graph thory
by o.k.oo, May 23, 2025, 5:14 PM
There are 10 people at a party. None of the 3 friends of each person are friends with each other. What is the maximum number of friends at this party?
This post has been edited 1 time. Last edited by o.k.oo, an hour ago
Consecutive squares are floors
by ICE_CNME_4, May 22, 2025, 1:50 PM
Determine how many positive integers
have the property that both
are consecutive perfect squares.

![\[
\left\lfloor \sqrt{2n - 1} \right\rfloor \quad \text{and} \quad \left\lfloor \sqrt{3n + 2} \right\rfloor
\]](http://latex.artofproblemsolving.com/d/e/c/dec527c4673470e7023d6aed00fac90268bd2fcc.png)
Computing functions
by BBNoDollar, May 18, 2025, 5:25 PM
Let
,
, with
,
. Prove that there exists
such that for every 
(For
and
, the notation
represents
. )






![\[
f_n(x) = \frac{x}{1 + nx}, \quad \text{if and only if } f(x) = \frac{x}{1 + x}, \quad \forall x \geq 0.
\]](http://latex.artofproblemsolving.com/0/f/c/0fc36d9264eb7e103128c489aeae521a859c1fd4.png)




Simson lines on OH circle
by DVDTSB, May 13, 2025, 12:10 PM
Let
and
be two triangles inscribed in the same circle, centered at
, and sharing the same orthocenter
. The Simson lines of the points
with respect to triangle
form a non-degenerate triangle
.
Prove that the orthocenter of
lies on the circle with diameter
.
Note. Assume that the points
lie in this order on the circle and form a convex, non-degenerate hexagon.
Proposed by Andrei Chiriță







Prove that the orthocenter of


Note. Assume that the points

Proposed by Andrei Chiriță
This post has been edited 2 times. Last edited by DVDTSB, May 13, 2025, 12:24 PM
real functional equation
by DottedCaculator, Nov 2, 2023, 11:15 PM
Find all functions
(from the set of real numbers to itself) where
for all reals 
Proposed by cj13609517288



Proposed by cj13609517288
Fixed line
by TheUltimate123, Jun 29, 2023, 1:39 AM
Let
be a point on segment
. Let
be a fixed circle passing through
, and let
be a variable point on
. Let
be the intersection of the tangent to the circumcircle of
at
and the tangent to the circumcircle of
at
. Show that as
varies,
lies on a fixed line.
Proposed by Elliott Liu and Anthony Wang













Proposed by Elliott Liu and Anthony Wang
Finding all possible $n$ on a strange division condition!!
by MathLuis, Nov 12, 2021, 12:23 AM
Find the sum of all positive integers
such that
is an integer.


This post has been edited 2 times. Last edited by MathLuis, Nov 12, 2021, 12:24 AM
IMO Shortlist 2012, Geometry 8
by lyukhson, Jul 29, 2013, 12:30 PM
Let
be a triangle with circumcircle
and
a line without common points with
. Denote by
the foot of the perpendicular from the center of
to
. The side-lines
intersect
at the points
different from
. Prove that the circumcircles of the triangles
,
and
have a common point different from
or are mutually tangent at
.
Proposed by Cosmin Pohoata, Romania
















Proposed by Cosmin Pohoata, Romania
This post has been edited 3 times. Last edited by djmathman, Aug 11, 2017, 2:28 PM
Reason: added source
Reason: added source
IMO 2012 P5
by mathmdmb, Jul 11, 2012, 7:03 PM
Let
be a triangle with
, and let
be the foot of the altitude from
. Let
be a point in the interior of the segment
. Let
be the point on the segment
such that
. Similarly, let
be the point on the segment
such that
. Let
be the point of intersection of
and
.
Show that
.
Proposed by Josef Tkadlec, Czech Republic















Show that

Proposed by Josef Tkadlec, Czech Republic
This post has been edited 3 times. Last edited by Eternica, Jun 19, 2024, 10:03 AM
Robots in Space
by rrusczyk, Aug 25, 2011, 1:00 AM
I should have shared this sooner; crossposted from the AoPS blog:
AoPS blog wrote:
The folks at TopCoder are working with MIT, NASA, and DARPA to offer a robotics/programming competition for high school students. Winning teams may have their programs run by astronauts on the International Space Station! More details here. The deadline for registering is September 5.
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