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Consider five points on a circle. For every three of them, we draw the perpendicular from the centroid of the triangle they determine to the line through the remaining two points. Prove that the ten lines thus formed are concurrent.
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points on sides of a triangle, intersections, extensions, ratio of areas wanted
parmenides51 2
N
by MathIQ.
Source: Mexican Mathematical Olympiad 1997 OMM P5
Let 
be points on the sides 
respectively of a triangle 
. Suppose that 
and 
meet at 
and meet at 
, and 
and meet at 
, such that 
, and 
. Compute the ratio of the area of 
to the area of 
.
be points on the sides 
respectively of a triangle 
. Suppose that 
and 
meet at 
and meet at 
, and 
and meet at 
, such that 
, and 
. Compute the ratio of the area of 
to the area of 
.
2 replies
Three numbers cannot be squares simultaneously
WakeUp 38
N
by SomeonecoolLovesMaths
Source: APMO 2011
Let 
be positive integers. Prove that it is impossible to have all of the three numbers 
to be perfect squares.
be positive integers. Prove that it is impossible to have all of the three numbers 
to be perfect squares.
38 replies
Primes and sets
mathisreaI 40
N
by Tinoba-is-emotional
Source: IMO 2022 Problem 3
Let 
be a positive integer and let 
be a finite set of odd prime numbers. Prove that there is at most one way (up to rotation and reflection) to place the elements of around the circle such that the product of any two neighbors is of the form 
for some positive integer 
.
be a positive integer and let 
be a finite set of odd prime numbers. Prove that there is at most one way (up to rotation and reflection) to place the elements of around the circle such that the product of any two neighbors is of the form 
for some positive integer 
.
40 replies
Problem G5 - IMO Shortlist 2007
April 30
N
by Double07
Source: ISL 2007, G5, AIMO 2008, TST 3, P2
Let 
be a fixed triangle, and let 
, 
, 
be the midpoints of sides 
, 
, 
, respectively. Let 
be a variable point on the circumcircle. Let lines 
, 
, 
meet the circumcircle again at 
, 
, 
, respectively. Assume that the points 
, 
, 
, , , are distinct, and lines 
, 
, 
form a triangle. Prove that the area of this triangle does not depend on .
Author: Christopher Bradley, United Kingdom
be a fixed triangle, and let 
, 
, 
be the midpoints of sides 
, 
, 
, respectively. Let 
be a variable point on the circumcircle. Let lines 
, 
, 
meet the circumcircle again at 
, 
, 
, respectively. Assume that the points 
, 
, 
, , , are distinct, and lines 
, 
, 
form a triangle. Prove that the area of this triangle does not depend on .Author: Christopher Bradley, United Kingdom
30 replies
Self-evident inequality trick
Lukaluce 16
N
by ErTeeEs06
Source: 2025 Junior Macedonian Mathematical Olympiad P4
Let 
, and 
be positive real numbers, such that 
. Prove the inequality
![\[\frac{x^3}{2 + x} + \frac{y^3}{2 + y} + \frac{z^3}{2 + z} \ge 1.\]](//latex.artofproblemsolving.com/b/e/5/be5819a67c3cd78f2dea35fdccf48688c720ce3c.png)
When does the equality hold?
, and 
be positive real numbers, such that 
. Prove the inequality![\[\frac{x^3}{2 + x} + \frac{y^3}{2 + y} + \frac{z^3}{2 + z} \ge 1.\]](http://latex.artofproblemsolving.com/b/e/5/be5819a67c3cd78f2dea35fdccf48688c720ce3c.png)
When does the equality hold?
16 replies
Help my diagram has too many points
MarkBcc168 29
N
by VideoCake
Source: IMO Shortlist 2023 G6
Let be an acute-angled triangle with circumcircle 
. A circle 
is internally tangent to at 
and also tangent to 
at 
. Let 
and 
intersect at 
and 
respectively. Let 
and 
be points on line such that 
is the midpoint of 
and 
is the midpoint of 
. Lines 
and 
meet at 
and intersect again at 
and 
respectively. The ray 
meets the circumcircle of triangle 
again at 
.
Prove that
.
Kian Moshiri, United Kingdom
be an acute-angled triangle with circumcircle 
. A circle 
is internally tangent to at 
and also tangent to 
at 
. Let 
and 
intersect at 
and 
respectively. Let 
and 
be points on line such that 
is the midpoint of 
and 
is the midpoint of 
. Lines 
and 
meet at 
and intersect again at 
and 
respectively. The ray 
meets the circumcircle of triangle 
again at 
.Prove that
.Kian Moshiri, United Kingdom
29 replies
A problem with series
Pena317 1
N
by venhancefan777
Source: P5, Mexico Center Regional Olympiad 2019
A serie of positive integers 
,
,. . . ,
is 
if for every index 
that holds 
, there exist at least 
terms of the serie such that they are all less or equal to .
Find the maximum value of the sum
, where ,,. . . , is an serie.
