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number theory diophantic with factorials and primes
skellyrah 4
N
by skellyrah
Source: by me
find all triplets of non negative integers (a,b,p) where p is prime such that 
4 replies
.![\[\sqrt{a+\frac{9}{b+2c}}+\sqrt{b+\frac{9}{c+2a}}+\sqrt{c+\frac{9}{a+2b}}+\frac{2(ab+bc+ca)}{9}\ge\frac{20}{3}\]](http://latex.artofproblemsolving.com/b/3/2/b32e0bcba967ca847e878ac186c3168cd6c5fc66.png)
primes,exponentials,factorials
skellyrah 6
N
by skellyrah
find all primes p,q such that 
is a prime number
is a prime number
6 replies
Serbian selection contest for the IMO 2025 - P5
OgnjenTesic 2
N
by GreenTea2593
Source: Serbian selection contest for the IMO 2025
Determine the smallest positive real number 
such that there exists a sequence of positive real numbers 
, 
, with the property that for every it holds that:
![\[
a_1 + \cdots + a_{n+1} < \alpha \cdot a_n.
\]](//latex.artofproblemsolving.com/9/b/7/9b73deb46032844b324288b02470ff0623ac0ff1.png)
Proposed by Pavle Martinović
such that there exists a sequence of positive real numbers 
, 
, with the property that for every it holds that:![\[
a_1 + \cdots + a_{n+1} < \alpha \cdot a_n.
\]](http://latex.artofproblemsolving.com/9/b/7/9b73deb46032844b324288b02470ff0623ac0ff1.png)
Proposed by Pavle Martinović
2 replies
centroid wanted, point that minimizes sum of squares of distances from sides
parmenides51 1
N
by SuperBarsh
Source: Oliforum Contest V 2017 p9 https://artofproblemsolving.com/community/c2487525_oliforum_contes
Given a triangle 
, let 
be the point which minimizes the sum of squares of distances from the sides of the triangle. Let 
the projections of on the sides of the triangle . Show that 
is the barycenter of 
.
(Jack D’Aurizio)
, let 
be the point which minimizes the sum of squares of distances from the sides of the triangle. Let 
the projections of on the sides of the triangle . Show that 
is the barycenter of 
.(Jack D’Aurizio)
1 reply
Strictly monotone polynomial with an extra condition
Popescu 11
N
by Iveela
Source: IMSC 2024 Day 2 Problem 2
Let 
be the set of all positive real numbers. Find all strictly monotone (increasing or decreasing) functions 
such that there exists a two-variable polynomial 
with real coefficients satisfying
for all 
.
Proposed by Navid Safaei, Iran
be the set of all positive real numbers. Find all strictly monotone (increasing or decreasing) functions 
such that there exists a two-variable polynomial 
with real coefficients satisfying
for all 
.Proposed by Navid Safaei, Iran
11 replies
Hard geometry
Lukariman 1
N
by ricarlos
Given triangle ABC, any line d intersects AB at D, intersects AC at E, intersects BC at F. Let O1,O2,O3 be the centers of the circles circumscribing triangles ADE, BDF, CFE. Prove that the orthocenter of triangle O1O2O3 lies on line d.
1 reply
Russian Diophantine Equation
LeYohan 1
N
by Natrium
Source: Moscow, 1963
Find all integer solutions to
.
.
1 reply
Simple Geometry
AbdulWaheed 6
N
by Gggvds1
Source: EGMO
Try to avoid Directed angles
Let ABC be an acute triangle inscribed in circle
. Let 
be the midpoint of the arc 
not containing 
and define 
similarly. Show that the orthocenter of 
is the incenter 
of .
Let ABC be an acute triangle inscribed in circle
. Let 
be the midpoint of the arc 
not containing 
and define 
similarly. Show that the orthocenter of 
is the incenter 
of .
6 replies
Bosnia and Herzegovina JBMO TST 2013 Problem 4
gobathegreat 4
N
by FishkoBiH
Source: Bosnia and Herzegovina Junior Balkan Mathematical Olympiad TST 2013
It is given polygon with 
sides 
. His vertices are marked with numbers such that sum of numbers marked by any 
consecutive vertices is constant and its value is 
. If we know that 
is marked with 
and 
is marked with 
, determine with which number is marked 
sides 
. His vertices are marked with numbers such that sum of numbers marked by any 
consecutive vertices is constant and its value is 
. If we know that 
is marked with 
and 
is marked with 
, determine with which number is marked 
4 replies
