Prove that
.![$\sqrt[3]{ab}+ \sqrt[3]{cd} < \sqrt[3]{(a+b+c)(b+c+d)}$](http://latex.artofproblemsolving.com/0/5/0/0502ba65f0545b3af84c41b2549d6256550b473b.png)
.
be real numbers less than 2.Then prove that 
with 
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.
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Hard to approach it !
BogG 128
N
by alexanderchew
Source: Swiss Imo Selection 2006
Let 
be an acute-angled triangle with 
. Let 
be the orthocenter of triangle 
, and let 
be the midpoint of the side 
. Let 
be a point on the side 
and 
a point on the side 
such that 
and the points , , are on the same line. Prove that the line 
is perpendicular to the common chord of the circumscribed circles of triangle and triangle 
.
be an acute-angled triangle with 
. Let 
be the orthocenter of triangle 
, and let 
be the midpoint of the side 
. Let 
be a point on the side 
and 
a point on the side 
such that 
and the points , , are on the same line. Prove that the line 
is perpendicular to the common chord of the circumscribed circles of triangle and triangle 
.
128 replies
Popular children at camp with algebra and geometry
Assassino9931 1
N
by internationalnick123456
Source: RMM Shortlist 2024 C3
Fix an odd integer 
. At a maths camp, there are 
children, each of whom selects
either algebra or geometry as their favourite topic. At lunch, they sit at
tables, with children
on each table, and start talking about mathematics. A child is said to be popular if their favourite
topic has a majority at their table. For dinner, the students again sit at tables, with children
on each table, such that no two children share a table at both lunch and dinner. Determine the
minimal number of young mathematicians who are popular at both mealtimes. (The minimum is across all sets of topic preferences and seating arrangements.)
. At a maths camp, there are 
children, each of whom selectseither algebra or geometry as their favourite topic. At lunch, they sit at
tables, with childrenon each table, and start talking about mathematics. A child is said to be popular if their favourite
topic has a majority at their table. For dinner, the students again sit at
tables, with childrenon each table, such that no two children share a table at both lunch and dinner. Determine the
minimal number of young mathematicians who are popular at both mealtimes. (The minimum is across all sets of topic preferences and seating arrangements.)
1 reply
Inspired by giangtruong13
sqing 3
N
by sqing
Source: Own
Let 
and 
.Prove that ![$$ a^2-ab+b^2 \geq \sqrt[3]{2} $$](//latex.artofproblemsolving.com/7/9/8/798484c6b985ebbace2a3dde816f11a46635feba.png)
Let and 
.Prove that ![$$ 3\geq a^2+ab+b^2 \geq \sqrt[3]{4} $$](//latex.artofproblemsolving.com/d/7/7/d77fbdc50a88395259e58d31413b9ac453e5dfd0.png)
and 
.Prove that ![$$ a^2-ab+b^2 \geq \sqrt[3]{2} $$](http://latex.artofproblemsolving.com/7/9/8/798484c6b985ebbace2a3dde816f11a46635feba.png)
Let and 
.Prove that ![$$ 3\geq a^2+ab+b^2 \geq \sqrt[3]{4} $$](http://latex.artofproblemsolving.com/d/7/7/d77fbdc50a88395259e58d31413b9ac453e5dfd0.png)
3 replies
Very easy case of a folklore polynomial equation
Assassino9931 2
N
by luutrongphuc
Source: Bulgaria EGMO TST 2025 P6
Determine all polynomials 
of odd degree with real coefficients such that 
.
of odd degree with real coefficients such that 
.
2 replies
Something nice
KhuongTrang 30
N
by KhuongTrang
Source: own
Problem. Given 
be non-negative real numbers such that 
Prove that
be non-negative real numbers such that 
Prove that
30 replies
Tangency geo
Assassino9931 1
N
by sami1618
Source: RMM Shortlist 2024 G1
Let be an acute triangle with 
and 
. Let be the midpoint of the side . The circumcircle of triangle 
intersects the side again at 
and the circumcircle of triangle 
intersects the side again at 
. The point 
lies on the perpendicular bisector of the segment so that the points and 
lie on the same side of 
and 
. Prove that the circumcircles of triangles 
and 
are tangent.
be an acute triangle with 
and 
. Let be the midpoint of the side . The circumcircle of triangle 
intersects the side again at 
and the circumcircle of triangle 
intersects the side again at 
. The point 
lies on the perpendicular bisector of the segment so that the points and 
lie on the same side of 
and 
. Prove that the circumcircles of triangles 
and 
are tangent.
1 reply
Removing cell to tile with L tetromino
ItzsleepyXD 1
N
by internationalnick123456
Source: [not own] , Mock Thailand Mathematic Olympiad P4
Consider 
Define a cell with 
property if after remove that cell from the board The board can be tile with 
tetromino.
Find the number of position of cell 
Note: tetromino can be rotated but not flipped
Define a cell with 
property if after remove that cell from the board The board can be tile with 
tetromino.Find the number of position of
cell 
Note: tetromino can be rotated but not flipped
1 reply
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
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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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K
Y
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