,
,. . . ,
is 
if for every index 
that holds 
, there exist at least 
terms of the serie such that they are all less or equal to .Find the maximum value of the sum
, where ,,. . . , is an serie.
1 reply
IMO 2009, problem 4
ZetaX 60
N
by FarrukhBurzu
Let be a triangle with 
. The angle bisectors of 
and 
meet the sides 
and 
at 
and 
, respectively. Let 
be the incentre of triangle 
. Suppose that 
. Find all possible values of .
Jan Vonk, Belgium, Peter Vandendriessche, Belgium and Hojoo Lee, Korea
be a triangle with 
. The angle bisectors of 
and 
meet the sides 
and 
at 
and 
, respectively. Let 
be the incentre of triangle 
. Suppose that 
. Find all possible values of .Jan Vonk, Belgium, Peter Vandendriessche, Belgium and Hojoo Lee, Korea
60 replies
Tennis tournament with rotating courts
v_Enhance 6
N
by Blast_S1
Source: ELMO Shortlist 2013: Problem C10, by Ray Li
Let 
be a fixed positive integer. There are 
people, numbered 
, participating in a tennis tournament. For any two positive integers 
with 
, player has a higher skill level than player 
. Prior to the first round, the players are paired arbitrarily and each pair is assigned a unique court among courts, numbered 
.
During a round, each player plays against the other person assigned to his court (so that exactly one match takes place per court), and the player with higher skill wins the match (in other words, there are no upsets). Afterwards, for
, the winner of court moves to court 
and the loser of court stays on court ; however, the winner of court 1 stays on court 1 and the loser of court 1 moves to court .
Find all positive integers such that, regardless of the initial pairing, the players 
all change courts immediately after the th round.
Proposed by Ray Li
be a fixed positive integer. There are 
people, numbered 
, participating in a tennis tournament. For any two positive integers 
with 
, player has a higher skill level than player 
. Prior to the first round, the players are paired arbitrarily and each pair is assigned a unique court among courts, numbered 
.During a round, each player plays against the other person assigned to his court (so that exactly one match takes place per court), and the player with higher skill wins the match (in other words, there are no upsets). Afterwards, for
, the winner of court moves to court 
and the loser of court stays on court ; however, the winner of court 1 stays on court 1 and the loser of court 1 moves to court .Find all positive integers
such that, regardless of the initial pairing, the players 
all change courts immediately after the th round.Proposed by Ray Li
6 replies
∑(a-b)(a-c)/(2a^2 + (b+c)^2) >= 0
Zhero 24
N
by RevolveWithMe101
Source: ELMO Shortlist 2010, A2
Let be positive reals. Prove that
![\[ \frac{(a-b)(a-c)}{2a^2 + (b+c)^2} + \frac{(b-c)(b-a)}{2b^2 + (c+a)^2} + \frac{(c-a)(c-b)}{2c^2 + (a+b)^2} \geq 0. \]](//latex.artofproblemsolving.com/6/9/3/6937791380a8642edca3e773dc72dbf3f0e0f206.png)
Calvin Deng.
be positive reals. Prove that![\[ \frac{(a-b)(a-c)}{2a^2 + (b+c)^2} + \frac{(b-c)(b-a)}{2b^2 + (c+a)^2} + \frac{(c-a)(c-b)}{2c^2 + (a+b)^2} \geq 0. \]](http://latex.artofproblemsolving.com/6/9/3/6937791380a8642edca3e773dc72dbf3f0e0f206.png)
Calvin Deng.
24 replies
i am not abel to prove or disprove
frost23 8
N
by frost23
Source: made on my own
sorrrrrry
8 replies
Concurrency with 10 lines
oVlad 1
N
by kokcio
Source: Romania EGMO TST 2017 Day 1 P1
Consider five points on a circle. For every three of them, we draw the perpendicular from the centroid of the triangle they determine to the line through the remaining two points. Prove that the ten lines thus formed are concurrent.
1 reply
Concurrency with 10 lines
G
H
G
H
Source: Romania EGMO TST 2017 Day 1 P1
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Y by
Let be center of mass of this 
points. Let 
be centroid of some triangle and 
be midpoint of chord with two other points. Then if line 
intersects line drawn through at point , then 
, so position of is uniquely determined by the position of points 
. (if 
, then 
).
Generalization of this problem is in plane geometry by V. Prasolov (problem 14.13): on a circle,
points are given. Through the center of mass of 
points a straight line is drawn perpendicularly to the chord that connects the two remaining points. Prove that all such straight lines intersect at one point.
be center of mass of this 
points. Let 
be centroid of some triangle and 
be midpoint of chord with two other points. Then if line 
intersects line drawn through at point , then 
, so position of is uniquely determined by the position of points 
. (if 
, then 
).Generalization of this problem is in plane geometry by V. Prasolov (problem 14.13): on a circle,
points are given. Through the center of mass of 
points a straight line is drawn perpendicularly to the chord that connects the two remaining points. Prove that all such straight lines intersect at one point.
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