Cool functional equation
by Rayanelba, Apr 30, 2025, 7:39 PM
Find all functions 
that verify the following equation for all 
:
that verify the following equation for all 
:
primes,exponentials,factorials
by skellyrah, Apr 30, 2025, 6:31 PM
find all primes p,q such that 
is a prime number
is a prime numberThis post has been edited 3 times. Last edited by skellyrah, 4 hours ago
Queue geo
by vincentwant, Apr 30, 2025, 3:54 PM
Let 
be an acute scalene triangle with circumcenter 
. Let 
be the feet of the altitudes from 
to 
respectively. Let 
be the midpoint of 
. Let 
be the circle with diameter 
. Let 
be the intersection of 
and 
. Let 
be the orthocenter of . Let 
be the intersection of 
and . Let 
be the lines through 
tangent to 
respectively. Let 
be the intersection of 
and 
. Let 
be the intersection of 
and 
. Let 
be the line through parallel to 
and let 
be the reflection of across . Prove that 
is tangent to 
.
be an acute scalene triangle with circumcenter 
. Let 
be the feet of the altitudes from 
to 
respectively. Let 
be the midpoint of 
. Let 
be the circle with diameter 
. Let 
be the intersection of 
and 
. Let 
be the orthocenter of . Let 
be the intersection of 
and . Let 
be the lines through 
tangent to 
respectively. Let 
be the intersection of 
and 
. Let 
be the intersection of 
and 
. Let 
be the line through parallel to 
and let 
be the reflection of across . Prove that 
is tangent to 
.This post has been edited 1 time. Last edited by vincentwant, Today at 3:55 PM
Very easy NT
by GreekIdiot, Apr 30, 2025, 2:49 PM
Prove that there exists no natural number 
such that 
.
such that 
.Functional Geometry
by GreekIdiot, Apr 27, 2025, 1:08 PM
Let 
be a function from the Euclidean plane to the real numbers such that 
for any acute triangle 
with circumcenter , centroid 
and orthocenter . Prove that 
is constant.
be a function from the Euclidean plane to the real numbers such that 
for any acute triangle 
with circumcenter , centroid 
and orthocenter . Prove that 
is constant.This post has been edited 1 time. Last edited by GreekIdiot, Apr 27, 2025, 1:08 PM
Can you construct the incenter of a triangle ABC?
by PennyLane_31, Oct 29, 2023, 1:53 AM
Given points and , Jaqueline has a ruler that allows tracing the line 
. Jaqueline also has a special object that allows the construction of a circle of diameter . Also, always when two circles (or a circle and a line, or two lines) intersect, she can mark the points of the intersection with a pencil and trace more lines and circles using these dispositives by the points marked. Initially, she has an acute scalene triangle . Show that Jaqueline can construct the incenter of .
and , Jaqueline has a ruler that allows tracing the line 
. Jaqueline also has a special object that allows the construction of a circle of diameter . Also, always when two circles (or a circle and a line, or two lines) intersect, she can mark the points of the intersection with a pencil and trace more lines and circles using these dispositives by the points marked. Initially, she has an acute scalene triangle . Show that Jaqueline can construct the incenter of .This post has been edited 1 time. Last edited by PennyLane_31, Oct 26, 2024, 3:18 PM
Right-angled triangle if circumcentre is on circle
by liberator, Jan 4, 2016, 9:41 PM
Let the excircle of triangle opposite the vertex 
be tangent to the side at the point 
. Define the points 
on 
and 
on 
analogously, using the excircles opposite 
and 
, respectively. Suppose that the circumcentre of triangle 
lies on the circumcircle of triangle . Prove that triangle is right-angled.
Proposed by Alexander A. Polyansky, Russia
opposite the vertex 
be tangent to the side at the point 
. Define the points 
on 
and 
on 
analogously, using the excircles opposite 
and 
, respectively. Suppose that the circumcentre of triangle 
lies on the circumcircle of triangle . Prove that triangle is right-angled.Proposed by Alexander A. Polyansky, Russia
Rectangle EFGH in incircle, prove that QIM = 90
by v_Enhance, Jul 18, 2014, 7:48 PM
Let be a triangle with incenter , and suppose the incircle is tangent to and at 
and 
. Denote by and the reflections of and over . Let be the intersection of with 
, and let 
be the midpoint of . Prove that 
and 
are perpendicular.
be a triangle with incenter , and suppose the incircle is tangent to and at 
and 
. Denote by and the reflections of and over . Let be the intersection of with 
, and let 
be the midpoint of . Prove that 
and 
are perpendicular.Another quadrilateral in a circle
by v_Enhance, May 3, 2013, 8:09 PM
Let 
be a quadrilateral inscribed in a circle , and let be a point on the extension of 
such that 
and 
are tangent to . The tangent at intersects at and the line at 
. Let be the second point of intersection between and . Prove that , , are collinear.
be a quadrilateral inscribed in a circle , and let be a point on the extension of 
such that 
and 
are tangent to . The tangent at intersects at and the line at 
. Let be the second point of intersection between and . Prove that , , are collinear.af(a)+bf(b)+2ab=x^2 for all natural a, b - show that f(a)=a
by shoki, May 14, 2011, 3:53 PM
Suppose that 
is a function for which the expression 
for all 
is always a perfect square. Prove that 
for all 
.
is a function for which the expression 
for all 
is always a perfect square. Prove that 
for all 
.The oldest, shortest words — "yes" and "no" — are those which require the most thought.
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