[/quote]
,
be real numbers such that 
and 
and determine the equality cases.
be positive real integer such that 
Prove that
such that 
and 
satisfying :![\[n^{2p}=m^2+n^2+p+1\]](http://latex.artofproblemsolving.com/7/4/e/74ee592cb8b3f3a3996ea34f40b78fde9d3672fc.png)
and 
are integers and 
is a prime number.
Approach: I'm trying to factor 
to get 
.
such that 
divides 
.
![$a,b,c \in [-\frac{1}{2}, \frac{1}{2}]$](http://latex.artofproblemsolving.com/5/0/b/50bc691baa4e4bbef868e4799929c8c93d1b6f1a.png)
.
Let ![$a,b,c \in [-2,2]$](http://latex.artofproblemsolving.com/8/0/6/8066d4353091772c401e3e34f5499a8fa2bd9991.png)
.
be positive real numbers such that 
Prove that 
be prime numbers such that their sum isn't divisible by 
.
positive integer quadruples satisfy:
![\[\sum_{k=1}^n\frac{\phi(k)}k\ln\frac{2^k}{2^k-1}\ge 1-\frac 1{2^n}.\]](http://latex.artofproblemsolving.com/3/d/3/3d32207efff8052214e65aca2505c9e4e7bb581c.png)
be the set of positive real numbers. Find all functions 
such that, for all 
,
Y by arfekete, aidan0626, vadava_lx, eg4334, scannose, OronSH, Tem8, BananaPenguin, zhenghua, ruihl123, happypi31415, Jyuan26
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
