Reducibility of 2x^2 cyclotomic

by vincentwant, Apr 30, 2025, 3:50 PM

Let $S$ denote the set of all positive integers less than $1020$ that are relatively prime to $1020$ . Let $\omega=\cos\frac{\pi}{510}+i\sin\frac{\pi}{510}$ . Is the polynomial $\prod_{n\in S}(2x^2-\omega^n)$ reducible over the rational numbers, given that it has integer coefficients?

This post has been edited 1 time. Last edited by vincentwant, 3 hours ago

algebra

Very easy NT

by GreekIdiot, Apr 30, 2025, 2:49 PM

Prove that there exists no natural number $n>1$ such that $n \mid 2^n-1$ .

number theory

bezout

Elementary

Fermat s Little Theorem

Azer and Babek playing a game on a chessboard

by Nuran2010, Apr 29, 2025, 5:03 PM

Azer and Babek have a $8 \times 8$ chessboard. Initially, Azer colors all cells of this chessboard with some colors. Then, Babek takes $2$ rows and $2$ columns and looks at the $4$ cells in the intersection. Babek wants to have all these $4$ cells in a same color, but Azer doesn't. With at least how many colors, Azer can reach his goal?

This post has been edited 1 time. Last edited by Nuran2010, Yesterday at 6:16 PM

combinatorics

Coloring

Chessboard

Hard inequality

by JK1603JK, Apr 29, 2025, 4:24 AM

Let $a,b,c>0$ and $a^2+b^2+c^2=2(a+b+c).$ Find the minimum $P=(a+b+c)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)$

inequalities

Geometric inequality with Fermat point

by Assassino9931, Apr 27, 2025, 10:21 PM

Let $ABC$ be an acute triangle and let $P$ be an interior point for it such that $\angle APB = \angle BPC = \angle CPA$ . Prove that
$\frac{PA^2 + PB^2 + PC^2}{2S} + \frac{4}{\sqrt{3}} \leq \frac{1}{\sin \alpha} + \frac{1}{\sin \beta} + \frac{1}{\sin \gamma}.$ When does equality hold?

inequalities proposed

Something nice

by KhuongTrang, Nov 1, 2023, 12:56 PM

Problem. Given $a,b,c$ be non-negative real numbers such that $ab+bc+ca=1.$ Prove that

$\sqrt{a+1}+\sqrt{b+1}+\sqrt{c+1}\le 1+2\sqrt{a+b+c+abc}.$

This post has been edited 2 times. Last edited by KhuongTrang, Nov 19, 2023, 11:59 PM

inequalities

Straight-edge is a social construct

by anantmudgal09, Jan 15, 2023, 11:41 AM

Euclid has a tool called cyclos which allows him to do the following:

Given three non-collinear marked points, draw the circle passing through them.
Given two marked points, draw the circle with them as endpoints of a diameter.
Mark any intersection points of two drawn circles or mark a new point on a drawn circle.

Show that given two marked points, Euclid can draw a circle centered at one of them and passing through the other, using only the cyclos.

Proposed by Rohan Goyal, Anant Mudgal, and Daniel Hu

This post has been edited 1 time. Last edited by anantmudgal09, Jan 15, 2023, 11:43 AM

An I for an I

by Eyed, Jul 20, 2021, 8:54 PM

Let $ABC$ be a triangle with incenter $I$ and circumcircle $\Gamma$ . Circles $\omega_{B}$ passing through $B$ and $\omega_{C}$ passing through $C$ are tangent at $I$ . Let $\omega_{B}$ meet minor arc $AB$ of $\Gamma$ at $P$ and $AB$ at $M\neq B$ , and let $\omega_{C}$ meet minor arc $AC$ of $\Gamma$ at $Q$ and $AC$ at $N\neq C$ . Rays $PM$ and $QN$ meet at $X$ . Let $Y$ be a point such that $YB$ is tangent to $\omega_{B}$ and $YC$ is tangent to $\omega_{C}$ .

Show that $A,X,Y$ are collinear.

This post has been edited 1 time. Last edited by Eyed, Jul 20, 2021, 9:59 PM

Weighted Blocks

by ilovemath04, Sep 22, 2020, 11:48 PM

You are given a set of $n$ blocks, each weighing at least $1$ ; their total weight is $2n$ . Prove that for every real number $r$ with $0 \leq r \leq 2n-2$ you can choose a subset of the blocks whose total weight is at least $r$ but at most $r + 2$ .