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Calculating sum of the numbers
Sadigly 5
N
an hour ago
by aokmh3n2i2rt
Source: Azerbaijan Junior MO 2025 P4
A
square is filled with numbers
.The numbers inside four
squares is summed,and arranged in an increasing order. Is it possible to obtain the following sequences as a result of this operation?








5 replies
Swap to the symmedian
Noob_at_math_69_level 7
N
an hour ago
by awesomeming327.
Source: DGO 2023 Team P1
Let
be a triangle with points
lie on the perpendicular bisector of
such that
lie on a circle. Suppose
are perpendicular to sides
at points
The tangent lines from points
to the circumcircle of
intersects at point
Prove that:
are parallel.
Proposed by Paramizo Dicrominique











Proposed by Paramizo Dicrominique
7 replies
Find (AB * CD) / (AC * BD) & prove orthogonality of circles
Maverick 15
N
an hour ago
by Ilikeminecraft
Source: IMO 1993, Day 1, Problem 2
Let
,
,
,
be four points in the plane, with
and
on the same side of the line
, such that
and
. Find the ratio
![\[\frac{AB \cdot CD}{AC \cdot BD}, \]](//latex.artofproblemsolving.com/f/3/c/f3cf9729b32b210faeea6e2d23da3582be2ae061.png)
and prove that the circumcircles of the triangles
and
are orthogonal. (Intersecting circles are said to be orthogonal if at either common point their tangents are perpendicuar. Thus, proving that the circumcircles of the triangles
and
are orthogonal is equivalent to proving that the tangents to the circumcircles of the triangles
and
at the point
are perpendicular.)









![\[\frac{AB \cdot CD}{AC \cdot BD}, \]](http://latex.artofproblemsolving.com/f/3/c/f3cf9729b32b210faeea6e2d23da3582be2ae061.png)
and prove that the circumcircles of the triangles







15 replies
f(x+f(x)+f(y))=x+f(x+y)
dangerousliri 10
N
2 hours ago
by jasperE3
Source: FEOO, Shortlist A5
Find all functions
such that for any positive real numbers
and
,
Proposed by Athanasios Kontogeorgis, Grecce, and Dorlir Ahmeti, Kosovo




10 replies
n-variable inequality
ABCDE 66
N
2 hours ago
by ND_
Source: 2015 IMO Shortlist A1, Original 2015 IMO #5
Suppose that a sequence
of positive real numbers satisfies
for every positive integer
. Prove that
for every
.

![\[a_{k+1}\geq\frac{ka_k}{a_k^2+(k-1)}\]](http://latex.artofproblemsolving.com/0/3/9/03909a5f60d944063182ec97cf73f283178e8d4d.png)



66 replies

Euler Line Madness
raxu 75
N
3 hours ago
by lakshya2009
Source: TSTST 2015 Problem 2
Let ABC be a scalene triangle. Let
,
and
be the respective intersections with BC of the internal angle bisector, external angle bisector, and the median from A. The circumcircle of
intersects
a second time at point
different from A. Define
and
analogously. Prove that the circumcenter of
lies on the Euler line of ABC.
(The Euler line of ABC is the line passing through the circumcenter, centroid, and orthocenter of ABC.)
Proposed by Ivan Borsenco









(The Euler line of ABC is the line passing through the circumcenter, centroid, and orthocenter of ABC.)
Proposed by Ivan Borsenco
75 replies
Own made functional equation
Primeniyazidayi 8
N
3 hours ago
by MathsII-enjoy
Source: own(probably)
Find all functions
such that
for all



8 replies

IMO ShortList 2002, geometry problem 7
orl 110
N
4 hours ago
by SimplisticFormulas
Source: IMO ShortList 2002, geometry problem 7
The incircle
of the acute-angled triangle
is tangent to its side
at a point
. Let
be an altitude of triangle
, and let
be the midpoint of the segment
. If
is the common point of the circle
and the line
(distinct from
), then prove that the incircle
and the circumcircle of triangle
are tangent to each other at the point
.















110 replies

Cute NT Problem
M11100111001Y1R 6
N
4 hours ago
by X.Allaberdiyev
Source: Iran TST 2025 Test 4 Problem 1
A number
is called lucky if it has at least two distinct prime divisors and can be written in the form:
where
are distinct prime numbers that divide
. (Note: it is possible that
has other prime divisors not among
.) Prove that for every prime number
, there exists a lucky number
such that
.

![\[
n = p_1^{\alpha_1} + \cdots + p_k^{\alpha_k}
\]](http://latex.artofproblemsolving.com/7/4/4/744a5ccaeb9476ebd7d999c395762cb6e99a7a71.png)







6 replies
China MO 2021 P6
NTssu 23
N
4 hours ago
by bin_sherlo
Source: CMO 2021 P6
Find
, such that for any
,



23 replies
