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Functional xf(x+f(y))=(y-x)f(f(x)) for all reals x,y
cretanman   65
N 37 minutes ago by youochange
Source: BMO 2023 Problem 1
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
65 replies
cretanman
May 10, 2023
youochange
37 minutes ago
3 var inequality
ehuseyinyigit   1
N 44 minutes ago by ehuseyinyigit
Source: Own
Let $x,y,z$ be positive real numbers. Prove that

$$\dfrac{x^3+72xy^2}{z^3+x^2y}+\dfrac{y^3+72yz^2}{x^3+y^2z}+\dfrac{z^3+72zx^2}{y^3+z^2x}\geq \dfrac{15}{2}+\dfrac{102xyz(x+y+z)}{x^3y+y^3z+z^3x}$$
1 reply
ehuseyinyigit
4 hours ago
ehuseyinyigit
44 minutes ago
trig manipulation with sum of sines identity
ACalculationError   0
an hour ago
Source: PMO 2018 Qualifying Stage Part I. 10
Problem Statement: In triangle $\triangle ABC$, suppose
$$5 \sin A + 12 \cos B = 15, \quad 12 \sin B + 5 \cos A = 2.$$What is the measure of angle $C$?
Answer Confirmation
Solution

0 replies
ACalculationError
an hour ago
0 replies
Periodic sequence
EeEeRUT   9
N an hour ago by Supertinito
Source: Isl 2024 A5
Find all periodic sequence $a_1,a_2,\dots$ of real numbers such that the following conditions hold for all $n\geqslant 1$:$$a_{n+2}+a_{n}^2=a_n+a_{n+1}^2\quad\text{and}\quad |a_{n+1}-a_n|\leqslant 1.$$
Proposed by Dorlir Ahmeti, Kosovo
9 replies
EeEeRUT
Jul 16, 2025
Supertinito
an hour ago
trig basic identities
ACalculationError   0
an hour ago
Source: Sipnayan 2017 Junior High School Average #1
Problem Statement: Let $x,y\in[0,\frac{\pi}{2}]$ satisfy $\sin x = \frac{5}{13}$ and $\sin y = \frac{15}{17}$. Find $\tan(x+y)$
Answer Confirmation
Solution
0 replies
ACalculationError
an hour ago
0 replies
I am [not] a parallelogram
peppapig_   18
N an hour ago by endless_abyss
Source: ISL 2024/G4
Let $ABCD$ be a quadrilateral with $AB$ parallel to $CD$ and $AB<CD$. Lines $AD$ and $BC$ intersect at a point $P$. Point $X$ distinct from $C$ lies on the circumcircle of triangle $ABC$ such that $PC=PX$. Point $Y$ distinct from $D$ lies on the circumcircle of triangle $ABD$ such that $PD=PY$. Lines $AX$ and $BY$ intersect at $Q$.

Prove that $PQ$ is parallel to $AB$.

Fedir Yudin, Mykhailo Shtandenko, Anton Trygub, Ukraine
18 replies
peppapig_
Jul 16, 2025
endless_abyss
an hour ago
Problem 3 IMO 2005 (Day 1)
Valentin Vornicu   125
N an hour ago by blueprimes
Let $x,y,z$ be three positive reals such that $xyz\geq 1$. Prove that
\[ \frac { x^5-x^2 }{x^5+y^2+z^2} + \frac {y^5-y^2}{x^2+y^5+z^2} + \frac {z^5-z^2}{x^2+y^2+z^5} \geq 0 . \]
Hojoo Lee, Korea
125 replies
Valentin Vornicu
Jul 13, 2005
blueprimes
an hour ago
An easy symmetric inequality
seoneo   14
N an hour ago by blueprimes
Source: kjmo 2012 pr 1
Prove the following inequality where positive reals $a$, $b$, $c$ satisfies $ab+bc+ca=1$.
\[
        \frac{a+b}{\sqrt{ab(1-ab)}} + \frac{b+c}{\sqrt{bc(1-bc)}} + \frac{c+a}{\sqrt{ca(1-ca)}} \le \frac{\sqrt{2}}{abc}
    \]
14 replies
seoneo
Sep 21, 2017
blueprimes
an hour ago
A functional equation
joybangla   14
N an hour ago by Lyte188
Source: Switzerland Math Olympiad, Final round 2014, P3
Find all such functions $f :\mathbb{R}\to \mathbb{R}$ such that for all $x,y\in\mathbb{R}$ the following holds :
\[ f(x^2)+f(xy)=f(x)f(y)+yf(x)+xf(x+y) \]
14 replies
joybangla
Jun 2, 2014
Lyte188
an hour ago
Rectangle EFGH in incircle, prove that QIM = 90
v_Enhance   70
N an hour ago by hectorleo123
Source: Taiwan 2014 TST1, Problem 3
Let $ABC$ be a triangle with incenter $I$, and suppose the incircle is tangent to $CA$ and $AB$ at $E$ and $F$. Denote by $G$ and $H$ the reflections of $E$ and $F$ over $I$. Let $Q$ be the intersection of $BC$ with $GH$, and let $M$ be the midpoint of $BC$. Prove that $IQ$ and $IM$ are perpendicular.
70 replies
v_Enhance
Jul 18, 2014
hectorleo123
an hour ago
a