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, 15 minutes 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

PoP+Parallel

by Solilin, Apr 5, 2025, 6:10 AM

Let ABC be a triangle and let D, E, F be the feet of the altitudes, with D on BC, E on CA, and F on AB. Let the parallel through D to EF meet AB at X and AC at Y. Let T be the intersection of EF with BC and let M be the midpoint of side BC. Prove that the points T, M, X, Y are concyclic.

Range of ab + bc + ca

by bamboozled, Apr 5, 2025, 2:12 AM

Let $(a^2+1)(b^2+1)(c^2+1) = 9$ , where $a, b, c \in R$ , then the number of integers in the range of $ab + bc + ca$ is __

inequalities

Functional Equation

by AnhQuang_67, Apr 4, 2025, 4:50 PM

Find all functions $f: \mathbb{R} \to \mathbb{R}$ satisfying $2\cdot f\Big(\dfrac{-xy}{2}+f(x+y)\Big)=xf(y)+yf(x), \forall x, y \in \mathbb{R}$

This post has been edited 1 time. Last edited by AnhQuang_67, Today at 2:25 AM
Reason: oops my bad

function

algebra

functional equation

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

gcd of coefficients of polynomial

by QueenArwen, Mar 11, 2025, 11:18 AM

Given a polynomial with integer coefficients, which has at least one integer root. The greatest common divisor of all its integer roots equals $1$ . Prove that if the leading coefficient of the polynomial equals $1$ then the greatest common divisor of the other coefficients also equals $1$ .

algebra

ToT

number theory

greatest common divisor

polynomial

Chessboard pattern

by BR1F1SZ, Dec 27, 2024, 6:41 PM

In a $100 \times 100$ board, each square is colored either white or black, with all the squares on the border of the board being black. Additionally, no $2 \times 2$ square within the board has all four squares of the same color. Prove that the board contains a $2 \times 2$ square colored like a chessboard.

combinatorics

Special line through antipodal

by Phorphyrion, Oct 28, 2024, 8:05 PM

Triangle $\triangle ABC$ is inscribed in circle $\Omega$ . Let $I$ denote its incenter and $I_A$ its $A$ -excenter. Let $N$ denote the midpoint of arc $BAC$ . Line $NI_A$ meets $\Omega$ a second time at $T$ . The perpendicular to $AI$ at $I$ meets sides $AC$ and $AB$ at $E$ and $F$ respectively. The circumcircle of $\triangle BFT$ meets $BI_A$ a second time at $P$ , and the circumcircle of $\triangle CET$ meets $CI_A$ a second time at $Q$ . Prove that $PQ$ passes through the antipodal to $A$ on $\Omega$ .