such that ![\[f(2x+y) + f(x+f(2y)) = f(x)f(y) - xy\]](http://latex.artofproblemsolving.com/0/6/2/0623b7c328af6d088cfc5dd288afe3e17a94ff29.png)
for all reals 
and 
.
Y by Adventure10, Mango247
inscribe 
. 
is the midpoint of 
. 
is the intersection of the tangent of at 
and 
. The tangent of at 
intersects 
at 
. Simillarity, we have 
.Prove that
have second common point (not 
).
G
Topic
First Poster
Last Poster
pretty well known
dotscom26 0
Let 
be a scalene triangle such that 
is its incircle. 
is tangent to at 
. A point 
(
) is located on 
.
Let
, 
, and 
be the incircles of the triangles 
, 
, and 
, respectively.
Show that the common tangent to and is also tangent to .
be a scalene triangle such that 
is its incircle. 
is tangent to at 
. A point 
(
) is located on 
.Let
, 
, and 
be the incircles of the triangles 
, 
, and 
, respectively.Show that the common tangent to
and is also tangent to .0 replies
Modular NT
oVlad 3
N
by EVKV
Source: Romania JBMO TST 2024 Day 1 P1
Find all the positive integers 
and 
such that 
is a prime number.
Cosmin Manea and Dragoș Petrică
and 
such that 
is a prime number.Cosmin Manea and Dragoș Petrică
3 replies
+1 wRatio conditions; prove angle XPA = angle AQY
MellowMelon 15
N
by cj13609517288
Source: USA TSTST 2011/2012 P2
Two circles and intersect at points 
and 
. Line 
is tangent to at 
and to at 
so that is closer to than . Let 
and 
be points on major arcs 
(on ) and 
(on ), respectively, such that 
. Extend segments 
and 
through to 
and 
, respectively, such that 
. Given that the circumcenter of triangle 
lies on line 
, prove that 
.
and intersect at points and . Line 
is tangent to at 
and to at 
so that is closer to than . Let 
and 
be points on major arcs 
(on ) and 
(on ), respectively, such that 
. Extend segments 
and 
through to 
and 
, respectively, such that 
. Given that the circumcenter of triangle 
lies on line 
, prove that 
.
15 replies
IMO 2017 Problem 1
cjquines0 154
N
by blueprimes
Source: IMO 2017 Problem 1
For each integer 
, define the sequence 
for 
as
Determine all values of 
such that there exists a number such that 
for infinitely many values of 
.
Proposed by Stephan Wagner, South Africa
, define the sequence 
for 
as
Determine all values of 
such that there exists a number such that 
for infinitely many values of 
.Proposed by Stephan Wagner, South Africa
154 replies
IMO 2018 Problem 5
orthocentre 76
N
by Maximilian113
Source: IMO 2018
Let 
, 
, 
be an infinite sequence of positive integers. Suppose that there is an integer 
such that, for each 
, the number
is an integer. Prove that there is a positive integer 
such that 
for all 
.
Proposed by Bayarmagnai Gombodorj, Mongolia
, 
, 
be an infinite sequence of positive integers. Suppose that there is an integer 
such that, for each 
, the number
is an integer. Prove that there is a positive integer 
such that 
for all 
.Proposed by Bayarmagnai Gombodorj, Mongolia
76 replies
Junior Balkan Mathematical Olympiad 2024- P3
Lukaluce 13
N
by EVKV
Source: JBMO 2024
Find all triples of positive integers 
that satisfy the equation
Proposed by Ognjen Tešić, Serbia
that satisfy the equation
Proposed by Ognjen Tešić, Serbia
13 replies
Geometry :3c
popop614 2
N
by Ianis
Source: MINE :<
Quadrilateral 
has an incenter 
Suppose 
. Let be the midpoint of 
. Suppose that 
. 
meets 
again at point 
. Let points and be such that is the midpoint of 
and is the midpoint of 
. Point lies on the plane such that 
is a parallelogram, and suppose the angle bisectors of 
and 
concur on 
.
