Integer polynomial

by luutrongphuc, Apr 5, 2025, 8:25 AM

For every integer $n \geq 3$ , let $S_n$ be the set of all positive integers not exceeding $n$ that are relatively prime to $n$ . Consider the polynomial
$P_n(x) = \sum_{k \in S_n} x^{k - 1}$
a) Prove that there exists a positive integer $r_n$ and a polynomial $Q_n(x)$ with integer coefficients such that
$P_n(x) = (x^{r_n} + 1) Q_n(x).$
b)Find all integers $n$ such that $P_n(x)$ is irreducible in $\mathbb{Z}[x]$ .

This post has been edited 2 times. Last edited by luutrongphuc, 4 hours ago

Integer Polynomial

algebra

polynomial

Uhhhhhhhhhh

by sealight2107, Apr 5, 2025, 8:24 AM

Let $x,y,z$ be reals such that $0<x,y,z<\frac{1}{2}$ and $x+y+z=1$ .Prove that:
$4(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}) - \frac{1}{xyz} >8$

inequalities

Prime number and composite number

by mingzhehu, Apr 5, 2025, 8:06 AM

I have one topic on how to identify Prime Number and Composite Number quickly? Maybe the number is more than 100 or 1000.......!
If there are some formula that can be used to verify the number easily, it will be highly appreciated.
Does anybody has any good idea for that?

number theory

prime numbers

a hard geometry problen

by Tuguldur, Apr 4, 2025, 3:56 PM

Let $ABCD$ be a convex quadrilateral. Suppose that the circles with diameters $AB$ and $CD$ intersect at points $X$ and $Y$ . Let $P=AC\cap BD$ and $Q=AD\cap BC$ . Prove that the points $P$ , $Q$ , $X$ and $Y$ are concyclic.
( $AB$ and $CD$ are not the diagnols)

geometry

Regarding Maaths olympiad prepration

by omega2007, Apr 4, 2025, 3:13 PM

<Hey Everyone'>
I'm 10 grader student and Im starting prepration for maths olympiad..>>> From scratch (not 2+2=4 )

Do you haves compiled resources of Handouts,
PDF,
Links,
List of books topic wise
which are shared on AOPS (and from your perspective) for maths olympiad and any useful thing, which will help me in boosting Maths olympiad prepration.

This post has been edited 1 time. Last edited by omega2007, Today at 2:48 AM
Reason: Spelling error

Problem 6

by blug, Apr 4, 2025, 12:17 PM

A strictly decreasing function $f:(0, \infty)\Rightarrow (0, \infty)$ attaining all positive values and positive numbers $a_1\ne b_1$ are given. Numbers $a_2, b_2, a_3, b_3, ...$ satisfy
$a_{n+1}=a_n+f(b_n),\;\;\;\;\;\;\;b_{n+1}=b_n+f(a_n)$ for every $n\geq 1$ . Prove that there exists a positive integer $n$ satisfying $|a_n-b_n| >2025$ .

algebra

functions

function

Problem 4

by blug, Apr 4, 2025, 11:59 AM

A positive integer $n\geq 2$ and a set $S$ consisting of $2n$ disting positive integers smaller than $n^2$ are given. Prove that there exists a positive integer $r\in \{1, 2, ..., n\}$ that can be written in the form $r=a-b$ , for $a, b\in \mathbb{S}$ in at least $3$ different ways.

combinatorics

Problem 3

by blug, Apr 4, 2025, 11:55 AM

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.

combinatorics

Problem 2

by blug, Apr 4, 2025, 11:49 AM

Positive integers $k, m, n ,p$ integers are such that $p=2^{2^n}+1$ is prime and $p\mid 2^k-m$ . Prove that there exists a positive integer $l$ such that $p^2\mid 2^l-m$ .

number theory

prime numbers

D1010 : How it is possible ?

by Dattier, Mar 10, 2025, 10:49 AM

Is it true that $\forall n \in \mathbb N^*, (24^n \times B \mod A) \mod 2 = 0$ ?

A=1728400904217815186787639216753921417860004366580219212750904
024377969478249664644267971025952530803647043121025959018172048
336953969062151534282052863307398281681465366665810775710867856
720572225880311472925624694183944650261079955759251769111321319
421445397848518597584590900951222557860592579005088853698315463
815905425095325508106272375728975

B=2275643401548081847207782760491442295266487354750527085289354
965376765188468052271190172787064418854789322484305145310707614
546573398182642923893780527037224143380886260467760991228567577
953725945090125797351518670892779468968705801340068681556238850
340398780828104506916965606659768601942798676554332768254089685
307970609932846902

This post has been edited 6 times. Last edited by Dattier, Mar 16, 2025, 10:10 AM

number theory

Computer solutions