A refinement of IMO SHL 2002

Four real numbers $p$ , $q$ , $r$ , $s$ satisfy $p+q+r+s = 9$ and $p^{2}+q^{2}+r^{2}+s^{2}= 21$ . Prove that there exists a permutation $\left(a,b,c,d\right)$ of $\left(p,q,r,s\right)$ such that $ab-cd \geq 2$ .

28 replies

who

Jul 8, 2006

asdf334

13 minutes ago

chat gpt

fuv870 2

N an hour ago by fuv870

The chat gpt alreadly knows how to solve the problem of IMO USAMO and AMC?

2 replies

fuv870

an hour ago

fuv870

an hour ago

Inequality with wx + xy + yz + zw = 1

Fermat -Euler 23

N an hour ago by hgomamogh

Source: IMO ShortList 1990, Problem 24 (THA 2)

Let $w, x, y, z$ are non-negative reals such that $wx + xy + yz + zw = 1$ .
Show that $\frac {w^3}{x + y + z} + \frac {x^3}{w + y + z} + \frac {y^3}{w + x + z} + \frac {z^3}{w + x + y}\geq \frac {1}{3}$ .

23 replies

Fermat -Euler

Nov 2, 2005

hgomamogh

an hour ago

Waiting for a dm saying me again "old geometry"

drmzjoseph 0

2 hours ago

Source: Idk easy

Given $ABCD$ a tangencial quadrilateral that is not a rhombus, let $a,b,c,d$ be lengths of tangents from $A,B,C,D$ to the incircle of the quadrilateral which center is $I$ . Let $M,N$ be the midpoints of $AC,BD$ resp. Prove that
$\frac{MI}{IN}=\frac{a+c}{b+d}$

0 replies

drmzjoseph

2 hours ago

0 replies

Finally hard NT on UKR MO from NT master

mshtand1 2

N 2 hours ago by IAmTheHazard

Source: Ukrainian Mathematical Olympiad 2025. Day 1, Problem 11.4

A pair of positive integer numbers $(a, b)$ is given. It turns out that for every positive integer number $n$ , for which the numbers $(n - a)(n + b)$ and $n^2 - ab$ are positive, they have the same number of divisors. Is it necessarily true that $a = b$ ?

Proposed by Oleksii Masalitin

2 replies

mshtand1

Mar 13, 2025

IAmTheHazard

2 hours ago

IMOC 2017 G5 (<A=120 => E, F, Y,Z are concyclic, incenter related)

parmenides51 4

N 2 hours ago by ehuseyinyigit

Source: https://artofproblemsolving.com/community/c6h1740077p11309077

We have $\vartriangle ABC$ with $I$ as its incenter. Let $D$ be the intersection of $AI$ and $BC$ and define $E, F$ in a similar way. Furthermore, let $Y = CI \cap DE, Z = BI \cap DF$ . Prove that if $\angle BAC = 120^o$ , then $E, F, Y,Z$ are concyclic.
IMAGE

4 replies

parmenides51

Mar 20, 2020

ehuseyinyigit

2 hours ago

Bosnia and Herzegovina JBMO TST 2013 Problem 1

gobathegreat 3

N 3 hours ago by DensSv

Source: Bosnia and Herzegovina Junior Balkan Mathematical Olympiad TST 2013

It is given $n$ positive integers. Product of any one of them with sum of remaining numbers increased by $1$ is divisible with sum of all $n$ numbers. Prove that sum of squares of all $n$ numbers is divisible with sum of all $n$ numbers

3 replies

gobathegreat

Sep 16, 2018

DensSv

3 hours ago

D1015 : A strange EF for polynomials

Dattier 0

3 hours ago

Source: les dattes à Dattier

Find all $P \in \mathbb R[x,y]$ with $P \not\in \mathbb R[x] \cup \mathbb R[y]$ and $\forall g,f$ homeomorphismes of $\mathbb R$ , $P(f,g)$ is an homoemorphisme too.

0 replies

Dattier

3 hours ago

0 replies

P, Q,R collinear and U, R, O, V concyclic wanted, cyclic ABCD, circumcenters

parmenides51 2

N 3 hours ago by DensSv

Source: 2012 Romania JBMO TST2 P4

The quadrilateral $ABCD$ is inscribed in a circle centered at $O$ , and $\{P\} = AC \cap BD, \{Q\} = AB \cap CD$ . Let $R$ be the second intersection point of the circumcircles of the triangles $ABP$ and $CDP$ .
a) Prove that the points $P, Q$ , and $R$ are collinear.
b) If $U$ and $V$ are the circumcenters of the triangles $ABP$ , and $CDP$ , respectively, prove that the points $U, R, O, V$ are concyclic.

2 replies

parmenides51

May 29, 2020

DensSv

3 hours ago

Unsolved Diophantine(I think)

Nuran2010 1

N 3 hours ago by Nuran2010

Find all solutions for the equation $2^n=p+3^p$ where $n$ is a positive integer and $p$ is a prime.(Don't get mad at me,I've used the search function and did not see a correct and complete solution anywhere.)

1 reply

Nuran2010

Mar 14, 2025

Nuran2010

3 hours ago

2^a + 3^b + 1 = 6^c

togrulhamidli2011 1

N 3 hours ago by CM1910

Find all positive integers (a, b, c) such that:

$2^a + 3^b + 1 = 6^c$

1 reply

togrulhamidli2011

Today at 12:34 PM

CM1910

3 hours ago

A refinement of IMO SHL 2002

mihaig 2

N Nov 19, 2021 by mihaig

Source: Own

Let $ABC$ be a triangle for which there exists an interior point $F$ such that $\angle AFB=\angle BFC=\angle CFA$ . Let the lines $BF$ and $CF$ meet the sides $AC$ and $AB$ at $D$ and $E$ respectively. Prove that
$\sqrt{AB\cdot AC}\geq2DE.$