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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
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
Rubik's cube problem
ilikejam 20
N
by jasperE3
If I have a solved Rubik's cube, and I make a finite sequence of (legal) moves repeatedly, prove that I will eventually resolve the puzzle.
(this wording is kinda goofy but i hope its sorta intuitive)
(this wording is kinda goofy but i hope its sorta intuitive)
20 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
Problem 2
Functional_equation 15
N
by basilis
Source: Azerbaijan third round 2020
are positive integer.Solve the equation:
15 replies
VERY HARD MATH PROBLEM!
slimshadyyy.3.60 14
N
by GreekIdiot
Let a ≥b ≥c ≥0 be real numbers such that a^2 +b^2 +c^2 +abc = 4. Prove that
a+b+c+(√a−√c)^2 ≥3.
a+b+c+(√a−√c)^2 ≥3.
14 replies
Intersection of a cevian with the incircle
djb86 24
N
by Ilikeminecraft
Source: South African MO 2005 Q4
The inscribed circle of triangle touches the sides , 
and at 
, 
and 
respectively. Let 
denote the other point of intersection of 
and the inscribed circle. Prove that 
extended passes through the midpoint of 
if and only if 
.
touches the sides , 
and at 
, 
and 
respectively. Let 
denote the other point of intersection of 
and the inscribed circle. Prove that 
extended passes through the midpoint of 
if and only if 
.
24 replies
Polynomials and their shift with all real roots and in common
Assassino9931 2
N
by AshAuktober
Source: Bulgaria Spring Mathematical Competition 2025 11.4
We call two non-constant polynomials friendly if each of them has only real roots, and every root of one polynomial is also a root of the other. For two friendly polynomials 
and a constant 
, it is given that 
and 
are also friendly polynomials. Prove that 
.
and a constant 
, it is given that 
and 
are also friendly polynomials. Prove that 
.
2 replies
