3 var with parameter (on GMA 575)
by mihaig, Jul 28, 2025, 7:34 PM
Find the least constant 
such that
for all 
with 
such that
for all 
with 
be with no obtuse angles.
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
, 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
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
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
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)
such that there exist four complex numbers 
with 
, and for any complex number 
, if 
, then![\[|az^3+bz^2+cz+d|\le M.\]](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 
.
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 
.
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)
it holds that 
divides at least one of![\[n^2-1, n^3-1, n^4-1\]](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.
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
Let 
, 
, be 3 complex numbers with 
. Prove that 
, 
, be 3 complex numbers with 
. Prove that 
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