Found problems: 5802
It is known that for every integer $n > 1$ there is a prime number among the numbers $n+1,n+2,...,2n-1.$ Determine all positive integers $n$ with the following property: Every integer $m > 1$ less than $n$ and coprime to $n$ is prime.
Determine all integers $ n\geq 2$ having the following property: for any integers $a_1,a_2,\ldots, a_n$ whose sum is not divisible by $n$, there exists an index $1 \leq i \leq n$ such that none of the numbers $$a_i,a_i+a_{i+1},\ldots,a_i+a_{i+1}+\ldots+a_{i+n-1}$$ is divisible by $n$. Here, we let $a_i=a_{i-n}$ when $i >n$.
[i]Proposed by Warut Suksompong, Thailand[/i]
Let $ A_1,A_2,\dots,A_n$ be idempotent matrices with real entries. Prove that:
\[ \mbox{N}(A_1)\plus{}\mbox{N}(A_2)\plus{}\dots\plus{}\mbox{N}(A_n)\geq \mbox{rank}(I\minus{}A_1A_2\dots A_n)\]
$ \mbox{N}(A)$ is $ \mbox{dim}(\mbox{ker(A)})$
Let $ A$ be the subset of the set of positive integers, having the following $ 2$ properties:
1) If $ a$ belong to $ A$,than all of the divisors of $ a$ also belong to $ A$;
2) If $ a$ and $ b$, $ 1 < a < b$, belong to $ A$, than $ 1 \plus{} ab$ is also in $ A$;
Prove that if $ A$ contains at least $ 3$ positive integers, than $ A$ contains all positive integers.
Let $n \geq 3$ be an integer. Let $t_1$, $t_2$, ..., $t_n$ be positive real numbers such that \[n^2 + 1 > \left( t_1 + t_2 + \cdots + t_n \right) \left( \frac{1}{t_1} + \frac{1}{t_2} + \cdots + \frac{1}{t_n} \right).\] Show that $t_i$, $t_j$, $t_k$ are side lengths of a triangle for all $i$, $j$, $k$ with $1 \leq i < j < k \leq n$.
Suppose that $a_0=1$ and that $a_{n+1}=a_n+e^{-a_n}$ for $n=0,1,2,\dots.$ Does $a_n-\log n$ have a finite limit as $n\to\infty?$ (Here $\log n=\log_en=\ln n.$)
In a school, every pair of students are either friends or strangers. Friendship is mutual, and no student is friends with themselves. A sequence of (not necessarily distinct) students $A_1, A_2, \dots, A_{2023}$ is called [i]mischievous[/i] if
$\bullet$ Total number of friends of $A_1$ is odd.
$\bullet$ $A_i$ and $A_{i+1}$ are friends for $i=1, 2, \dots, 2022$.
$\bullet$ Total number of friends of $A_{2023}$ is even.
Prove that the total number of [i]mischievous[/i] sequences is even.
Let $n$ be a positive integer and $u_1,u_2,\cdots ,u_n$ be positive integers not larger than $2^k, $ for some integer $k\geq 3.$ A representation of a non-negative integer $t$ is a sequence of non-negative integers $a_1,a_2,\cdots ,a_n$ such that $t=a_1u_1+a_2u_2+\cdots +a_nu_n.$
Prove that if a non-negative integer $t$ has a representation,then it also has a representation where less than $2k$ of numbers $a_1,a_2,\cdots ,a_n$ are non-zero.
Let $A$ be a real $n\times n$ matrix such that $A^3=0$
a) prove that there is unique real $n\times n$ matrix $X$ that satisfied the equation
$X+AX+XA^2=A$
b) Express $X$ in terms of $A$
Let $\vec{v_1},\vec{v_2},\vec{v_3},\vec{v_4}$ and $\vec{v_5}$ be vectors in three dimensions. Show that for some $i,j$ in $1,2,3,4,5$, $\vec{v_i}\cdot \vec{v_j}\ge 0$.
Let $ \mathcal{P}$ be a convex polygon with $ n$ sides, $ n\ge3$. Any set of $ n \minus{} 3$ diagonals of $ \mathcal{P}$ that do not intersect in the interior of the polygon determine a [i]triangulation[/i] of $ \mathcal{P}$ into $ n \minus{} 2$ triangles. If $ \mathcal{P}$ is regular and there is a triangulation of $ \mathcal{P}$ consisting of only isosceles triangles, find all the possible values of $ n$.
