.
and 
,
Y by cubres
Find all polynomials 
with real coefficients obeying
![\[P(x) P(x+1) = P(x^2 + x + 1)\]](//latex.artofproblemsolving.com/b/e/f/beff3420fa8394f4f98c031d81eea6a5a3b8d28c.png)
for all real numbers 
.
with real coefficients obeying![\[P(x) P(x+1) = P(x^2 + x + 1)\]](http://latex.artofproblemsolving.com/b/e/f/beff3420fa8394f4f98c031d81eea6a5a3b8d28c.png)
for all real numbers 
.G
Topic
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Last Poster
CSMGO P3: A problem on the infamous line XH
amar_04 12
N
by WLOGQED1729
Source: https://artofproblemsolving.com/community/c594864h2372843p19407517
Let 
be a scalene triangle with the orthocenter 
. Let 
be the reflection of 
over 
and 
be the reflection of 
over 
. Let the tangents to the circumcircle of at points and meet at a point 
. Suppose that the lines 
and 
meet at a point 
. Prove that 
is perpendicular to 
.
be a scalene triangle with the orthocenter 
. Let 
be the reflection of 
over 
and 
be the reflection of 
over 
. Let the tangents to the circumcircle of at points and meet at a point 
. Suppose that the lines 
and 
meet at a point 
. Prove that 
is perpendicular to 
.
12 replies
1 viewing
Three mutually tangent circles
math154 8
N
by lakshya2009
Source: ELMO Shortlist 2011, G2
Let 
be three mutually tangent circles such that 
are externally tangent at , 
are internally tangent at 
, and 
are internally tangent at . Let 
be the centers of , respectively. Given that is the foot of the perpendicular from to , prove that 
.
David Yang.
be three mutually tangent circles such that 
are externally tangent at , 
are internally tangent at 
, and 
are internally tangent at . Let 
be the centers of , respectively. Given that is the foot of the perpendicular from to , prove that 
.David Yang.
8 replies
Line AT passes through either S_1 or S_2
v_Enhance 89
N
by zuat.e
Source: USA December TST for 57th IMO 2016, Problem 2
Let 
be a scalene triangle with circumcircle 
, and suppose the incircle of touches at 
. The angle bisector of 
meets and at 
and 
. The circumcircle of 
intersects the -excircle at 
, 
, and at 
. Prove that line passes through either or .
Proposed by Evan Chen
be a scalene triangle with circumcircle 
, and suppose the incircle of touches at 
. The angle bisector of 
meets and at 
and 
. The circumcircle of 
intersects the -excircle at 
, 
, and at 
. Prove that line passes through either or .Proposed by Evan Chen
89 replies
Easy geo
kooooo 3
N
by Blackbeam999
Source: own
In triangle , let 
and be the circumcenter and orthocenter, respectively. Let 
and 
be the midpoints of and , respectively, and let and be the feet of the perpendiculars from and to the opposite sides, respectively. Show that if is the intersection of 
and 
, then 
is perpendicular to 
.
, let 
and be the circumcenter and orthocenter, respectively. Let 
and 
be the midpoints of and , respectively, and let and be the feet of the perpendiculars from and to the opposite sides, respectively. Show that if is the intersection of 
and 
, then 
is perpendicular to 
.3 replies
Interesting
imnotgoodatmathsorry 0
Source: Own.
Problem 1. Let 
. Prove that:
Problem 2. Let
. Prove that:
![$(a+b+c)^4(ab+bc+ca) - 9\sum{\frac{a}{c}} \ge 54[(a+b)(b+c)(c+a)+abc-1]$](//latex.artofproblemsolving.com/d/3/c/d3cf439c79cb84ca23083e51b6190ab5118f34e8.png)
. Prove that:
Problem 2. Let
. Prove that:![$(a+b+c)^4(ab+bc+ca) - 9\sum{\frac{a}{c}} \ge 54[(a+b)(b+c)(c+a)+abc-1]$](http://latex.artofproblemsolving.com/d/3/c/d3cf439c79cb84ca23083e51b6190ab5118f34e8.png)
0 replies
$n^{22}-1$ and $n^{40}-1$
v_Enhance 5
N
by Kempu33334
Source: OTIS Mock AIME 2024 #13
Let 
denote the sum of all integers 
such that 
and exactly one of 
and 
is divisible by 
. Compute the remainder when is divided by 
.
Raymond Zhu
denote the sum of all integers 
such that 
and exactly one of 
and 
is divisible by 
. Compute the remainder when is divided by 
.Raymond Zhu
5 replies
Annoying 2^x-5 = 11^y
Valentin Vornicu 38
N
by Kempu33334
Find all positive integer solutions to 
.
.38 replies
Polish MO Finals 2014, Problem 5
j___d 14
N
by Kempu33334
Source: Polish MO Finals 2014
Find all pairs 
of positive integers that satisfy
.
of positive integers that satisfy
.
14 replies
IMO LongList 1985 CYP2 - System of Simultaneous Equations
Amir Hossein 15
N
by Kempu33334
Solve the system of simultaneous equations
![\[\sqrt x - \frac 1y - 2w + 3z = 1,\]](//latex.artofproblemsolving.com/d/0/f/d0ff4819db851e6c12f738dcdb6b81bbd2ed0ca7.png)
![\[x + \frac{1}{y^2} - 4w^2 - 9z^2 = 3,\]](//latex.artofproblemsolving.com/6/5/3/653c1547620f85ff761817a5cc333f16bce9c4c6.png)
![\[x \sqrt x - \frac{1}{y^3} - 8w^3 + 27z^3 = -5,\]](//latex.artofproblemsolving.com/1/b/9/1b9c2ed383acc931aa3607ba82031e182e18f3d1.png)
![\[x^2 + \frac{1}{y^4} - 16w^4 - 81z^4 = 15.\]](//latex.artofproblemsolving.com/e/0/3/e032ce109cd74bbae47dc6bca04dcab2d8f9ef29.png)
![\[\sqrt x - \frac 1y - 2w + 3z = 1,\]](http://latex.artofproblemsolving.com/d/0/f/d0ff4819db851e6c12f738dcdb6b81bbd2ed0ca7.png)
![\[x + \frac{1}{y^2} - 4w^2 - 9z^2 = 3,\]](http://latex.artofproblemsolving.com/6/5/3/653c1547620f85ff761817a5cc333f16bce9c4c6.png)
![\[x \sqrt x - \frac{1}{y^3} - 8w^3 + 27z^3 = -5,\]](http://latex.artofproblemsolving.com/1/b/9/1b9c2ed383acc931aa3607ba82031e182e18f3d1.png)
![\[x^2 + \frac{1}{y^4} - 16w^4 - 81z^4 = 15.\]](http://latex.artofproblemsolving.com/e/0/3/e032ce109cd74bbae47dc6bca04dcab2d8f9ef29.png)
15 replies
Fneqn or Realpoly?
G
H
G
H
Source: India, not sure which year. Found in OTIS pset
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Y by cubres
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