My Unsolved FE in R+

by ZeltaQN2008, May 6, 2025, 3:17 AM

Find all functions $f:\mathbb{R}^+ \rightarrow \mathbb{R}^+$ such that for all any $x,y\in (0,\infty):$
$f(1+xf(y))=yf(x+y)$

This post has been edited 1 time. Last edited by ZeltaQN2008, 2 hours ago
Reason: Wrong source

function

algebra

Inspired by Austria 2025

by sqing, May 6, 2025, 2:01 AM

Let $a,b\geq 0 ,a,b\neq 1$ and $a^2+b^2=1.$ Prove that $(a + b ) \left( \frac{a}{(b -1)^2} + \frac{b}{(a - 1)^2} \right) \geq 12+8\sqrt 2$

inequalities proposed

algebra

IMO Genre Predictions

by ohiorizzler1434, May 3, 2025, 6:51 AM

Everybody, with IMO upcoming, what are you predictions for the problem genres?

Personally I predict: predict

Parallelograms and concyclicity

by Lukaluce, Apr 14, 2025, 10:59 AM

Let $ABC$ be an acute triangle with incentre $I$ and $AB \neq AC$ . Let lines $BI$ and $CI$ intersect the circumcircle of $ABC$ at $P \neq B$ and $Q \neq C$ , respectively. Consider points $R$ and $S$ such that $AQRB$ and $ACSP$ are parallelograms (with $AQ \parallel RB, AB \parallel QR, AC \parallel SP$ , and $AP \parallel CS$ ). Let $T$ be the point of intersection of lines $RB$ and $SC$ . Prove that points $R, S, T$ , and $I$ are concyclic.

Inequality with a,b,c

by GeoMorocco, Apr 11, 2025, 10:05 PM

Let $a,b,c$ be positive real numbers such that : $ab+bc+ca=3$ . Prove that : $\frac{\sqrt{1+a^2}}{1+ab}+\frac{\sqrt{1+b^2}}{1+bc}+\frac{\sqrt{1+c^2}}{1+ca}\ge \sqrt{\frac{3(a+b+c)}{2}}$

inequalities

Property of the divisors of k^3 - 2

by Scilyse, Jan 13, 2025, 9:36 AM

Given two integers, $k$ and $d$ such that $d$ divides $k^3 - 2$ . Show that there exists integers $a$ , $b$ , $c$ satisfying $d = a^3 + 2b^3 + 4c^3 - 6abc$ .

Proposed by Csongor Beke and László Bence Simon, Cambridge

This post has been edited 1 time. Last edited by Scilyse, Jan 13, 2025, 1:34 PM

number theory

komal

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

Concurrency from isogonal Mittenpunkt configuration

by MarkBcc168, Apr 28, 2020, 7:07 AM

Let $\triangle ABC$ be a scalene triangle with circumcenter $O$ , incenter $I$ , and incircle $\omega$ . Let $\omega$ touch the sides $\overline{BC}$ , $\overline{CA}$ , and $\overline{AB}$ at points $D$ , $E$ , and $F$ respectively. Let $T$ be the projection of $D$ to $\overline{EF}$ . The line $AT$ intersects the circumcircle of $\triangle ABC$ again at point $X\ne A$ . The circumcircles of $\triangle AEX$ and $\triangle AFX$ intersect $\omega$ again at points $P\ne E$ and $Q\ne F$ respectively. Prove that the lines $\text{[math]}$ , $FP$ , and $OI$ are concurrent.

Proposed by MarkBcc168.

This post has been edited 1 time. Last edited by MarkBcc168, Apr 28, 2020, 7:08 AM

geometry

Infimum of decreasing sequence b_n/n^2

by a1267ab, Dec 16, 2019, 5:08 PM

Choose positive integers $b_1, b_2, \dotsc$ satisfying
$1=\frac{b_1}{1^2} > \frac{b_2}{2^2} > \frac{b_3}{3^2} > \frac{b_4}{4^2} > \dotsb$ and let $r$ denote the largest real number satisfying $\tfrac{b_n}{n^2} \geq r$ for all positive integers $n$ . What are the possible values of $r$ across all possible choices of the sequence $(b_n)$ ?

Carl Schildkraut and Milan Haiman

This post has been edited 3 times. Last edited by a1267ab, Dec 16, 2019, 6:11 PM

algebra

\sqrt{a^2+b^2+2}+\sqrt{b^2+c^2+2 }+\sqrt{c^2+a^2+2}\ge 6

by parmenides51, Jul 25, 2018, 2:53 PM

Let $a, b, c$ be positive real numbers such that $a + b + c + ab + bc + ca + abc = 7$ . Prove
that $\sqrt{a^2 + b^2 + 2 }+\sqrt{b^2 + c^2 + 2 }+\sqrt{c^2 + a^2 + 2 } \ge 6$ .

algebra

JBMO Shortlist

Inequality

Robots in Space

by rrusczyk, Aug 25, 2011, 1:00 AM

I should have shared this sooner; crossposted from the AoPS blog:

AoPS blog wrote:

The folks at TopCoder are working with MIT, NASA, and DARPA to offer a robotics/programming competition for high school students. Winning teams may have their programs run by astronauts on the International Space Station! More details here. The deadline for registering is September 5.

contests

programming

robotics