Difficult lattice point coloring problem
by CBMaster, Apr 11, 2025, 5:29 PM
This post has been edited 5 times. Last edited by CBMaster, 14 minutes ago
Reason: .
Reason: .
TST Junior Romania 2025
by ant_, Apr 11, 2025, 5:01 PM
Consider the isosceles triangle 
, with 
, and the circle 
with center 
and radius 
. Denote by 
the midpoint of side . The line 
intersects the circle for the second time in 
. Let 
be a point on the circle such that 
and 
. Show that 
.
, with 
, and the circle 
with center 
and radius 
. Denote by 
the midpoint of side . The line 
intersects the circle for the second time in 
. Let 
be a point on the circle such that 
and 
. Show that 
.Line passes through the fix point
by moony_, Apr 11, 2025, 1:44 PM
Let be a triangle. 
and 
are points, such that 
, 
= 
and 
( - inside 
and - oitside). Proove that line 
passes through the fix point.
be a triangle. 
and 
are points, such that 
, 
= 
and 
( - inside 
and - oitside). Proove that line 
passes through the fix point.This post has been edited 5 times. Last edited by moony_, 5 hours ago
R+ Functional Equation
by Mathdreams, Apr 11, 2025, 1:27 PM
Find all functions 
such that ![\[f(f(x)) + xf(xy) = x + f(y)\]](//latex.artofproblemsolving.com/a/6/f/a6f84dd53eec985b90bad3df6580eeaf29d73195.png)
for all positive real numbers 
and 
.
(Andrew Brahms, USA)
such that ![\[f(f(x)) + xf(xy) = x + f(y)\]](http://latex.artofproblemsolving.com/a/6/f/a6f84dd53eec985b90bad3df6580eeaf29d73195.png)
for all positive real numbers 
and 
.(Andrew Brahms, USA)
This post has been edited 1 time. Last edited by Mathdreams, 6 hours ago
evan chen??
by Captainscrubz, Apr 11, 2025, 3:50 AM
Let point and be on sides 
and respectively in 
such that 
. Let 
be the circumcenter of 
and let 
. Prove that 
and be on sides 
and respectively in 
such that 
. Let 
be the circumcenter of 
and let 
. Prove that 
This post has been edited 1 time. Last edited by Captainscrubz, Today at 3:51 AM
Reason: resp.
Reason: resp.
be the n-th prime number in increasing order for 
.
satisfying 
for all 
Stop Projecting your insecurities
by naman12, Dec 12, 2022, 5:06 PM
Let be an acute triangle. Let be the midpoint of side 
, and let and 
be the feet of the altitudes from 
and 
, respectively. Suppose that the common external tangents to the circumcircles of triangles 
and 
intersect at a point 
, and that lies on the circumcircle of . Prove that line 
is perpendicular to line .
Kevin Cong
be an acute triangle. Let be the midpoint of side 
, and let and 
be the feet of the altitudes from 
and 
, respectively. Suppose that the common external tangents to the circumcircles of triangles 
and 
intersect at a point 
, and that lies on the circumcircle of . Prove that line 
is perpendicular to line .Kevin Cong
This post has been edited 2 times. Last edited by v_Enhance, Dec 19, 2022, 4:04 AM
Functional equation
by Pmshw, May 8, 2022, 3:57 PM
Find all functions 
such that for any real value of 
we have:
such that for any real value of 
we have:
This post has been edited 1 time. Last edited by Pmshw, May 8, 2022, 3:58 PM
SMO 2015 open q3
by dominicleejun, Mar 31, 2018, 9:17 AM
Find all functions 
, where 
is the set of real numbers, such that
, where 
is the set of real numbers, such that
This post has been edited 2 times. Last edited by dominicleejun, Aug 16, 2019, 9:16 PM
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