#12 on this year’s AIME, could it possibly be a simple problem with a simple solution?
Let 
be a right triangle with right angle at 
Let 
and 
be points on 
with between 
and such that 
and 
trisect 
If 
then 
can be written as 
where 
and 
are relatively prime positive integers, and 
is a positive integer not divisible by the square of any prime. Find 
This month has been crazy! It’s almost February 29th, since 2012 is a leap year. About a month ago I attended SUMiT and had an amazing time. I met many people my age interested in math like me. It was organized very well, thanks to GirlAngle and others.
Also in the past month were the AMC’s. On the AMC12 A, there was one problem that particularly intrigued me problem 2: See if you find a nice solution
Consider the polynomial
The coefficient of 
is equal to 
. What is 
?
Happy new year everyone! It’s 2012=2^2*503! Let’s start the year off with a simple problem: A palindrome is a number that reads the same forwards and backwards. How many digits does the 2012th palindrome have?
Please keep working on the last couple problems. I posted the solution to the first problem. Please comment if you want a hint!
The following problem is also a telescoping problem which is found on the 2005 Purple Comet Competition, #20. Please comment if you have any ideas! Note that it uses similar techniques as some of the previous problems that I posted, so please work on them also!
The summation 
Please continue to work on the previous problem; I will post hints or solutions to that in the near future!
Regardless: A cool pattern of polynomials goes as follows-
(x+1)^2=x^2+2x+1
(x^2+x+1)^2=x^4+2x^3+3x^2+2x+1
(x^3+x^2+x+1)^2=x^6+2x^5+3x^4+4x^3+3x^2+2x+1
….
See if you can use this little tip to simplify the following problem (which comes from AIME 2003, number 15!). It combines this tip and the ideas emphasized in the previous post!
Let
Let
Let
Note that this problem involves multiple steps; even if you don’t know the entire solution to it, feel free to comment with intermediate steps or ideas you may have.
Topic 1: Summations
When I was younger, I always believed that working with problems that looked short and that did not have many scary symbols was easier than dealing with long, complex looking questions. Concepts which especially intimidated me were summations and floor functions. Just by the look of them, they seem s like a mess. However, in reality there are problems in those topics that are very simple!
In summations, often times a concept called telescoping comes into play. Telescoping is when you are able to cancel out most terms in the summation. In summations it is also helpful to sometimes reorganize your problem.
Summations problems come in all different shapes and sizes. The following problem is #14 on the AIME which I spoke about in my previous post.
For each positive integer n, let
Note:
Please comment if you have any ideas for creative solutions! I will give hints for this problem in later posts! Have fun with this problem!
Also good luck to all of the girls participating in the Math Prize for Girls next weekend!
Welcome to the wonderful world of mathematics problem solving!
Often times, mathematics problems seem like enormous messes. There are certain questions that we, problem solvers, encounter that do not seem to ever terminate. However, with the knowledge of a few special tricks they fall to pieces and become very elementary. This blog is for people who are interested in messy, difficult-looking, yet very interesting problems. They are not your typical high-school math, “plug and chug” questions. They delve deeper into the concepts that you learn in high-school mathematics. This blog creates an opportunity for you to witness creative solutions and interact with others!
With my personal experience, I have encountered many challenging questions. However, even now I still get discouraged when a problem seems frightening or very messy at first sight. The first time that I took the AIME, question number 14, was a relatively simple problem. However, since it was #14 (out of 15 questions on the AIME contest) and I was still relatively inexperienced, I barely even looked at this problem when I took the contest. The problem seemed challenging with “scary summations”. However, as it turned out, this problem was not bad at all. When I got home, I solved it in about five minutes. Since then, I have been very cautious when it came to “messy-looking” problems because I didn’t want to miss such a chance again.
This blog is for people like me who seek an extension to their mathematics knowledge and the confidence to approach such “scary, messy” problems. I will periodically present tough problems, and explain their solutions showing how the mess turns into a simple, beautiful work of art.
Come join the fun!
