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Find (AB * CD) / (AC * BD) & prove orthogonality of circles
Maverick 15
N
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.)
, 
, 
, 
be four points in the plane, with and on the same side of the line 
, such that 
and 
. Find the ratio![\[\frac{AB \cdot CD}{AC \cdot BD}, \]](http://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.)
15 replies
f(x+f(x)+f(y))=x+f(x+y)
dangerousliri 10
N
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
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
by ND_
Source: 2015 IMO Shortlist A1, Original 2015 IMO #5
Suppose that a sequence 
of positive real numbers satisfies ![\[a_{k+1}\geq\frac{ka_k}{a_k^2+(k-1)}\]](//latex.artofproblemsolving.com/0/3/9/03909a5f60d944063182ec97cf73f283178e8d4d.png)
for every positive integer 
. Prove that 
for every 
.
of positive real numbers satisfies ![\[a_{k+1}\geq\frac{ka_k}{a_k^2+(k-1)}\]](http://latex.artofproblemsolving.com/0/3/9/03909a5f60d944063182ec97cf73f283178e8d4d.png)
for every positive integer 
. Prove that 
for every 
.
66 replies
1 viewing
Euler Line Madness
raxu 75
N
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
, 
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
75 replies
Own made functional equation
Primeniyazidayi 8
N
by MathsII-enjoy
Source: own(probably)
Find all functions 
such that 
for all 
such that 
for all 
8 replies
IMO ShortList 2002, geometry problem 7
orl 110
N
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 .
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
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:
![\[
n = p_1^{\alpha_1} + \cdots + p_k^{\alpha_k}
\]](//latex.artofproblemsolving.com/7/4/4/744a5ccaeb9476ebd7d999c395762cb6e99a7a71.png)
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 
.
is called lucky if it has at least two distinct prime divisors and can be written in the form:![\[
n = p_1^{\alpha_1} + \cdots + p_k^{\alpha_k}
\]](http://latex.artofproblemsolving.com/7/4/4/744a5ccaeb9476ebd7d999c395762cb6e99a7a71.png)
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 
.
6 replies
,
,
Prove that the circumcentres of the triangles are collinear
orl 19
N
by Ilikeminecraft
Source: IMO Shortlist 1997, Q9
Let 
be a non-isosceles triangle with incenter 
Let 
be the smaller circle through 
tangent to 
and 
(the addition of indices being mod 3). Let 
be the second point of intersection of 
and 
Prove that the circumcentres of the triangles 
are collinear.
be a non-isosceles triangle with incenter 
Let 
be the smaller circle through 
tangent to 
and 
(the addition of indices being mod 3). Let 
be the second point of intersection of 
and 
Prove that the circumcentres of the triangles 
are collinear.
19 replies
such that 
coprime, 
and 
.
