Perfect Squares, Infinite Integers and Integers
by steven_zhang123, Mar 16, 2025, 12:06 PM
For which integer
, are there infinitely many positive integers
such that
is a perfect square? (Here
denotes the integer part of the real number
?





Unsolved Diophantine(I think)
by Nuran2010, Mar 14, 2025, 4:41 PM
Find all solutions for the equation
where
is a positive integer and
is a prime.(Don't get mad at me,I've used the search function and did not see a correct and complete solution anywhere.)



Another NT FE
by nukelauncher, Sep 22, 2020, 11:58 PM
Find all functions
such that
divides
for all positive integers
and
with
.






another geometry problem with sharky-devil point
by anyone__42, Jun 27, 2020, 6:59 PM
Let
be an acute triangle with
, Let
be the intouch triangle with
,
,
,, let
be the intersecttion of the perpendicular from
to
with
, and
.
Prove that
and
are concylic











Prove that


This post has been edited 1 time. Last edited by anyone__42, Jun 28, 2020, 8:35 PM
Foot from vertex to Euler line
by cjquines0, Jul 19, 2017, 4:36 PM
Let
be the foot of perpendicular from
to the Euler line (the line passing through the circumcentre and the orthocentre) of an acute scalene triangle
. A circle
with centre
passes through
and
, and it intersects sides
and
at
and
respectively. Let
be the foot of altitude from
to
, and let
be the midpoint of
. Prove that the circumcentre of triangle
is equidistant from
and
.



















Line Perpendicular to Euler Line
by tastymath75025, Jun 29, 2017, 2:55 AM
Let
be a triangle with circumcircle
, circumcenter
, and orthocenter
. Assume that
and that
. Let
and
be the midpoints of sides
and
, respectively, and let
and
be the feet of the altitudes from
and
in
, respectively. Let
be the intersection of line
with the tangent line to
at
. Let
be the intersection point, other than
, of
with the circumcircle of
. Let
be the intersection of lines
and
. Prove that
.
Proposed by Ray Li



























Proposed by Ray Li
IMO Shortlist 2011, Algebra 5
by orl, Jul 11, 2012, 10:23 PM
Prove that for every positive integer
the set
can be partitioned into
triples in such a way that the numbers from each triple are the lengths of the sides of some obtuse triangle.
Proposed by Canada



Proposed by Canada
p + q + r + s = 9 and p^2 + q^2 + r^2 + s^2 = 21
by who, Jul 8, 2006, 4:51 PM
Four real numbers
,
,
,
satisfy
and
. Prove that there exists a permutation
of
such that
.









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.



To share with readers my favorite problem I came across today :) (Shout for contrib.)
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