8/10/2023

I started doing math from the book "The Three Year MATHCOUNTS Marathon", by Karen Ge.

Don't be misled by the title though: this book is very hard and is at the AMC 10 - AIME level. It even has some problems from olympiads such as USAMO. (this is not just what I say, this is what a lot of other people say who have used this book)

Anyways, I just started this book, and I'm on Chapter 2. Today I read the theory and did the sample problems in this chapter. My favorite sample problem was an AIME problem: 

Problem:

Find the smallest positive integer for which the expansion of $(xy-3x+7y-21{)}^n$, after like terms have been collected, has at least $1996$ terms.

My Solution:

Note that $$(xy-3x+7y-21{)}^n=[(x+7)(y-3){]}^n=(x+7{)}^n(y-3{)}^n.$$ The number of terms in this expression is the number of terms in $(x+7{)}^n$ times the number of terms in $(y-3{)}^n$. This is because in the product of $(x+7{)}^n$ and $(y-3{)}^n$, every term has a unique power of $x$ and a unique power of $y$, meaning that there are no "like terms".

By the Binomial Theorem, the number of terms in $(x+7{)}^n$ is $n+1$. Similarly, the number of terms in $(y-3{)}^n$ is also $n+1$. 

Therefore, we want $$(n+1)(n+1)\ge 1996$$ $$⟹(n+1{)}^2\ge 1996$$ $$n\ge 44.$$

So our answer is $\boxed{{\displaystyle 44}}$.