.
.
such that 
.![$$2 \sqrt[4]{\frac{a}{b+c-a}}\ge 2 +\frac{2a^2-b^2-c^2}{(a+b)(a+c)}.$$](http://latex.artofproblemsolving.com/f/0/7/f0731fbe5c83aedc8474975f930f7194c08cc070.png)
and 
Prove that![$$3\left(\sqrt[3] 2+\frac{1}{\sqrt[3] 2} -1\right) \geq a+b+c\geq \frac{3+\sqrt{11}}{2}$$](http://latex.artofproblemsolving.com/f/6/6/f66042b4d07212741245953a8940280c94a52988.png)
![$$\frac{3}{2}\left(4+\sqrt[3] 4-\sqrt[3] 2\right) \geq a+b+c+ab+bc+ca\geq \frac{3(1+\sqrt{11})}{2}$$](http://latex.artofproblemsolving.com/0/5/6/056db8bf4d80242d705c265b6a64e063c8fca15a.png)
Prove that
Y by Adventure10
O -circumcenter, H - ortocenter and (OD) perpendicular (DE).
Prove that angle(DHE)=angle(ABC).
Prove that angle(DHE)=angle(ABC).
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In Cyclic Quadrilateral ABCD, find AB^2+BC^2-CD^2-AD^2
Darealzolt 0
Source: KTOM April 2025 P8
Given Cyclic Quadrilateral 
with an area of 
, with 
. If 
, Hence find the value of 
.
with an area of 
, with 
. If 
, Hence find the value of 
.
0 replies
Inequality by Po-Ru Loh
v_Enhance 57
N
by Learning11
Source: ELMO 2003 Problem 4
Let 
be real numbers such that ![\[ \frac{1}{x^2-1} + \frac{1}{y^2-1} + \frac{1}{z^2-1} = 1. \]](//latex.artofproblemsolving.com/8/f/5/8f5d49c9b44e3db3cd85fea7b6521e1dc02d4269.png)
Prove that ![\[ \frac{1}{x+1} + \frac{1}{y+1} + \frac{1}{z+1} \le 1. \]](//latex.artofproblemsolving.com/c/0/4/c0438d6eba686d0a11fd4decb4afa9a14aa50bac.png)
be real numbers such that ![\[ \frac{1}{x^2-1} + \frac{1}{y^2-1} + \frac{1}{z^2-1} = 1. \]](http://latex.artofproblemsolving.com/8/f/5/8f5d49c9b44e3db3cd85fea7b6521e1dc02d4269.png)
Prove that ![\[ \frac{1}{x+1} + \frac{1}{y+1} + \frac{1}{z+1} \le 1. \]](http://latex.artofproblemsolving.com/c/0/4/c0438d6eba686d0a11fd4decb4afa9a14aa50bac.png)
57 replies
2024 IMO P1
EthanWYX2009 104
N
by SYBARUPEMULA
Source: 2024 IMO P1
Determine all real numbers 
such that, for every positive integer 
the integer
is a multiple of 
(Note that 
denotes the greatest integer less than or equal to 
For example, 
and 
)
Proposed by Santiago Rodríguez, Colombia
such that, for every positive integer 
the integer
is a multiple of 
(Note that 
denotes the greatest integer less than or equal to 
For example, 
and 
)Proposed by Santiago Rodríguez, Colombia
104 replies
ai+aj is the multiple of n
Jackson0423 0
Consider an increasing sequence of integers
.For every positive integer
, there exist indices 
such that 
is divisible by .Given that
, find the minimum possible value of 
.0 replies
Addition on the IMO
naman12 139
N
by ezpotd
Source: IMO 2020 Problem 1
Consider the convex quadrilateral 
. The point 
is in the interior of . The following ratio equalities hold:
![\[\angle PAD:\angle PBA:\angle DPA=1:2:3=\angle CBP:\angle BAP:\angle BPC\]](//latex.artofproblemsolving.com/9/3/8/9388a91325842f1ec67a4693f0f10af46ed66c6d.png)
Prove that the following three lines meet in a point: the internal bisectors of angles 
and 
and the perpendicular bisector of segment 
.
Proposed by Dominik Burek, Poland
. The point 
is in the interior of . The following ratio equalities hold:![\[\angle PAD:\angle PBA:\angle DPA=1:2:3=\angle CBP:\angle BAP:\angle BPC\]](http://latex.artofproblemsolving.com/9/3/8/9388a91325842f1ec67a4693f0f10af46ed66c6d.png)
Prove that the following three lines meet in a point: the internal bisectors of angles 
and 
and the perpendicular bisector of segment 
.Proposed by Dominik Burek, Poland
139 replies
IMO ShortList 1998, number theory problem 5
orl 66
N
by lksb
Source: IMO ShortList 1998, number theory problem 5
Determine all positive integers 
for which there exists an integer 
such that 
is a divisor of 
.
for which there exists an integer 
such that 
is a divisor of 
.
66 replies
ANGLE EQUAL
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are two chords of a circle and 
,
is another chord of that circle which passes through 
,
,
if and only if 
.
intersect the circumcircle of 
again in 
,
.
,
intersects the circumcircle in 
,
,
)
and 
.
is a parallelogramm, and consequently
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Y by PrimeSol, Adventure10
Dear Mathlinkers,
if you like butterflies, you can see
http://jl.ayme.pagesperso-orange.fr/Docs/Papillon.pdf
Sincerely
Jean-Louis
if you like butterflies, you can see
http://jl.ayme.pagesperso-orange.fr/Docs/Papillon.pdf
Sincerely
Jean-Louis
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Y by Adventure10, Mango247
Dear Mathlinkers,
just an interesting link with
http://www.artofproblemsolving.com/community/c6t48f6h1283140_two_perpendiculars
Sincerely
Jean-Louis
just an interesting link with
http://www.artofproblemsolving.com/community/c6t48f6h1283140_two_perpendiculars
Sincerely
Jean-Louis
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