Old or new triangle
by mihaig, Jul 28, 2025, 3:18 PM
One of the Craziest Problem I've ever seen (see the proposers)
by EthanWYX2009, Jul 17, 2025, 1:56 PM
For a positive integer
, let
be the minimal positive integer, such that for any
positive integers
,
,
,
,
takes at most
distinct integer values as
ranges over all non-empty subsets of
.
Determine the value of
.
Proposed by Zhenyu Dong from Hangzhou Xuejun High School, Chunji Wang from Shanghai High School, and Cheng Jiang from Tsinghua University











Determine the value of

Proposed by Zhenyu Dong from Hangzhou Xuejun High School, Chunji Wang from Shanghai High School, and Cheng Jiang from Tsinghua University
This post has been edited 1 time. Last edited by EthanWYX2009, Jul 17, 2025, 1:56 PM
Midpoint of arc black magic
by MarkBcc168, Jul 16, 2025, 3:01 AM
Let
be a triangle with incenter
such that
. The second intersections of
,
, and
with the circumcircle of triangle
are
,
, and
, respectively. Lines
and
intersect at
and lines
and
intersect at
. Suppose the circumcircle of triangles
and
intersect again at
. Lines
and
intersect the circumcircle of triangle
again at
and
, respectively.
Prove that the circumcenter of triangle
lies on
.
Proposed by Thailand
























Prove that the circumcenter of triangle


Proposed by Thailand
Complex number inequalities
by jhz, Mar 26, 2025, 12:28 AM
Find the smallest real number
such that there exist four complex numbers
with
, and for any complex number
, if
, then![\[|az^3+bz^2+cz+d|\le M.\]](//latex.artofproblemsolving.com/a/9/b/a9b4f124bf07c5a5aaa2d2e46c249ac44b7aaf57.png)





![\[|az^3+bz^2+cz+d|\le M.\]](http://latex.artofproblemsolving.com/a/9/b/a9b4f124bf07c5a5aaa2d2e46c249ac44b7aaf57.png)
This post has been edited 1 time. Last edited by jhz, Mar 26, 2025, 12:28 AM
Parallel lines in two-circle configuration
by Tintarn, Apr 4, 2024, 6:43 PM
Let
be an acute triangle,
its circumcircle and
its circumcenter. The altitude from
intersects
in a point
and the segment
intersects the circumcircle of
in a point
. Finally, let
be the midpoint of
. Show that
is parallel to
.













Nice sequence bound
by VicKmath7, Feb 13, 2024, 11:21 AM
Let
be a positive integer. The sequence
of non-negative reals is defined by
for all positive integers
. Show that there exists a constant
, such that
for all positive integers
.







402 divides at least one of n^2-1, n^3-1, n^4-1
by Demetres, Feb 21, 2022, 4:12 PM
Determine for how many positive integers
it holds that
divides at least one of
![\[n^2-1, n^3-1, n^4-1\]](//latex.artofproblemsolving.com/9/a/c/9acc81f21cdbb844acd1d2db370e1ed68bfcbb0c.png)


![\[n^2-1, n^3-1, n^4-1\]](http://latex.artofproblemsolving.com/9/a/c/9acc81f21cdbb844acd1d2db370e1ed68bfcbb0c.png)
Strange Conditional Sequence
by MarkBcc168, Jun 11, 2019, 12:18 AM
Let
be a fixed positive integer. The infinite sequence
is defined in the following way:
is a positive integer, and for every integer
we have
For each
, determine all possible values of
such that every term in the sequence is an integer.







This post has been edited 1 time. Last edited by MarkBcc168, Jun 12, 2019, 2:42 PM
complex numbers inequality
by LeXuS, Mar 8, 2019, 9:07 AM
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