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satisfying the equation![\[
(f(x+y))^2= f(x^2) + f(2xf(y) + y^2), \quad \forall x, y \in \mathbb{R}.
\]](http://latex.artofproblemsolving.com/2/5/e/25eed911f303ed7cc4e03765a2f943ad4641c451.png)
Geometry
youochange 5
N
by lolsamo
m:}
Let
be a triangle inscribed in a circle, where the tangents to the circle at points 
and 
intersect at the point 
. Let 
be a point on the arc 
(not containing ) such that 
and 
. Let the lines 
and 
intersect at point 
. Let 
be the reflection of with respect to the line . The lines 
and 
intersect at point 
, and intersects the circumcircle of again at point 
.
Prove that the point lies on the circumcircle of 
.
Let
be a triangle inscribed in a circle, where the tangents to the circle at points 
and 
intersect at the point 
. Let 
be a point on the arc 
(not containing ) such that 
and 
. Let the lines 
and 
intersect at point 
. Let 
be the reflection of with respect to the line . The lines 
and 
intersect at point 
, and intersects the circumcircle of again at point 
.Prove that the point
lies on the circumcircle of 
.
5 replies
Something nice
KhuongTrang 25
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
25 replies
Two Functional Inequalities
Mathdreams 6
N
by Assassino9931
Source: 2025 Nepal Mock TST Day 2 Problem 2
Determine all functions 
such that 
and 
for any real numbers 
and 
.
(Miroslav Marinov, Bulgaria)
such that 
and 
for any real numbers 
and 
.(Miroslav Marinov, Bulgaria)
6 replies
Pythagorean new journey
XAN4 2
N
by mathprodigy2011
Source: Inspired by sarjinius
The number 
is written on the blackboard. Every time, Carmela can erase the number 
on the black board and replace it with a new number 
, if and only if 
is a perfect square. Prove or disprove that all positive integers 
can be written exactly once on the blackboard.
is written on the blackboard. Every time, Carmela can erase the number 
on the black board and replace it with a new number 
, if and only if 
is a perfect square. Prove or disprove that all positive integers 
can be written exactly once on the blackboard.
2 replies
sqrt(2) and sqrt(3) differ in at least 1000 digits
Stuttgarden 2
N
by straight
Source: Spain MO 2025 P3
We write the decimal expressions of 
and 
as ![\[\sqrt{2}=1.a_1a_2a_3\dots\quad\quad\sqrt{3}=1.b_1b_2b_3\dots\]](//latex.artofproblemsolving.com/4/d/5/4d5049d5c3b26757a25c5dd34462c5462228b83d.png)
where each 
or 
is a digit between 0 and 9. Prove that there exist at least 1000 values of 
between 
and 
such that 
.
and 
as ![\[\sqrt{2}=1.a_1a_2a_3\dots\quad\quad\sqrt{3}=1.b_1b_2b_3\dots\]](http://latex.artofproblemsolving.com/4/d/5/4d5049d5c3b26757a25c5dd34462c5462228b83d.png)
where each 
or 
is a digit between 0 and 9. Prove that there exist at least 1000 values of 
between 
and 
such that 
.
2 replies
combinatorics and number theory beautiful problem
Medjl 2
N
by mathprodigy2011
Source: Netherlands TST for BxMo 2017 problem 4
A quadruple 
of positive integers with 
is called good if we can colour each integer red, blue, green or purple, in such a way that
of each 
consecutive integers at least one is coloured red;
of each 
consecutive integers at least one is coloured blue;
of each 
consecutive integers at least one is coloured green;
of each 
consecutive integers at least one is coloured purple.
Determine all good quadruples with
of positive integers with 
is called good if we can colour each integer red, blue, green or purple, in such a way that of each 
consecutive integers at least one is coloured red;
of each 
consecutive integers at least one is coloured blue;
of each 
consecutive integers at least one is coloured green;
of each 
consecutive integers at least one is coloured purple.Determine all good quadruples with
2 replies
Squence problem
AlephG_64 1
N
by RagvaloD
Source: 2025 Finals Portuguese Math Olympiad P1
Francisco wrote a sequence of numbers starting with 
. From the fourth term of the sequence onwards, each term of the sequence is the average of the previous three. Given that the first six terms of the sequence are natural numbers and that the sixth number written was 
, what is the fifth term of the sequence?
. From the fourth term of the sequence onwards, each term of the sequence is the average of the previous three. Given that the first six terms of the sequence are natural numbers and that the sixth number written was 
, what is the fifth term of the sequence?
1 reply
50 points in plane
pohoatza 12
N
by de-Kirschbaum
Source: JBMO 2007, Bulgaria, problem 3
Given are 
points in the plane, no three of them belonging to a same line. Each of these points is colored using one of four given colors. Prove that there is a color and at least 
scalene triangles with vertices of that color.
points in the plane, no three of them belonging to a same line. Each of these points is colored using one of four given colors. Prove that there is a color and at least 
scalene triangles with vertices of that color.
12 replies
beautiful functional equation problem
Medjl 6
N
by Sadigly
Source: Netherlands TST for BxMO 2017 problem 2
Let define a function 
such that :
for all prime numbers 
.
for all positive integers 
find the smallest
such that 
such that :
for all prime numbers 
.
for all positive integers 
find the smallest
such that 
6 replies
