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Topic
First Poster
Last Poster
Inequality with a,b,c,d
GeoMorocco 2
N
by sqing
Source: Moroccan Training 2025
Let 
positive real numbers such that 
. Prove that :
positive real numbers such that 
. Prove that :
2 replies
angle wanted, right ABC, AM=CB , CN=MB
parmenides51 2
N
by Tsikaloudakis
Source: 2022 European Math Tournament - Senior First + Grand League - Math Battle 1.3
In a right-angled triangle 
, points 
and 
are taken on the legs 
and 
, respectively, so that 
and 
. Find the acute angle between line segments 
and 
.
, points 
and 
are taken on the legs 
and 
, respectively, so that 
and 
. Find the acute angle between line segments 
and 
.
2 replies
Inspired by Czech-Polish-Slovak 2017
sqing 4
N
by sqing
Let 
be real numbers. Prove that
be real numbers. Prove that
4 replies
+1 winteresting inequality
pennypc123456789 1
N
by Quantum-Phantom
Let 
be real numbers satisfying 
. Find the maximum value of
![\[P = \dfrac{a(b+c)}{a^2+2bc+3} + \dfrac{b(a+c) }{b^2+2ca +3 } + \dfrac{c(a+b)}{c^2+2ab+3}.\]](//latex.artofproblemsolving.com/7/b/2/7b2ec0a87ca05d3cb29da888a4dfa73e10f238f8.png)
be real numbers satisfying 
. Find the maximum value of![\[P = \dfrac{a(b+c)}{a^2+2bc+3} + \dfrac{b(a+c) }{b^2+2ca +3 } + \dfrac{c(a+b)}{c^2+2ab+3}.\]](http://latex.artofproblemsolving.com/7/b/2/7b2ec0a87ca05d3cb29da888a4dfa73e10f238f8.png)
1 reply
Jbmo 2015 problem 1
neverlose 11
N
by MATHS_ENTUSIAST
Find all prime numbers 
and positive integers 
satisfying the equation ![\[a^2+b^2+16c^2 = 9k^2 + 1.\]](//latex.artofproblemsolving.com/5/4/5/545f8edbdebe4e759c5a70bd3d489e5d53c9147c.png)
Proposed by Moldova
and positive integers 
satisfying the equation ![\[a^2+b^2+16c^2 = 9k^2 + 1.\]](http://latex.artofproblemsolving.com/5/4/5/545f8edbdebe4e759c5a70bd3d489e5d53c9147c.png)
Proposed by Moldova
11 replies
Colors over colors in a grid
FAA2533 1
N
by Rohit-2006
Source: BdMO 2025 Higher Secondary P3
Two player are playing in an 
grid. Initially the whole board is black. On 
's move, he selects 
subgrid and color it white. On 
's move, he selects a 
subgrid and colors it black. wants to make the whole board white. Can he do it?
Proposed by S M A Nahian
grid. Initially the whole board is black. On 
's move, he selects 
subgrid and color it white. On 
's move, he selects a 
subgrid and colors it black. wants to make the whole board white. Can he do it?Proposed by S M A Nahian
1 reply
1 viewing
AD and BC meet MN at P and Q
WakeUp 6
N
by Nari_Tom
Source: Baltic Way 2005
Let 
be a convex quadrilateral such that 
. Let and be the midpoints of and 
, respectively. The lines 
and meet the line 
at 
and 
, respectively. Prove that 
.
be a convex quadrilateral such that 
. Let and be the midpoints of and 
, respectively. The lines 
and meet the line 
at 
and 
, respectively. Prove that 
.
6 replies
thank you
Piwbo 0
Let 
be the n-th prime number in increasing order for 
. Prove that there exists a sequence of distinct prime numbers 
satisfying 
for all 
be the n-th prime number in increasing order for 
. Prove that there exists a sequence of distinct prime numbers 
satisfying 
for all 
0 replies
IMO ShortList 2002, geometry problem 4
orl 24
N
by Avron
Source: IMO ShortList 2002, geometry problem 4
Circles 
and 
intersect at points and . Distinct points 
and 
(not at or ) are selected on . The lines 
and 
meet again at 
and 
respectively, and the lines 
and 
meet at 
. Prove that, as and vary, the circumcentres of triangles 
all lie on one fixed circle.
and 
intersect at points and . Distinct points 
and 
(not at or ) are selected on . The lines 
and 
meet again at 
and 
respectively, and the lines 
and 
meet at 
. Prove that, as and vary, the circumcentres of triangles 
all lie on one fixed circle.
24 replies
comb and nt mixed
MR.1 1
N
by MR.1
Source: own
in antarctica there is 
penguin and each one is numbered using numbers 
. penguin 
is called 
if 
where 
is 
prime divisor of 
(
(where k is number of 's prime divisors). what is maximum ratio of 
where is number of in antarctica?(
)
penguin and each one is numbered using numbers 
. penguin 
is called 
if 
where 
is 
prime divisor of 
(
(where k is number of 's prime divisors). what is maximum ratio of 
where is number of in antarctica?(
)
1 reply
