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Midpoints of chords on a circle
AwesomeToad 38
N
by LeYohan
Source: 0
Let 
be a circle and 
a given point in the plane. Each line through which intersects determines a chord of . Show that the midpoints of these chords lie on a circle.
be a circle and 
a given point in the plane. Each line through which intersects determines a chord of . Show that the midpoints of these chords lie on a circle.
38 replies
Polish MO finals, problem 1
michaj 4
N
by AshAuktober
In each cell of a matrix 
a number from a set 
is written --- in the first row numbers 
, in the second 
and so on. Exactly 
of them have been chosen, no two from the same row or the same column. Let us denote by 
a number chosen from row number 
. Show that:
![\[ \frac{1^2}{a_1}+\frac{2^2}{a_2}+\ldots +\frac{n^2}{a_n}\geq \frac{n+2}{2}-\frac{1}{n^2+1}\]](//latex.artofproblemsolving.com/a/d/5/ad5ae6861da79a08d32093f63827016868856a3a.png)
a number from a set 
is written --- in the first row numbers 
, in the second 
and so on. Exactly 
of them have been chosen, no two from the same row or the same column. Let us denote by 
a number chosen from row number 
. Show that:![\[ \frac{1^2}{a_1}+\frac{2^2}{a_2}+\ldots +\frac{n^2}{a_n}\geq \frac{n+2}{2}-\frac{1}{n^2+1}\]](http://latex.artofproblemsolving.com/a/d/5/ad5ae6861da79a08d32093f63827016868856a3a.png)
4 replies
2025 Caucasus MO Seniors P7
BR1F1SZ 1
N
by X.Luser
Source: Caucasus MO
From a point 
lying outside the circle 
, two tangents are drawn touching at points 
and 
. A point 
is chosen on the segment 
. Let points and 
be the midpoints of segments 
and 
respectively. The circumcircle of triangle 
intersects again at point 
(
). Prove that the line 
passes through the centroid of triangle 
.
lying outside the circle 
, two tangents are drawn touching at points 
and 
. A point 
is chosen on the segment 
. Let points and 
be the midpoints of segments 
and 
respectively. The circumcircle of triangle 
intersects again at point 
(
). Prove that the line 
passes through the centroid of triangle 
.
1 reply
Easy geometry
Bluesoul 13
N
by AshAuktober
Source: CJMO 2022 P1
Let 
has circumcircle 
, drop the perpendicular line from 
to 
and meet at point 
, similarly, altitude from 
to 
meets at 
. Prove that if 
(sorry it is from my memory I can't remember the exact problem, but it means the same)
has circumcircle 
, drop the perpendicular line from 
to 
and meet at point 
, similarly, altitude from 
to 
meets at 
. Prove that if 
(sorry it is from my memory I can't remember the exact problem, but it means the same)
13 replies
IMO Shortlist 2013, Geometry #2
lyukhson 77
N
by endless_abyss
Source: IMO Shortlist 2013, Geometry #2
Let be the circumcircle of a triangle 
. Denote by and the midpoints of the sides 
and , respectively, and denote by 
the midpoint of the arc of not containing . The circumcircles of the triangles 
and 
intersect the perpendicular bisectors of and at points 
and 
, respectively; assume that and lie inside the triangle . The lines and 
intersect at . Prove that 
.
be the circumcircle of a triangle 
. Denote by and the midpoints of the sides 
and , respectively, and denote by 
the midpoint of the arc of not containing . The circumcircles of the triangles 
and 
intersect the perpendicular bisectors of and at points 
and 
, respectively; assume that and lie inside the triangle . The lines and 
intersect at . Prove that 
.
77 replies
1 viewing
f(x*f(y)) = f(x)/y
orl 23
N
by Maximilian113
Source: IMO 1990, Day 2, Problem 4, IMO ShortList 1990, Problem 25 (TUR 4)
Let 
be the set of positive rational numbers. Construct a function 
such that
![\[ f(xf(y)) = \frac {f(x)}{y}
\]](//latex.artofproblemsolving.com/5/9/1/59172e31cf6b7d885295f6e1daf6891a71739247.png)
for all
, 
in .
be the set of positive rational numbers. Construct a function 
such that![\[ f(xf(y)) = \frac {f(x)}{y}
\]](http://latex.artofproblemsolving.com/5/9/1/59172e31cf6b7d885295f6e1daf6891a71739247.png)
for all
, 
in .
23 replies
Heavy config geo involving mixtilinear
Assassino9931 2
N
by Assassino9931
Source: Bulgaria Spring Mathematical Competition 2025 12.4
Let be an acute-angled triangle 
with 
and incenter 
. Let 
be the mixtilinear circle at vertex 
, i.e. the circle internally tangent to the circumcircle of 
and also tangent to lines 
and 
. A circle 
passes through points 
and 
and is tangent to at point 
, with 
and being inside 
. Prove that:
be an acute-angled triangle 
with 
and incenter 
. Let 
be the mixtilinear circle at vertex 
, i.e. the circle internally tangent to the circumcircle of 
and also tangent to lines 
and 
. A circle 
passes through points 
and 
and is tangent to at point 
, with 
and being inside 
. Prove that:
2 replies
Guess the leader's binary string!
cjquines0 78
N
by de-Kirschbaum
Source: 2016 IMO Shortlist C1
The leader of an IMO team chooses positive integers 
and 
with 
, and announces them to the deputy leader and a contestant. The leader then secretly tells the deputy leader an -digit binary string, and the deputy leader writes down all -digit binary strings which differ from the leader’s in exactly positions. (For example, if 
and 
, and if the leader chooses 
, the deputy leader would write down 
and 
.) The contestant is allowed to look at the strings written by the deputy leader and guess the leader’s string. What is the minimum number of guesses (in terms of and ) needed to guarantee the correct answer?
and 
with 
, and announces them to the deputy leader and a contestant. The leader then secretly tells the deputy leader an -digit binary string, and the deputy leader writes down all -digit binary strings which differ from the leader’s in exactly positions. (For example, if 
and 
, and if the leader chooses 
, the deputy leader would write down 
and 
.) The contestant is allowed to look at the strings written by the deputy leader and guess the leader’s string. What is the minimum number of guesses (in terms of and ) needed to guarantee the correct answer?78 replies
Monkeys have bananas
nAalniaOMliO 5
N
by jkim0656
Source: Belarusian National Olympiad 2025
Ten monkeys have 60 bananas. Each monkey has at least one banana and any two monkeys have different amounts of bananas.
Prove that any six monkeys can distribute their bananas between others such that all 4 remaining monkeys have the same amount of bananas.
Prove that any six monkeys can distribute their bananas between others such that all 4 remaining monkeys have the same amount of bananas.
5 replies
Fixed point config on external similar isosceles triangles
Assassino9931 1
N
by E50
Source: Bulgaria Spring Mathematical Competition 2025 10.2
Let be an acute scalene triangle. A point 
varies on its side . The points 
and 
are the midpoints of the arcs 
and 
(not containing ) of the circumcircles of triangles 
and 
, respectively. Prove that the circumcircle of triangle 
passes through a fixed point, independent of the choice of on .
be an acute scalene triangle. A point 
varies on its side . The points 
and 
are the midpoints of the arcs 
and 
(not containing ) of the circumcircles of triangles 
and 
, respectively. Prove that the circumcircle of triangle 
passes through a fixed point, independent of the choice of on .
1 reply
