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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
Impossible to search, classic graph problem
AshAuktober 0
Source: Classic
Prove that any graph 
with 
has at least two cycles in it.
with 
has at least two cycles in it.
0 replies
Functional equation
Dadgarnia 11
N
by jasperE3
Source: Iranian TST 2018, second exam day 1, problem 1
Find all functions 
that satisfy the following conditions:
a.
b. The set
is an interval.
Proposed by Navid Safaei
that satisfy the following conditions:a.
b. The set
is an interval.Proposed by Navid Safaei
11 replies
Geo challenge on finding simple ways to solve it
Assassino9931 2
N
by Assassino9931
Source: Bulgaria Spring Mathematical Competition 2025 9.2
Let be an acute scalene triangle inscribed in a circle 
. The angle bisector of 
intersects 
at 
and at 
. The point 
is the midpoint of 
. Let 
be the altitude in 
, and the circumcircle of 
intersects again at 
. Let 
be the midpoint of , and let 
be the reflection of 
with respect to . Prove that the triangles 
and 
are similar.
be an acute scalene triangle inscribed in a circle 
. The angle bisector of 
intersects 
at 
and at 
. The point 
is the midpoint of 
. Let 
be the altitude in 
, and the circumcircle of 
intersects again at 
. Let 
be the midpoint of , and let 
be the reflection of 
with respect to . Prove that the triangles 
and 
are similar.
2 replies
is a pair of real numbers such that
.
Train yourself on folklore NT FE ideas
Assassino9931 2
N
by bo18
Source: Bulgaria Spring Mathematical Competition 2025 9.4
Determine all functions 
such that 
divides 
for any positive integers 
and 
.
such that 
divides 
for any positive integers 
and 
.
2 replies
1 viewing
When is this well known sequence periodic?
Assassino9931 2
N
by Assassino9931
Source: Bulgaria Spring Mathematical Competition 2025 12.2
Determine all values of 
for which the sequence of real numbers with 
for all 
is periodic from the beginning.
for which the sequence of real numbers with 
for all 
is periodic from the beginning.
2 replies
Concurrence of angle bisectors
proglote 65
N
by smbellanki
Source: Brazil MO #5
Let be an acute triangle and 
is orthocenter. Let be the intersection of 
and 
and be the intersection of 
and . The circumcircle of 
cuts the circumcircle of at 
. Prove that the angle bisectors of 
and 
concur at a point on 
be an acute triangle and 
is orthocenter. Let be the intersection of 
and 
and be the intersection of 
and . The circumcircle of 
cuts the circumcircle of at 
. Prove that the angle bisectors of 
and 
concur at a point on 
65 replies
determinant of matrix
jokerjoestar 1
N
by paxtonw
Calculate the determinant of the matrix below: ![\[
A =
\begin{bmatrix}
1 & 2 & 3 & \cdots & n-1 & n \\
2 & 3 & 4 & \cdots & n & 1 \\
3 & 4 & 5 & \cdots & 1 & 2 \\
\vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\
n-1 & n & 1 & \cdots & n-3 & n-2 \\
n & 1 & 2 & \cdots & n-2 & n-1
\end{bmatrix}
\]](//latex.artofproblemsolving.com/7/e/5/7e564602cc0cc14e1501f6654b00a4dfeac50f70.png)
![\[
A =
\begin{bmatrix}
1 & 2 & 3 & \cdots & n-1 & n \\
2 & 3 & 4 & \cdots & n & 1 \\
3 & 4 & 5 & \cdots & 1 & 2 \\
\vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\
n-1 & n & 1 & \cdots & n-3 & n-2 \\
n & 1 & 2 & \cdots & n-2 & n-1
\end{bmatrix}
\]](http://latex.artofproblemsolving.com/7/e/5/7e564602cc0cc14e1501f6654b00a4dfeac50f70.png)
1 reply