The angle bisectors of
and 
meet 
at and . Prove that 
.
has an incenter 
Suppose 
. Let be the midpoint of 
. Suppose that 
. 
meets 
again at point 
. Let points and be such that is the midpoint of 
and is the midpoint of 
. Point lies on the plane such that 
is a parallelogram, and suppose the angle bisectors of 
and 
concur on 
.The angle bisectors of
and 
meet 
at and . Prove that 
.
2 replies
cursed tangent is xiooix
TestX01 2
N
by TestX01
Source: xiooix and i
Let 
be a triangle. Let and 
be the intersections of the and 
angle bisectors with the opposite sides. Let 
. Let 
where 
is the major arc midpiont.
i)Show that points
and are coconic on a conic 
ii) If intersects 
again at , not equal to 
or , then prove 
and 
concur on 
I will post solution in ~1 week if noone solves.
be a triangle. Let and 
be the intersections of the and angle bisectors with the opposite sides. Let 
. Let 
where 
is the major arc midpiont.i)Show that points
and are coconic on a conic 
ii) If
intersects 
again at , not equal to 
or , then prove 
and 
concur on 
I will post solution in ~1 week if noone solves.
2 replies
Game on a row of 9 squares
EmersonSoriano 2
N
by Mr.Sharkman
Source: 2018 Peru TST Cono Sur P10
Let be a positive integer. Alex plays on a row of 9 squares as follows. Initially, all squares are empty. In each turn, Alex must perform exactly one of the following moves:
Choose a number of the form 
, with 
a non-negative integer, and place it in an empty square.
Choose two (not necessarily consecutive) squares containing the same number, say . Replace the number in one of the squares with 
and erase the number in the other square.
At the end of the game, one square contains the number
, while the other squares are empty. Determine, as a function of , the maximum number of turns Alex can make.
be a positive integer. Alex plays on a row of 9 squares as follows. Initially, all squares are empty. In each turn, Alex must perform exactly one of the following moves:
Choose a number of the form 
, with 
a non-negative integer, and place it in an empty square.
Choose two (not necessarily consecutive) squares containing the same number, say . Replace the number in one of the squares with 
and erase the number in the other square.At the end of the game, one square contains the number
, while the other squares are empty. Determine, as a function of , the maximum number of turns Alex can make.
2 replies
Guessing Point is Hard
MarkBcc168 30
N
by Circumcircle
Source: IMO Shortlist 2023 G5
Let be an acute-angled triangle with circumcircle 
and circumcentre 
. Points 
and 
lie on such that 
and 
. Let 
meet at , and 
meet at .
Prove that the circumcircles of triangles
and 
have an intersection lie on line 
.
Ivan Chan Kai Chin, Malaysia
be an acute-angled triangle with circumcircle 
and circumcentre . Points 
and 
lie on such that 
and 
. Let 
meet at , and 
meet at .Prove that the circumcircles of triangles
and 
have an intersection lie on line 
.Ivan Chan Kai Chin, Malaysia
30 replies
Geometry
G
H
G
H
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Y by Adventure10
noway wrote:
LKira wrote:
My geogebra say no, it not true, may you check again your problem ?
