where 
denotes the Fibonacci sequence.**
is defined by the conditions
and 
for a set of indices 
.
satisfying the equation![\[
(I_p + A)^n = B^n + C^n.
\]](http://latex.artofproblemsolving.com/2/d/7/2d75d47203353160ccac356251659bd16a4cb5c8.png)
is a postive integer.
be a function that is at least twice differentiable on an open interval containing ![\( [0, 2\pi] \)](http://latex.artofproblemsolving.com/9/7/7/977adaa4f83dc8164975a3e78539f27cbb354adf.png)
.![\[
f(0) = f(2\pi) = f'(0) = f'(2\pi) = 0
\]](http://latex.artofproblemsolving.com/d/9/9/d998bdd53ffec1fa4c88688d7402ad25efa31112.png)
and![\[
f(x) + f''(x) \geq 0, \quad \forall x \in [0,2\pi].
\]](http://latex.artofproblemsolving.com/a/1/c/a1c24260aa02cd989630295d9de8741af659e958.png)
Prove that 
for all ![\( x \in [0,2\pi] \)](http://latex.artofproblemsolving.com/d/c/5/dc59f105fb3b9f6be175d2c035ba2ead4b9d984a.png)
.
,
and Moivre's formula.
.
,
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The MAA may not be perfect, but in my view, their handling of the 2024–2025 competition cycle was truly commendable. Not a single leak of the AMC 8/10/12, AIME I/II, or USA(J)MO appeared on AoPS. This is an impressive feat in an environment where information travels fast and far.
Other supposed leaks remain nothing more than rumors; they are unverifiable, anecdotal, and almost certainly unfounded.
Let’s be realistic: the MAA did everything reasonably possible to protect the integrity of these contests. Leaks exist in many corners of life. Take government (look: https://www.washingtonpost.com/national-security/2025/03/24/trump-leak-signal-jeffrey-goldberg-atlantic/), corporations, and yes, even standardized testing. But here’s the difference: in those other domains, people don’t constantly talk about it. When someone scores a 1600 on the SAT or gets a 5 on the AP Calculus exam, no one jumps to accuse them of cheating. There's a presumption of merit and legitimacy.
Sadly, the same can’t be said for AMC competitions. Ironically, it's not cheating that undermines the credibility of the AMC; it’s the constant speculation about it. Even when there's no clear evidence, the persistent noise about leaks creates doubt. And that doubt sticks. It cheapens success.
You might feel justified in complaining about alleged cheating because you didn’t qualify this year. But what happens next year, when you make USA(J)MO? If the narrative continues to be “oh, everyone just cheats anyway,” your own hard-earned success will be met with skepticism. Not because you cheated—but because you've helped build a culture where perception overrides reality.
I’ve heard plenty of damaging and unfounded accusations just being an active member in my local math community. Teachers claiming a student with a B+ in Algebra II must have cheated to make AIME. Parents whispering that a “Chinese teacher” who runs group classes must be leaking problems because several of their students got DHR on the AMC 8. These accusations are unfair, baseless, and rooted in misunderstanding. School math is not competition math, and success often comes from years of focused training, not shortcuts.
So let’s stop. The more we talk about unverified cheating, the more we hurt ourselves and our peers. It’s time we protect the prestige of these contests—for the sake of those who’ve earned it, and for those who will in the future.
If you’re frustrated by your results, let that energy drive your preparation—not blame. Focus on growth, not gossip.
Other supposed leaks remain nothing more than rumors; they are unverifiable, anecdotal, and almost certainly unfounded.
Let’s be realistic: the MAA did everything reasonably possible to protect the integrity of these contests. Leaks exist in many corners of life. Take government (look: https://www.washingtonpost.com/national-security/2025/03/24/trump-leak-signal-jeffrey-goldberg-atlantic/), corporations, and yes, even standardized testing. But here’s the difference: in those other domains, people don’t constantly talk about it. When someone scores a 1600 on the SAT or gets a 5 on the AP Calculus exam, no one jumps to accuse them of cheating. There's a presumption of merit and legitimacy.
Sadly, the same can’t be said for AMC competitions. Ironically, it's not cheating that undermines the credibility of the AMC; it’s the constant speculation about it. Even when there's no clear evidence, the persistent noise about leaks creates doubt. And that doubt sticks. It cheapens success.
You might feel justified in complaining about alleged cheating because you didn’t qualify this year. But what happens next year, when you make USA(J)MO? If the narrative continues to be “oh, everyone just cheats anyway,” your own hard-earned success will be met with skepticism. Not because you cheated—but because you've helped build a culture where perception overrides reality.
I’ve heard plenty of damaging and unfounded accusations just being an active member in my local math community. Teachers claiming a student with a B+ in Algebra II must have cheated to make AIME. Parents whispering that a “Chinese teacher” who runs group classes must be leaking problems because several of their students got DHR on the AMC 8. These accusations are unfair, baseless, and rooted in misunderstanding. School math is not competition math, and success often comes from years of focused training, not shortcuts.
So let’s stop. The more we talk about unverified cheating, the more we hurt ourselves and our peers. It’s time we protect the prestige of these contests—for the sake of those who’ve earned it, and for those who will in the future.
If you’re frustrated by your results, let that energy drive your preparation—not blame. Focus on growth, not gossip.
