on a single line (vertices of some triangles might lie inside the polygon). We say that a trilateration is [i]good[/i] if its segments can be replaced with one-way arrows in such a way that the arrows along every triangle of the trilateration form a cycle and the arrows along the whole convex polygon also form a cycle. Find all $n\ge 3$ such that the regular $n$-gon has a good trilateration.
Let $\mathbb N$ be the set of positive integers. Find all functions $f: \mathbb N \to \mathbb N$ that satisfy the equation
\[ f^{abc-a}(abc) + f^{abc-b}(abc) + f^{abc-c}(abc) = a + b + c \]
for all $a,b,c \ge 2$.
(Here $f^1(n) = f(n)$ and $f^k(n) = f(f^{k-1}(n))$ for every integer $k$ greater than $1$.)
For each integer $n\ge 1,$ compute the smallest possible value of \[\sum_{k=1}^{n}\left\lfloor\frac{a_k}{k}\right\rfloor\] over all permutations $(a_1,\dots,a_n)$ of $\{1,\dots,n\}.$
[i]Proposed by Shahjalal Shohag, Bangladesh[/i]
There are $n$ boys and $n$ girls in a school class, where $n$ is a positive integer. The heights of all the children in this class are distinct. Every girl determines the number of boys that are taller than her, subtracts the number of girls that are taller than her, and writes the result on a piece of paper. Every boy determines the number of girls that are shorter than him, subtracts the number of boys that are shorter than him, and writes the result on a piece of paper.
Prove that the numbers written down by the girls are the same as the numbers written down by the boys (up to a permutation).
[i]Proposed by Stephan Wagner, Austria[/i]
Given $2n$ natural numbers, so that the average arithmetic of those $2n$ number is $2$. If all the number is not more than $2n$. Prove we can divide those $2n$ numbers into $2$ sets, so that the sum of each set to be the same.
Does there exists a function $f : \mathbb{N} \longrightarrow \mathbb{N}$
\begin{align*} f \left( m+ f(n) \right) = f(m) +f(n) + f(n+1) \end{align*}
for all $m,n \in \mathbb{N}$ ?
Let $\mathbb{N} = \{1, 2, 3, \ldots\}$ be the set of positive integers. Find all functions $f$, defined on $\mathbb{N}$ and taking values in $\mathbb{N}$, such that $(n-1)^2< f(n)f(f(n)) < n^2+n$ for every positive integer $n$.
Call a triple of numbers [b]Nice[/b] if one of them is the average of the other two. Assume that we have $2k+1$ distinct real numbers with $k^2$ [b] Nice[/b] triples. Prove that these numbers can be devided into two arithmetic progressions with equal ratios
Proposed by [i]Morteza Saghafian[/i]
Find all positive integers $(r,s)$ such that there is a non-constant sequence $a_n$ os positive integers such that for all $n=1,2,\dots$
\[ a_{n+2}= \left(1+\frac{{a_2}^r}{{a_1}^s} \right ) \left(1+\frac{{a_3}^r}{{a_2}^s} \right ) \dots \left(1+\frac{{a_{n+1}}^r}{{a_n}^s} \right ).\]
Proposed by Navid Safaei, Iran
Determine all real numbers $\alpha$ such that, for every positive integer $n,$ the integer
$$\lfloor\alpha\rfloor +\lfloor 2\alpha\rfloor +\cdots +\lfloor n\alpha\rfloor$$
is a multiple of $n.$ (Note that $\lfloor z\rfloor$ denotes the greatest integer less than or equal to $z.$ For example, $\lfloor -\pi\rfloor =-4$ and $\lfloor 2\rfloor= \lfloor 2.9\rfloor =2.$)
[i]Proposed by Santiago Rodríguez, Colombia[/i]
Let $A=\{1,2,\ldots,2012\}, \: B=\{1,2,\ldots,19\}$ and $S$ be the set of all subsets of $A.$ Find the number of functions $f : S\to B$ satisfying $f(A_1\cap A_2)=\min\{f(A_1),f(A_2)\}$ for all $A_1, A_2 \in S.$
Let $x_0,x_1,x_2,\dots$ be a infinite sequence of real numbers, such that the following three equalities are true:
I- $x_{2k}=(4x_{2k-1}-x_{2k-2})^2$, for $k\geq 1$
II- $x_{2k+1}=|\frac{x_{2k}}{4}-k^2|$, for $k\geq 0$
III- $x_0=1$
a) Determine the value of $x_{2022}$
b) Prove that there are infinite many positive integers $k$, such that $2021|x_{2k+1}$
Post your solutions below! :D
[b]Also, I think it is beneficial to everyone if you all attempt to comment on each other's solutions.[/b]
5/1/31. Let $n$ be a positive integer. For integers a, b with $0 \leq a b \leq n - 1$, let $r_n(a, b)$ denote
the remainder when $ab$ is divided by $n$. If $S_n$ denotes the sum of all $n^2$ remainders $r_n(a, b)$,
prove that
$\frac{1}{2}-\frac{1}{\sqrt{n}}\leq \frac{S_n}{n^3} \leq \frac{1}{2}$
$S$ be a set of ordered triples $(a,b,c)$ of distinct elements of a finite set $A$. Suppose that
[list=1]
[*] $(a,b,c)\in S\iff (b,c,a)\in S$
[*] $(a,b,c)\in S\iff (c,b,a)\not\in S$
[*] $(a,b,c),(c,d,a)\text{ both }\in S\iff (b,c,d),(d,a,b)\text{ both }\in S$[/list]
Prove there exists $g: A\to \mathbb{R}$, such that $g$ is one-one and $g(a)<g(b)<g(c)\implies (a,b,c)\in S$
Prove that for every positive integer $t$ there is a unique permutation $a_0, a_1, \ldots , a_{t-1}$ of $0, 1, \ldots , t-1$ such that, for every $0 \leq i \leq t-1$, the binomial coefficient $\binom{t+i}{2a_i}$ is odd and $2a_i \neq t+i$.
