[/quote]
,
be a sequence of positive integers with 
,
and![\[a_n = a_{n-1}^{a_{n-1}a_{n-2}}-1\]](http://latex.artofproblemsolving.com/1/b/f/1bf1a88979e7d2b33530de979d66a4e81032953d.png)
for all 
.
is a prime less than 
for some positive integer 
,
such that 
.
be a triangle with circumcircle 
,
.
,
,
with 
,
,
,
intersect at 
and lines 
and 
intersect at 
.
and 
intersect again at 
.
and 
intersect the circumcircle of triangle 
again at 
and 
,
lies on 
.
for which
be an acute angled scalene triangle with circumcircle 
.
and 
be the circumcenter and orthocenter of 
respectively. Let 
and 
be points on segments 
and 
Prove that 
and 
intersect on the circumcircle of 
.
grid with 
rows and 
columns, and each square of the grid is filled with a non-negative integer. Let 
be the average of all the numbers in the grid. Prove that if 
the there exist two identical rows (meaning same numbers in the same order).
is called feirense if it is possible to write on a sheet of paper some integers such that every positive divisor of 
be a square with 
side length and with center 
.
is drawn where point 
lies on line 
.
Y by arfekete, aidan0626, vadava_lx, eg4334, scannose, OronSH, Tem8, BananaPenguin, zhenghua, ruihl123, happypi31415
Hello AoPS,
A reputable source that is of a certain credibility has communicated me about details of Orange MOP, a new pathway to qualify for MOP. In particular, 3 rounds of a 10-problem proof-style examination, covering a variety of mathematical topics that requires proofs will be held from September 27, 2025 12:00 AM - November 8, 2025 11:59 PM EST. Each round will occur biweekly on a Saturday starting from September 27 as described above. The deadline for late submissions will be November 20, 2025 11:59 PM EST.
Solutions can be either handwritten or typed digitally with
. If you are sending solutions digitally through physical scan, please make sure your handwriting is eligible. Inability to discern hand-written solutions may warrant point deductions.
As for rules, digital resources and computational intelligence systems are allowed. Textbooks, reference handouts, and calculators are also a freedom provided by the MAA.
The link is said to be posted on the MAA website during the summer, and invites aspiring math students of all grade levels to participate. As for scoring, solutions will be graded on a
-point scale, and solutions will be graded in terms of both elegance and correctness.
As for qualification for further examinations, the Orange MOP examination passes both the AIME and USAJMO/USAMO requirement thresholds, and the top 5 scorers will receive the benefits and prestige of participating at the national level in the MOP program, and possibly the USA TST and the USA IMO team.
I implore you to consider this rare oppourtunity.
Warm wishes.
A reputable source that is of a certain credibility has communicated me about details of Orange MOP, a new pathway to qualify for MOP. In particular, 3 rounds of a 10-problem proof-style examination, covering a variety of mathematical topics that requires proofs will be held from September 27, 2025 12:00 AM - November 8, 2025 11:59 PM EST. Each round will occur biweekly on a Saturday starting from September 27 as described above. The deadline for late submissions will be November 20, 2025 11:59 PM EST.
Solutions can be either handwritten or typed digitally with
. If you are sending solutions digitally through physical scan, please make sure your handwriting is eligible. Inability to discern hand-written solutions may warrant point deductions.As for rules, digital resources and computational intelligence systems are allowed. Textbooks, reference handouts, and calculators are also a freedom provided by the MAA.
The link is said to be posted on the MAA website during the summer, and invites aspiring math students of all grade levels to participate. As for scoring, solutions will be graded on a
-point scale, and solutions will be graded in terms of both elegance and correctness.As for qualification for further examinations, the Orange MOP examination passes both the AIME and USAJMO/USAMO requirement thresholds, and the top 5 scorers will receive the benefits and prestige of participating at the national level in the MOP program, and possibly the USA TST and the USA IMO team.
I implore you to consider this rare oppourtunity.
Warm wishes.
This post has been edited 3 times. Last edited by blueprimes, Apr 2, 2025, 3:32 AM
