weird symmetric equation

by giangtruong13, Apr 27, 2025, 4:29 AM

Solve the equation: $8x^2-11x+1=(1-x)\sqrt{4x^2-6x+5}$

Interesting inequalities

by sqing, Apr 27, 2025, 4:16 AM

Let $a,b\geq 0$ and $a+b=2.$ Prove that
$ab( a^2+ b^2)^2 \leq \frac{128}{27}$ $ab( a^2-ab+ b^2)^2 \leq \frac{256}{81}$ $ab\sqrt{ab}( a^2+ b^2)^2 \leq \frac{1536}{343}\sqrt{\frac{6}{7}}$ $ab\sqrt{ab}( a^2-ab+ b^2)^2 \leq \frac{2048}{343\sqrt{7}}$

inequalities proposed

inequalities

Interesting inequalities

by sqing, Apr 27, 2025, 3:12 AM

Let $a,b\geq 0$ and $a+b+ab=3.$ Prove that
$ab^2( b +1) \leq 4$ $ab( b +1) \leq \frac{9}{4}$ $a^2b ( a+b^2 ) \leq \frac{76}{27}$ $a^2b( b +1 ) \leq \frac{3(69-11\sqrt{33})}{8}$ $a^2b^2( b +1 ) \leq \frac{2(73\sqrt{73}-595)}{27}$

This post has been edited 1 time. Last edited by sqing, 6 hours ago

inequalities proposed

inequalities

easy geo

by ErTeeEs06, Apr 26, 2025, 11:13 AM

Let $ABC$ be a triangle with incentre $I$ and circumcircle $\Omega$ . Let $D, E, F$ be the midpoints of the arcs $\stackrel{\frown}{BC}, \stackrel{\frown}{CA}, \stackrel{\frown}{AB}$ of $\Omega$ not containing $A, B, C$ respectively. Let $D'$ be the point of $\Omega$ diametrically opposite to $D$ . Show that $I, D'$ and the midpoint $M$ of $EF$ lie on a line.

This post has been edited 1 time. Last edited by ErTeeEs06, Yesterday at 11:15 AM
Reason: latex

Benelux fe

by ErTeeEs06, Apr 26, 2025, 11:05 AM

Does there exist a function $f:\mathbb{R}\to \mathbb{R}$ such that $f(x^2+f(y))=f(x)^2-y$ for all $x, y\in \mathbb{R}$ ?

This post has been edited 1 time. Last edited by ErTeeEs06, Yesterday at 11:05 AM
Reason: latex mistake

function

algebra

BxMO

easy functional

by B1t, Apr 26, 2025, 6:45 AM

Denote the set of real numbers by $\mathbb{R}$ . Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that for all $x, y, z \in \mathbb{R}$ ,
$f(xf(x+y)+z) = f(z) + f(x)y + f(xf(x)).$

This post has been edited 2 times. Last edited by B1t, Yesterday at 7:01 AM

algebra

P(x) | P(x^2-2)

by GreenTea2593, Apr 22, 2025, 3:27 AM

Let $P(x)$ be a monic polynomial with complex coefficients such that there exist a polynomial $Q(x)$ with complex coefficients for which $P(x^2-2)=P(x)Q(x).$ Determine all complex numbers that could be the root of $P(x)$ .

Proposed by Valentio Iverson, Indonesia

This post has been edited 1 time. Last edited by GreenTea2593, Apr 22, 2025, 3:27 AM

polynomial

algebra

2024 IMO P1

by EthanWYX2009, Jul 16, 2024, 1:13 PM

Determine all real numbers $\alpha$ such that, for every positive integer $n,$ the integer
$\lfloor\alpha\rfloor +\lfloor 2\alpha\rfloor +\cdots +\lfloor n\alpha\rfloor$ is a multiple of $n.$ (Note that $\lfloor z\rfloor$ denotes the greatest integer less than or equal to $z.$ For example, $\lfloor -\pi\rfloor =-4$ and $\lfloor 2\rfloor= \lfloor 2.9\rfloor =2.$ )

Proposed by Santiago Rodríguez, Colombia

This post has been edited 2 times. Last edited by EthanWYX2009, Jul 19, 2024, 5:33 AM
Reason: change to original text

algebra

IMO

Functional xf(x+f(y))=(y-x)f(f(x)) for all reals x,y

by cretanman, May 10, 2023, 3:50 PM

Find all functions $f\colon \mathbb{R} \rightarrow \mathbb{R}$ such that for all $x,y \in \mathbb{R}$ ,
$xf(x+f(y))=(y-x)f(f(x)).$
Proposed by Nikola Velov, Macedonia

This post has been edited 4 times. Last edited by Amir Hossein, May 13, 2023, 1:00 AM
Reason: Fixed source

Integer polynomial commutes with sum of digits

by cjquines0, Jul 19, 2017, 4:38 PM

For any positive integer $k$ , denote the sum of digits of $k$ in its decimal representation by $S(k)$ . Find all polynomials $P(x)$ with integer coefficients such that for any positive integer $n \geq 2016$ , the integer $P(n)$ is positive and $S(P(n)) = P(S(n)).$
Proposed by Warut Suksompong, Thailand

This post has been edited 1 time. Last edited by Amir Hossein, Jul 19, 2017, 5:11 PM
Reason: Added the proposer.

Dealing with Hard Problems

by rrusczyk, Jan 15, 2012, 7:13 PM

A parent of one of our students wrote me an email today about his daughter's frustration with difficult problems. I started to write a reply, and it expanded into a new article about the importance of hard problems and how to deal with them in our articles section here.

articles