Call a polynomial $f(x)$ [i]excellent[/i] if its coefficients are all in [0, 1) and $f(x)$ is an integer for all integers $x$.
a) Compute the number of excellent polynomials with degree at most 3.
b) Compute the number of excellent polynomials with degree at most $n$, in terms of $n$.
c) Find the minimum $n\ge3$ for which there exists an excellent polynomial of the form $\frac{1}{n!}x^n+g(x)$, where $g(x)$ is a polynomial of degree at most $n-3$.
Determine all functions $f: \mathbb{Q} \to \mathbb{Q}$ such that
$$f(2xy + \frac{1}{2}) + f(x-y) = 4f(x)f(y) + \frac{1}{2}$$
for all $x,y \in \mathbb{Q}$.
Show that there are infinitely many rational triples $(a, b, c)$ such that $$a + b + c = abc = 6.$$
Prove that for any positive integer $n$, \[\left\lfloor \frac{n}{3}\right\rfloor+\left\lfloor \frac{n+2}{6}\right\rfloor+\left\lfloor \frac{n+4}{6}\right\rfloor = \left\lfloor \frac{n}{2}\right\rfloor+\left\lfloor \frac{n+3}{6}\right\rfloor .\]
Let $m$ and $n$ be positive integers with $\gcd(m, n) = 1$, and let
\[a_k = \left\lfloor \frac{mk}{n} \right\rfloor - \left\lfloor \frac{m(k-1)}{n} \right\rfloor\]
for $k = 1, 2, \dots, n$. Suppose that $g$ and $h$ are elements in a group $G$ and that
\[gh^{a_1} gh^{a_2} \cdots gh^{a_n} = e,\]
where $e$ is the identity element. Show that $gh = hg$. (As usual, $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$.)
The set of positive integers is partitioned into finitely many subsets. Show that some subset $S$ has the following property: for every positive integer $n$, $S$ contains infinitely many multiples of $n$.
Consider those functions $ f: \mathbb{N} \mapsto \mathbb{N}$ which satisfy the condition
\[ f(m \plus{} n) \geq f(m) \plus{} f(f(n)) \minus{} 1
\]
for all $ m,n \in \mathbb{N}.$ Find all possible values of $ f(2007).$
[i]Author: Nikolai Nikolov, Bulgaria[/i]
Let $\phi(n)$ be the number of positive integers less than $n$ that are relatively prime to $n$, where $n$ is a positive integer. Find all pairs of positive integers $(m,n)$ such that \[2^n + (n-\phi(n)-1)! = n^m+1.\]
Let $\mathbb R$ be the set of real numbers. Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ where there exist a real constant $c\ge 0$ such that $$x^3+y^2f(y)+zf(z^2)\ge cf(xyz)$$ holds for all reals $x$, $y$, $z$ that satisfy $x+y+z\ge 0$.
[i]Proposed by Ivan Chan Kai Chin[/i]
Given an integer $n\ge 2$. Prove that there only exist a finite number of n-tuples of positive integers $(a_1,a_2,\ldots,a_n)$ which simultaneously satisfy the following three conditions:
[list]
[*] $a_1>a_2>\ldots>a_n$;
[*] $\gcd (a_1,a_2,\ldots,a_n)=1$;
[*] $a_1=\sum_{i=1}^{n}\gcd (a_i,a_{i+1})$,where $a_{n+1}=a_1$.[/list]
Consider a $n$-sided polygon inscribed in a circle ($n \geq 4$). Partition the polygon into $n-2$ triangles using [b]non-intersecting[/b] diagnols. Prove that, irrespective of the triangulation, the sum of the in-radii of the triangles is a constant.
The sequence $\{x_{n}\}$ is defined by \[x_{0}\in [0, 1], \; x_{n+1}=1-\vert 1-2 x_{n}\vert.\] Prove that the sequence is periodic if and only if $x_{0}$ is irrational.
Let $f: \mathbb{R} \to \mathbb{R}$ be an infinitely differentiable function satisfying $f(0) = 0$, $f(1) = 1$, and $f(x) \ge 0$ for all $x \in \mathbb{R}$. Show that there exist a positive integer $n$ and a real number $x$ such that $f^{(n)}(x) < 0$.
Find all functions $f:\mathbb{N}\rightarrow\mathbb{N}$ such that for every positive integer $n$ the following is valid: If $d_1,d_2,\ldots,d_s$ are all the positive divisors of $n$, then $$f(d_1)f(d_2)\ldots f(d_s)=n.$$