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p + q + r + s = 9 and p^2 + q^2 + r^2 + s^2 = 21
who   28
N 13 minutes ago by asdf334
Source: IMO Shortlist 2005 problem A3
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
\[ \sqrt{AB\cdot AC}\geq2DE. \]
2 replies
mihaig
Jul 19, 2021
mihaig
Nov 19, 2021
A refinement of IMO SHL 2002
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G H BBookmark kLocked kLocked NReply
Source: Own
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mihaig
7339 posts
#1
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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
\[ \sqrt{AB\cdot AC}\geq2DE. \]
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mihaig
7339 posts
#2
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The problem is very easy
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mihaig
7339 posts
#3
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The bump
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N Quick Reply
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