if i had qualified for the usa(j)mo (i wish), would i have been flown out for free like mathcounts nationals or do you have to plan your own trip for going to the usamo

Let a,b,c\ge 0: a+b+c=3. Prove \frac{1}{abc}+\frac{12}{a^2b+b^2c+c^2a}\ge 5.

Find all polynomials with integer coefficients such that, for all .

Given m,n,p,q \in \mathbb{N+}, find all solutions to 2^{m}3^{n}+5^{p}=7^{q}$

One of the paths I've found is to boil it down to solving two non-simultaneous equations 2^{m_1}+5^{n_1}=7^{q_1} and
7^{m_1}+5^{n_1}=2^{q_1} but its too hard. Any other approaches/solutions or a continuation of this path?

Let and Prove that
Where

Theres a server where we are all like discussing problems+helping each other practice. Hopefully you guys can join.

https://discord.gg/6hN3w4eK

This took me quite a while to make, but enjoy!

Introduction to C&P (suitable for AMC 8, AMC 10/12)
Chapter 1 - This is like the "introduction", which is pretty easy and is not very important.
Chapter 2 - VERY important! Study this chapter closely, as it contains techniques that will be used again and again in harder problems.
Chapter 3 - Another quite important chapter, though not as important as chapter 2. This chapter covers some of the most confusing parts in C&P and even I can't distinguish that well in that chapter.
Chapter 4 - Interesting but very basic. Not that important, really.
Chapter 5 - Another interesting chapter, which should be studied in greater detail than Chapter 4. The distinguishability section is most important here.
Chapter 6 - Not much, but attempt the problems and read the examples since many of them are very interesting.
Chapter 7 - Pretty important chapter, make sure you read all the sections but not very interesting.
Chapter 8 - Another one of the VERY important sections - make sure read this section closely and do all the problems, since I still compare apples to oranges sometimes.
Chapter 9 - Interesting, but not very important. More important is the concept to "Think About It!"
Chapter 10 - The only topic in the entire C&P series that covers Geometric Probability, this chapter doesn't go into enough detail. Read it closely to get the basics, but I'd recommend doing more practice on Geometric Probability (I'll be making a handout!)
Chapter 11 - This chapter is not really important, reference the section in Intermediate C&P for a deeper understanding of Expected value.
Chapter 12 - Pretty important chapter, study it closely as it gives you the tools to prove combinatorial identities and Pascal's triangle is quite useful.
Chapter 13 - Just get the Hockey Stick Identity - not very useful chapter. Distributions will also be covered in Intermediate C&P.
Chapter 14 - A bit important, but not very - The binomial theorem is easy to master, but if you need more practice read the section in IA.
Chapter 15 - Similar to chapter 6, read all the examples and attempt all the problems here.

AMC 10/12 Chapters: 2, 3, 5, 6, 7, 8, 10, 12, 15

Intermediate C&P Suitable for late AMC 12, AIME + olympiads
Chapter 1 - Review this section thoroughly though there are no exercises here.
Chapter 2 - If you've learned set theory before, this chapter should be a review, but nonetheless skim over this chapter.
Chapter 3 - ANOTHER IMPORTANT CHAPTER! PIE is very important and might be a bit complicated, so study this chapter closely.
Chapter 4 - This chapter is also quite important - Make sure you master both parts of this chapter.
Chapter 5 - A good chapter, but it's a bit too short for my liking. Read extra handouts on the Pigeonhole Principle.
Chapter 6 - Another great chapter - attempt all the problems in this chapter!
Chapter 7 - Yet another very important chapter - distributions tend to pop up all over the place. Attempt all the problems here.
Chapter 8 - This isn't really a chapter - if you've mastered Mathematical Induction, you can just skip this but I recommend doing the problems.
Chapter 9 - This is really just the introduction to Chapter 10, but nonetheless do some of the problems to get a firm recursion basis.
Chapter 10 - Another VERY IMPORTANT CHAPTER! The recursion section is more important than the Catalan Number section unless you're preparing for olympiads.
Chapter 11 - Past this chapter, the concepts start to get quite advanced. This is an interesting chapter and is quite important, so do many of the problems here.
Chapter 12 - A great chapter! This chapter is quite general, but try to learn how to prove combinatorial identities on your own.
Chapter 13 - A quite complex chapter, not that important unless you're preparing for olympiads.
Chapter 14 - A hard but great chapter! GFs are hacks to many common counting problems.
Chapter 15 - Just skip this chapter unless you're doing the Putnam or olympiads, since it's basically nonexistent in the AMC/AIMEs.
Chapter 16 - Many of the problems here are very hard, but do as much as you can here! Try to attempt every single problem though they are very hard.

AMC 12 chapters: 1, 3, 4, 5, 6, 7, 9, 10
AIME chapters: 1, 3, 4, 5, 6, 7, 9, 10, 11
Olympiad chapters: 3, 4, 5, 6, 7, 10, 11, 12, 13, 14, 15 [basically almost all of them rip]

Let be the altitude and be the intersection point of the altitudes of triangle . Point is symmetric to with respect to . The circumcircle of triangle intersects at points and . Prove that .

(V. Starodub, Kyiv)