Maximum number of nice subsets

by FireBreathers, Apr 23, 2025, 10:27 PM

Given a set $M$ of natural numbers with $n$ elements with $n$ odd number. A nonempty subset $S$ of $M$ is called $nice$ if the product of the elements of $S$ divisible by the sum of the elements of $M$ , but not by its square. It is known that the set $M$ itself is good. Determine the maximum number of $nice$ subsets (including $M$ itself).

combinatorics

number thoery

Polynomial

by Z_., Apr 23, 2025, 9:21 PM

Let $m$ be an integer greater than zero. Then, the value of the sum of the reciprocals of the cubes of the roots of the equation
$mx^4 + 8x^3 - 139x^2 - 18x + 9 = 0$ is equal to:

This post has been edited 1 time. Last edited by Z_., 3 hours ago
Reason: .

Polynomials

algebra

polynomial

Inequalities

by Scientist10, Apr 23, 2025, 6:36 PM

If $x, y, z \in \mathbb{R}$ , then prove that the following inequality holds:
$\sum_{\text{cyc}} \sqrt{1 + \left(x\sqrt{1 + y^2} + y\sqrt{1 + x^2}\right)^2} \geq \sum_{\text{cyc}} xy + 2\sum_{\text{cyc}} x$

inequalities

interesting function equation (fe) in IR

by skellyrah, Apr 23, 2025, 9:51 AM

find all function F: IR->IR such that $xf(f(y)) + yf(f(x)) = f(xf(y)) + f(xy)$

Tangents forms triangle with two times less area

by NO_SQUARES, Apr 23, 2025, 9:08 AM

Let $DEF$ be triangle, inscribed in parabola. Tangents in points $D,E,F$ forms triangle $ABC$ . Prove that $S_{DEF}=2S_{ABC}$ . ( $S_T$ is area of triangle $T$ ).
From F.S.Macaulay's book «Geometrical Conics», suggested by M. Panov

Attachments:

geometry

conics

parabola

Existence of perfect squares

by egxa, Apr 18, 2025, 9:48 AM

Find all natural numbers $n$ for which there exists an even natural number $a$ such that the number
$(a - 1)(a^2 - 1)\cdots(a^n - 1)$ is a perfect square.

number theory

Perfect Squares

FE solution too simple?

by Yiyj1, Apr 9, 2025, 3:26 AM

Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that the equality $f(f(x)+y) = f(x^2-y)+4f(x)y$ holds for all pairs of real numbers $(x,y)$ .

My solution

I feel like my solution is too simple. Is there something I did wrong or something I missed?

Functional Equations

function

algebra

Number Theory

by fasttrust_12-mn, Aug 16, 2024, 10:21 AM

Find all integers $n$ for which $n^7-41$ is the square of an integer

number theory

PAMO 2024

Floor double summation

by CyclicISLscelesTrapezoid, Jul 12, 2022, 12:52 PM

Which positive integers $n$ make the equation $\sum_{i=1}^n \sum_{j=1}^n \left\lfloor \frac{ij}{n+1} \right\rfloor=\frac{n^2(n-1)}{4}$ true?

IMO 2014 Problem 4

by ipaper, Jul 9, 2014, 11:38 AM

Let $P$ and $Q$ be on segment $BC$ of an acute triangle $ABC$ such that $\angle PAB=\angle BCA$ and $\angle CAQ=\angle ABC$ . Let $M$ and $N$ be the points on $AP$ and $AQ$ , respectively, such that $P$ is the midpoint of $AM$ and $Q$ is the midpoint of $AN$ . Prove that the intersection of $BM$ and $CN$ is on the circumference of triangle $ABC$ .

Proposed by Giorgi Arabidze, Georgia.

barycentric coordinates

geometry solved

Submit

Hey, what a nice blog!!!!!
by Power_Set, Jun 24, 2018, 9:53 PM
Is your blog dead ? No new post in almost 2 months.

My blog has died due to malnutrition of problems.
by integrated_JRC, May 2, 2018, 5:52 AM
@ below, I will do it soon
by pro_4_ever, Mar 1, 2018, 10:37 AM
Hi, I think we should 'hide' our solutions, as it is difficult to go up and down the blog
by Drunken_Master, Mar 1, 2018, 10:36 AM
Yeah, I created a test blog, it seems to be an AoPS problem not CSS, can't do anything about it. I'll post it in site support forum later.
by Vrangr, Feb 20, 2018, 5:48 AM
The problem is not in the CSS, I believe, as the CSS only looks after the appearance of the blog, and not the $\text{\LaTeX}$ rendering.
by WizardMath, Feb 19, 2018, 11:34 PM
I'll fix the CSS after boards of need be (I'll probably be starting my own blog too then)
by Vrangr, Feb 19, 2018, 4:41 PM
@ below,
Pls ask WizardMath...
The CSS is taken from his amazing Blog, "An Olympiad Journey"...
I know nothing abt programming.
(BTW I got permission for taking the CSS. No Copyright Issues!)
by pro_4_ever, Feb 19, 2018, 4:16 PM
Huge flaw in website design: comment gets erased if while typing $\LaTeX$ has an error and needs to be fixed.
by Vrangr, Feb 19, 2018, 12:26 PM
Upon WizardMath's Request, let us all post synthetic solutions. Well, let's declare that this blog is intended to give the reader some good problems with simple synthetic solutions. All the best. Not trying to be rude,but I might delete upcoming Bashes
by pro_4_ever, Feb 13, 2018, 4:50 PM
Here's an idea. Ban all bashes on this blog from now on. Also try posting your own previous geo solutions as storage (I hope this is allowed). This would make the quality of the blog better. Bashes are basically write-once, read-never things. Why bother?
by WizardMath, Feb 13, 2018, 4:33 PM
Somebody type Egmo 2013 Problem 1.
No bash,purely troll problem.
by QWERTYphysics, Feb 13, 2018, 4:11 PM
I hate having to see a pure geometry blog being swamped to death with bashes.

Obviously a bash is a super terrible idea. Don't do one.
Unless you are bad at synthetic.
by WizardMath, Feb 13, 2018, 2:58 PM
@ayan Done:)
by pro_4_ever, Feb 13, 2018, 1:38 PM
9th shout
by MEGAKNIGHT, Feb 13, 2018, 11:23 AM
@below Pls don't post a bash if you are thinking about posting a new entry!
by pro_4_ever, Feb 13, 2018, 11:11 AM
Wait, I thought bash is frowned upon in this blog.
by WizardMath, Feb 12, 2018, 6:57 PM
Hmmm...
Combi Geometry is accepted...
by pro_4_ever, Feb 10, 2018, 7:32 AM
Can we post Combi here?
by ayan.nmath, Feb 10, 2018, 7:31 AM
Parag dey
by Paragdey12, Feb 6, 2018, 1:17 PM
More posts please
by AnArtist, Feb 6, 2018, 2:38 AM
Second Shout!
by ccx09, Feb 3, 2018, 9:20 PM
1st shout!
by AnArtist, Jan 31, 2018, 9:20 AM