F=(F^3+F^3)/9-2F^3
by Yiyj1, Apr 10, 2025, 3:12 AM
Define the Fibonacci sequence 
as 
fir 
. Prove that 
for all 
.
as 
fir 
. Prove that 
for all 
.Inspired by 2025 Nepal
by sqing, Apr 10, 2025, 2:14 AM
Let 
be positive reals such that 
. Prove that
be positive reals such that 
. Prove that
This post has been edited 1 time. Last edited by sqing, 2 hours ago
be reals such that 
Prove that
Funny function that there isn't exist
by ItzsleepyXD, Apr 10, 2025, 1:50 AM
Determine all functions 
such that, for all positive integers 
and 
,
such that, for all positive integers 
and 
,
NT function debut
by AshAuktober, Apr 9, 2025, 3:53 PM
Let 
be a function taking in positive integers and outputting nonnegative integers, defined as follows:
is the number of positive integers with 
such that the equation 
has an integer solution 
.
Find all positive integers
such that
and 
(Adit Aggarwal, India.)
be a function taking in positive integers and outputting nonnegative integers, defined as follows:
is the number of positive integers with 
such that the equation 
has an integer solution 
.Find all positive integers
such that
and 
(Adit Aggarwal, India.)This post has been edited 1 time. Last edited by AshAuktober, an hour ago
4 lines concurrent
by Zavyk09, Apr 9, 2025, 11:51 AM
Let 
be triangle with circumcenter 
and orthocenter 
. 
intersect again at 
respectively. Lines through parallel to 
intersects 
at 
respectively. Point 
such that 
is a parallelogram. Prove that lines 
and 
are concurrent at a point on 
.
be triangle with circumcenter 
and orthocenter 
. 
intersect again at 
respectively. Lines through parallel to 
intersects 
at 
respectively. Point 
such that 
is a parallelogram. Prove that lines 
and 
are concurrent at a point on 
.where a, b, c are positive real numbers
by eyesofgod1930, Jun 8, 2020, 2:06 PM
where 
are positive real numbers.Prove that
are positive real numbers.Prove that
Integral solutions
by KDS, Jul 12, 2009, 10:45 AM
Prove that the equation 
has a unique solution in integers with 
and 
.
has a unique solution in integers with 
and 
.Inequality => square
by Rushil, Oct 7, 2005, 5:12 AM
Suppose 
is a cyclic quadrilateral inscribed in a circle of radius one unit. If 
, prove that is a square.
is a cyclic quadrilateral inscribed in a circle of radius one unit. If 
, prove that is a square.Isos Trap
by MithsApprentice, Oct 3, 2005, 10:47 PM
Let be an isosceles trapezoid with 
. The inscribed circle 
of triangle 
meets 
at 
. Let 
be a point on the (internal) angle bisector of 
such that 
. Let the circumscribed circle of triangle 
meet line at 
and 
. Prove that the triangle 
is isosceles.
be an isosceles trapezoid with 
. The inscribed circle 
of triangle 
meets 
at 
. Let 
be a point on the (internal) angle bisector of 
such that 
. Let the circumscribed circle of triangle 
meet line at 
and 
. Prove that the triangle 
is isosceles.
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