I have checked carefully, it's true ^^
Then see VMO 2018 P7
This post has been edited 1 time. Last edited by LKira, Dec 5, 2019, 2:26 PM
Z
K
Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Y by Adventure10, Mango247
Your second common point of the three circles has the first baricentric coordinate
a^2 (2 a^16-6 a^14 (b^2+c^2)+a^12 (-4 b^4+38 b^2 c^2-4 c^4)+2 a^10 (9 b^6-20 b^4 c^2-20 b^2 c^4+9 c^6)+a^8 (-67 b^6 c^2+174 b^4 c^4-67 b^2 c^6)+4 a^4 (b^2-c^2)^2 (b^8+9 b^6 c^2-32 b^4 c^4+9 b^2 c^6+c^8)-(b^4-c^4)^2 (2 b^8-9 b^6 c^2+10 b^4 c^4-9 b^2 c^6+2 c^8)+2 a^2 (b^2-c^2)^2 (3 b^10-18 b^8 c^2+11 b^6 c^4+11 b^4 c^6-18 b^2 c^8+3 c^10)-2 a^6 (9 b^10-51 b^8 c^2+46 b^6 c^4+46 b^4 c^6-51 b^2 c^8+9 c^10))
This triangle center is not currently in https://faculty.evansville.edu/ck6/encyclopedia/ETC.html
a^2 (2 a^16-6 a^14 (b^2+c^2)+a^12 (-4 b^4+38 b^2 c^2-4 c^4)+2 a^10 (9 b^6-20 b^4 c^2-20 b^2 c^4+9 c^6)+a^8 (-67 b^6 c^2+174 b^4 c^4-67 b^2 c^6)+4 a^4 (b^2-c^2)^2 (b^8+9 b^6 c^2-32 b^4 c^4+9 b^2 c^6+c^8)-(b^4-c^4)^2 (2 b^8-9 b^6 c^2+10 b^4 c^4-9 b^2 c^6+2 c^8)+2 a^2 (b^2-c^2)^2 (3 b^10-18 b^8 c^2+11 b^6 c^4+11 b^4 c^6-18 b^2 c^8+3 c^10)-2 a^6 (9 b^10-51 b^8 c^2+46 b^6 c^4+46 b^4 c^6-51 b^2 c^8+9 c^10))
This triangle center is not currently in https://faculty.evansville.edu/ck6/encyclopedia/ETC.html
Z
K
Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Y by Adventure10, Mango247
Is this correct?
We begin with the main claim
: Given a and point on its euler line. Let line 
,
,
intersect the 
again at 
,
, and 
respectively. Define 
to be the midpoint of , and respectively and define also ,,
to be the midpont of 
,
and 
respectively. Then we have that 
,
and 
are concurrent.
:
Animate point on the euler line, therefore we will get that the degree of point 
are all 2. Hence the degree of 
are also 2. Since point are all fixed we must have that the degree of line 
are all 2. So if we want to prove that are concurrent we only need to check that for 
possibilities of point , are always concurrent. It's easy to check that for is the circumcenter, centroid the two intersections of euler line with the circumcircle of , intersection of euler line with ,
,, the three lines always concurrent. Therefore, 
We go back to the original problem
Define
be the point on 
different from 
such that 
, 
and 
are all tangent to . Now define 
to be the midpoint of 
, 
and 
respectively.
Invert the figure WRT to, then the original problem is equivalent to proving that 
,
and 
are concur at one point.
However, it's a classic result from IMO 2011 G4 that
and 
are concurrent at point 
, the isogonal conjugate of isotomic conjugate of orthocenter 
of . Moreover it's well known that this point is on the euler line. Therefore by the first claim, the problem is done.
We begin with the main claim
: Given a and point on its euler line. Let line 
,
,
intersect the 
again at 
,
, and 
respectively. Define 
to be the midpoint of , and respectively and define also ,,
to be the midpont of 
,
and 
respectively. Then we have that 
,
and 
are concurrent.
:Animate point
on the euler line, therefore we will get that the degree of point 
are all 2. Hence the degree of 
are also 2. Since point are all fixed we must have that the degree of line 
are all 2. So if we want to prove that are concurrent we only need to check that for 
possibilities of point , are always concurrent. It's easy to check that for is the circumcenter, centroid the two intersections of euler line with the circumcircle of , intersection of euler line with ,
,, the three lines always concurrent. Therefore, 
We go back to the original problem
Define
be the point on different from 
such that 
, 
and 
are all tangent to . Now define 
to be the midpoint of 
, 
and 
respectively.Invert the figure WRT to
, then the original problem is equivalent to proving that 
,
and 
are concur at one point.However, it's a classic result from IMO 2011 G4 that
and 
are concurrent at point 
, the isogonal conjugate of isotomic conjugate of orthocenter 
of . Moreover it's well known that this point is on the euler line. Therefore by the first claim, the problem is done.This post has been edited 5 times. Last edited by Sugiyem, Dec 23, 2019, 10:11 PM
Z
K
Y
N Quick Reply
G
H
