a perfect square?
Prove that
Y by
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.
G
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Arbitrary point on BC and its relation with orthocenter
falantrng 31
N
by NZP_IMOCOMP4
Source: Balkan MO 2025 P2
In an acute-angled triangle 
, 
be the orthocenter of it and 
be any point on the side 
. The points 
are on the segments 
, respectively, such that the points 
and 
are cyclic. The segments 
and 
intersect at 
is a point on 
such that 
is tangent to the circumcircle of triangle 
at 
and 
intersect at 
. Prove that the points 
and lie on the same line.
Proposed by Theoklitos Parayiou, Cyprus
, 
be the orthocenter of it and 
be any point on the side 
. The points 
are on the segments 
, respectively, such that the points 
and 
are cyclic. The segments 
and 
intersect at 
is a point on 
such that 
is tangent to the circumcircle of triangle 
at 
and 
intersect at 
. Prove that the points 
and lie on the same line.Proposed by Theoklitos Parayiou, Cyprus
31 replies
IMO Genre Predictions
ohiorizzler1434 23
N
by ohiorizzler1434
Everybody, with IMO upcoming, what are you predictions for the problem genres?
Personally I predict:
Personally I predict:
23 replies
Classic FE
BR1F1SZ 4
N
by User141208
Source: Argentina IberoAmerican TST 2024 P5
Let 
be the set of real numbers. Find all functions 
such that, for all real numbers 
and 
, the following equation holds:
be the set of real numbers. Find all functions 
such that, for all real numbers 
and 
, the following equation holds:
4 replies
Maybe LTE
navredras 2
N
by Blackbeam999
Source: Bulgaria 1997
Let 
be a positive integer. If 
is a power of a prime number, prove that is also prime.
be a positive integer. If 
is a power of a prime number, prove that is also prime.
2 replies
Sequence Gets Ratio’d
v4913 21
N
by cursed_tangent1434
Source: EGMO 2023/1
There are 
positive real numbers 
. For each 
we let 
(here we define 
to be 
and 
to be 
). Assume that for all 
and 
in the range 
to 
, we have 
if and only if 
.
Prove that
.
positive real numbers 
. For each 
we let 
(here we define 
to be 
and 
to be 
). Assume that for all 
and 
in the range 
to 
, we have 
if and only if 
.Prove that
.
21 replies
Functional equation on (0,infinity)
mathwizard888 56
N
by Adywastaken
Source: 2016 IMO Shortlist A4
Find all functions 
such that for any 
, 
such that for any 
, 
56 replies
Orthocenter
jayme 6
N
by Sadigly
Dear Mathlinkers,
1. ABC an acuatangle triangle
2. H the orthcenter of ABC
3. DEF the orthic triangle of ABC
4. A* the midpoint of AH
5. X the point of intersection of AH and EF.
Prove : X is the orthocenter of A*BC.
Sincerely
Jean-Louis
1. ABC an acuatangle triangle
2. H the orthcenter of ABC
3. DEF the orthic triangle of ABC
4. A* the midpoint of AH
5. X the point of intersection of AH and EF.
Prove : X is the orthocenter of A*BC.
Sincerely
Jean-Louis
6 replies
positive integers forming a perfect square
cielblue 2
N
by Pal702004
Find all positive integers such that 
is a perfect square.
such that 
is a perfect square.
2 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.
Z
K
Y
N Quick Reply
G
H
