Prove that
.![\[\sqrt{a+\frac{9}{b+2c}}+\sqrt{b+\frac{9}{c+2a}}+\sqrt{c+\frac{9}{a+2b}}+\frac{2(ab+bc+ca)}{9}\ge\frac{20}{3}\]](http://latex.artofproblemsolving.com/b/3/2/b32e0bcba967ca847e878ac186c3168cd6c5fc66.png)
Prove that
Prove that
Where 
and 
Prove that
satisfying
Prove
Y by arqady
Let 
satisfying
Prove
satisfying
Prove
G
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First Poster
Last Poster
A functional equation
super1978 1
N
by CheerfulZebra68
Source: Somewhere
Find all functions 
such that:
for all 
such that:
for all 
1 reply
Prove that IMO is isosceles
YLG_123 4
N
by Blackbeam999
Source: 2024 Brazil Ibero TST P2
Let 
be an acute-angled scalene triangle with circumcenter 
. Denote by 
, 
, and 
the midpoints of sides 
, 
, and 
, respectively. Let 
be the circle passing through 
and tangent to 
at . The circle intersects and 
at points 
and 
, respectively (where and are distinct from ). Let 
be the midpoint of segment 
, and let 
be the intersection of lines and 
. Prove that 
and that triangle 
is isosceles.
be an acute-angled scalene triangle with circumcenter 
. Denote by 
, 
, and 
the midpoints of sides 
, 
, and 
, respectively. Let 
be the circle passing through 
and tangent to 
at . The circle intersects and 
at points 
and 
, respectively (where and are distinct from ). Let 
be the midpoint of segment 
, and let 
be the intersection of lines and 
. Prove that 
and that triangle 
is isosceles.4 replies
Geometric mean of squares a knight's move away
Pompombojam 0
Source: Problem Solving Tactics Methods of Proof Q27
Each square of an 
chessboard has a real number written in it in such a way that each number is equal to the geometric mean of all the numbers a knight's move away from it.
Is it true that all of the numbers must be equal?
chessboard has a real number written in it in such a way that each number is equal to the geometric mean of all the numbers a knight's move away from it.Is it true that all of the numbers must be equal?
0 replies
Circumcircle of ADM
v_Enhance 71
N
by judokid
Source: USA TSTST 2012, Problem 7
Triangle 
is inscribed in circle 
. The interior angle bisector of angle 
intersects side 
and at 
and 
(other than ), respectively. Let 
be the midpoint of side . The circumcircle of triangle 
intersects sides 
and 
again at 
and 
(other than ), respectively. Let 
be the midpoint of segment 
, and let 
be the foot of the perpendicular from to line 
. Prove that line 
is tangent to the circumcircle of triangle 
.
is inscribed in circle 
. The interior angle bisector of angle 
intersects side 
and at 
and 
(other than ), respectively. Let 
be the midpoint of side . The circumcircle of triangle 
intersects sides 
and 
again at 
and 
(other than ), respectively. Let 
be the midpoint of segment 
, and let 
be the foot of the perpendicular from to line 
. Prove that line 
is tangent to the circumcircle of triangle 
.
71 replies
Three operations make any number
awesomeming327. 2
N
by happymoose666
Source: own
The number 
is written on the board. Anna, Boris, and Charlie can do the following actions: Anna can replace the number with its floor. Boris can replace any integer number with its factorial. Charlie can replace any nonnegative number with its square root. Prove that the three can work together to make any positive integer in finitely many steps.
is written on the board. Anna, Boris, and Charlie can do the following actions: Anna can replace the number with its floor. Boris can replace any integer number with its factorial. Charlie can replace any nonnegative number with its square root. Prove that the three can work together to make any positive integer in finitely many steps.
2 replies
IMO 2017 Problem 4
Amir Hossein 116
N
by cj13609517288
Source: IMO 2017, Day 2, P4
Let 
and 
be different points on a circle such that 
is not a diameter. Let 
be the tangent line to at . Point 
is such that is the midpoint of the line segment 
. Point 
is chosen on the shorter arc of so that the circumcircle 
of triangle 
intersects at two distinct points. Let be the common point of and that is closer to . Line 
meets again at 
. Prove that the line 
is tangent to .
Proposed by Charles Leytem, Luxembourg
and 
be different points on a circle such that 
is not a diameter. Let 
be the tangent line to at . Point 
is such that is the midpoint of the line segment 
. Point 
is chosen on the shorter arc of so that the circumcircle 
of triangle 
intersects at two distinct points. Let be the common point of and that is closer to . Line 
meets again at 
. Prove that the line 
is tangent to .Proposed by Charles Leytem, Luxembourg
116 replies
Another right angled triangle
ariopro1387 1
N
by lolsamo
Source: Iran Team selection test 2025 - P7
Let be a right angled triangle with 
. Point is the midpoint of side And be an arbitrary point on 
. The reflection of 
over intersects lines and at and , respectively. The circumcircles of 
and intersect again at 
. Prove that the center of the circumcircle of 
lies on 
.
be a right angled triangle with 
. Point is the midpoint of side And be an arbitrary point on 
. The reflection of 
over intersects lines and at and , respectively. The circumcircles of 
and intersect again at 
. Prove that the center of the circumcircle of 
lies on 
.
1 reply
4 variables with quadrilateral sides 2
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G
H
Source: Own
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