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Polynomials and powers
rmtf1111 27
N
by bjump
Source: RMM 2018 Day 1 Problem 2
Determine whether there exist non-constant polynomials 
and 
with real coefficients satisfying
and 
with real coefficients satisfying
27 replies
Equilateral triangle formed by circle and Fermat point
Mimii08 0
Source: Heard from a friend
Hi! I found this interesting geometry problem and I would really appreciate help with the proof.
Let ABC be an acute triangle, and let T be the Fermat (Torricelli) point of triangle ABC. Let A1, B1, and C1 be the feet of the perpendiculars from T to the sides BC, AC, and AB, respectively. Let ω be the circle passing through points A1, B1, and C1. Let A2, B2, and C2 be the second points where ω intersects the sides BC, AC, and AB, respectively (different from A1, B1, C1).
Prove that triangle A2B2C2 is equilateral.
Let ABC be an acute triangle, and let T be the Fermat (Torricelli) point of triangle ABC. Let A1, B1, and C1 be the feet of the perpendiculars from T to the sides BC, AC, and AB, respectively. Let ω be the circle passing through points A1, B1, and C1. Let A2, B2, and C2 be the second points where ω intersects the sides BC, AC, and AB, respectively (different from A1, B1, C1).
Prove that triangle A2B2C2 is equilateral.
0 replies
Problem 3 IMO 2005 (Day 1)
Valentin Vornicu 121
N
by Rayvhs
Let 
be three positive reals such that 
. Prove that
![\[ \frac { x^5-x^2 }{x^5+y^2+z^2} + \frac {y^5-y^2}{x^2+y^5+z^2} + \frac {z^5-z^2}{x^2+y^2+z^5} \geq 0 . \]](//latex.artofproblemsolving.com/a/1/4/a14adf0f1e35df57c734c7df701d5a67da22dc3c.png)
Hojoo Lee, Korea
be three positive reals such that 
. Prove that![\[ \frac { x^5-x^2 }{x^5+y^2+z^2} + \frac {y^5-y^2}{x^2+y^5+z^2} + \frac {z^5-z^2}{x^2+y^2+z^5} \geq 0 . \]](http://latex.artofproblemsolving.com/a/1/4/a14adf0f1e35df57c734c7df701d5a67da22dc3c.png)
Hojoo Lee, Korea
121 replies
geo problem saved from graveyard
CrazyInMath 1
N
by Curious_Droid
Source: 3rd KYAC Math-A P5
Given triangle 
and orthocenter 
. The foot from to 
is 
respectively. A point 
satisfies that 
and 
are both tangent to 
. A circle passing through 
and tangent to 
intesects at another point 
. 
is an arbitrary point on 
, and 
is the second intesection point of 
and 
.
Prove that
are concyclic.
Proposed by CrazyInMath.
and orthocenter 
. The foot from to 
is 
respectively. A point 
satisfies that 
and 
are both tangent to 
. A circle passing through 
and tangent to 
intesects at another point 
. 
is an arbitrary point on 
, and 
is the second intesection point of 
and 
.Prove that
are concyclic.Proposed by CrazyInMath.
1 reply
From a well-known prob
m4thbl3nd3r 3
N
by aaravdodhia
Find all primes 
so that 
can be a perfect square
so that 
can be a perfect square
3 replies
weird conditions in geo
Davdav1232 1
N
by NO_SQUARES
Source: Israel TST 7 2025 p1
Let 
be an isosceles triangle with 
. Let 
be a point on 
. Let 
be a point inside the triangle such that 
and
![\[
CL \cdot BD = BL \cdot CD.
\]](//latex.artofproblemsolving.com/1/6/7/16771838c95f86e92f79b8d049e46ab473e6287d.png)
Prove that the circumcenter of triangle 
lies on line 
.
be an isosceles triangle with 
. Let 
be a point on 
. Let 
be a point inside the triangle such that 
and![\[
CL \cdot BD = BL \cdot CD.
\]](http://latex.artofproblemsolving.com/1/6/7/16771838c95f86e92f79b8d049e46ab473e6287d.png)
Prove that the circumcenter of triangle 
lies on line 
.
1 reply
Functional equation on R
rope0811 15
N
by ezpotd
Source: IMO ShortList 2003, algebra problem 2
Find all nondecreasing functions 
such that
(i)
(ii)
for all real numbers 
such that 
.
Proposed by A. Di Pisquale & D. Matthews, Australia
such that(i)
(ii)
for all real numbers 
such that 
.Proposed by A. Di Pisquale & D. Matthews, Australia
15 replies
all functions satisfying f(x+yf(x))+y = xy + f(x+y)
falantrng 34
N
by LenaEnjoyer
Source: Balkan MO 2025 P3
Find all functions 
such that for all 
,
![\[f(x+yf(x))+y = xy + f(x+y).\]](//latex.artofproblemsolving.com/c/f/3/cf3d20a041c27244e90876119b4568b7a3e13c03.png)
Proposed by Giannis Galamatis, Greece
such that for all 
,![\[f(x+yf(x))+y = xy + f(x+y).\]](http://latex.artofproblemsolving.com/c/f/3/cf3d20a041c27244e90876119b4568b7a3e13c03.png)
Proposed by Giannis Galamatis, Greece
34 replies
Miklos Schweitzer 1971_7
ehsan2004 1
N
by pi_quadrat_sechstel
Let 
be an integer, let 
be a set of 
elements, and let 
, be distinct subsets of of size at least 
such that ![\[ A_i \cap A_j \not= \emptyset, A_i \cap A_k \not= \emptyset, A_j \cap A_k \not= \emptyset, \;\textrm{imply}\ \;A_i \cap A_j \cap A_k \not= \emptyset \ .\]](//latex.artofproblemsolving.com/c/1/1/c113f5cdce00dfec6714c382a8e1bc74f188f025.png)
Show that 
.
P. Erdos
be an integer, let 
be a set of 
elements, and let 
, be distinct subsets of of size at least 
such that ![\[ A_i \cap A_j \not= \emptyset, A_i \cap A_k \not= \emptyset, A_j \cap A_k \not= \emptyset, \;\textrm{imply}\ \;A_i \cap A_j \cap A_k \not= \emptyset \ .\]](http://latex.artofproblemsolving.com/c/1/1/c113f5cdce00dfec6714c382a8e1bc74f188f025.png)
Show that 
.P. Erdos
1 reply
Functional equation with a twist (it's number theory)
Davdav1232 0
Source: Israel TST 8 2025 p2
Prove that for all primes 
such that 
or 
, there exist integers
![\[
1 \leq a_1 < a_2 < \cdots < a_{(p-1)/2} < p
\]](//latex.artofproblemsolving.com/6/8/e/68e3d73edec43b2cc480832adf23bbbb5f4c7bee.png)
such that
![\[
\prod_{\substack{1 \leq i < j \leq (p-1)/2}} (a_i + a_j)^2 \equiv 1 \pmod{p}.
\]](//latex.artofproblemsolving.com/2/9/e/29edf6ab8cee2a526e7753a140c5a291e987856a.png)
such that 
or 
, there exist integers![\[
1 \leq a_1 < a_2 < \cdots < a_{(p-1)/2} < p
\]](http://latex.artofproblemsolving.com/6/8/e/68e3d73edec43b2cc480832adf23bbbb5f4c7bee.png)
such that![\[
\prod_{\substack{1 \leq i < j \leq (p-1)/2}} (a_i + a_j)^2 \equiv 1 \pmod{p}.
\]](http://latex.artofproblemsolving.com/2/9/e/29edf6ab8cee2a526e7753a140c5a291e987856a.png)
0 replies