For a fixed natural number $m \geq 2$, prove that
[b]a.)[/b] There exists integers $x_1, x_2, \ldots, x_{2m}$ such that \[x_i x_{m + i} = x_{i + 1} x_{m + i - 1} + 1, i = 1, 2, \ldots, m \hspace{2cm}(*)\]
[b]b.)[/b] For any set of integers $\lbrace x_1, x_2, \ldots, x_{2m}$ which fulfils (*), an integral sequence $\ldots, y_{-k}, \ldots, y_{-1}, y_0, y_1, \ldots, y_k, \ldots$ can be constructed such that $y_k y_{m + k} = y_{k + 1} y_{m + k - 1} + 1, k = 0, \pm 1, \pm 2, \ldots$ such that $y_i = x_i, i = 1, 2, \ldots, 2m$.
Let the sequence $\{a_n\}_{n\ge 1}$ be defined by
$$a_n=\tan(n\theta)$$
where $\tan\theta =2$. Show that for all $n$, $a_n$ is a rational number which can be written with an odd denominator.
At the start of a game there are three boxes with $2008, 2009$ and $2010$ game pieces
Anja and Bernd play in turns according to the following rule:
[i]When it is your turn, select two boxes, empty them and then distribute the pieces from the third box to the three boxes, such that no box may remain empty.If you can no longer complete a turn, you have lost. [/i]
Who has a winning strategy when Anja starts?
Each rational number is painted either white or red. Call such a coloring of the rationals [i]sanferminera[/i] if for any distinct rationals numbers $x$ and $y$ satisfying one of the following three conditions: [list=1][*]$xy=1$,
[*]$x+y=0$,
[*]$x+y=1$,[/list]we have $x$ and $y$ painted different colors. How many sanferminera colorings are there?
Let $n$ be positive integer, set $M = \{ 1, 2, \ldots, 2n \}$. Find the minimum positive integer $k$ such that for any subset $A$ (with $k$ elements) of set $M$, there exist four pairwise distinct elements in $A$ whose sum is $4n + 1$.
(a) Prove that, for any positive integers $m\le \ell$ given, there is a positive integer $n$ and positive integers $x_1,\cdots,x_n,y_1,\cdots,y_n$ such that the equality \[ \sum_{i=1}^nx_i^k=\sum_{i=1}^ny_i^k\] holds for every $k=1,2,\cdots,m-1,m+1,\cdots,\ell$, but does not hold for $k=m$.
(b) Prove that there is a solution of the problem, where all numbers $x_1,\cdots,x_n,y_1,\cdots,y_n$ are distinct.
[i]Proposed by Ilya Bogdanov and Géza Kós.[/i]
Suppose that $ s_1,s_2,s_3, \ldots$ is a strictly increasing sequence of positive integers such that the sub-sequences \[s_{s_1},\, s_{s_2},\, s_{s_3},\, \ldots\qquad\text{and}\qquad s_{s_1+1},\, s_{s_2+1},\, s_{s_3+1},\, \ldots\] are both arithmetic progressions. Prove that the sequence $ s_1, s_2, s_3, \ldots$ is itself an arithmetic progression.
[i]Proposed by Gabriel Carroll, USA[/i]
Denote by $\mathbb{Q}^+$ the set of all positive rational numbers. Determine all functions $f : \mathbb{Q}^+ \mapsto \mathbb{Q}^+$ which satisfy the following equation for all $x, y \in \mathbb{Q}^+:$ \[f\left( f(x)^2y \right) = x^3 f(xy).\]
[i]Proposed by Thomas Huber, Switzerland[/i]
Let $n$ be a positive integer. Answer the following questions.
(1) Find the maximum value of $f_n(x)=x^{n}e^{-x}$ for $x\geq 0$.
(2) Show that $\lim_{x\to\infty} f_n(x)=0$.
(3) Let $I_n=\int_0^x f_n(t)\ dt$. Find $\lim_{x\to\infty} I_n(x)$.
Let A be a symmetric matrix such that the sum of elements of any row is zero.
Show that all elements in the main diagonal of cofator matrix of A are equal.