Update on many things... posted under Math by George
So, updates on competitions and the like:
We actually placed at the Stanford Math Tournament this year
3rd place team at Cuyamaca a few weeks ago
Currently working on CPSC, in which we placed 3rd nationally last year
UCSD Classes posted under Home by George
The spring quarter is coming up, and classes start in two days. Currently Analysis B, First Order Logic, and Analytic Number Theory look appealing (assuming I can do the last one, as it is a graduate course and I am yet unsure as to the difficulty). Interestingly, the number theory class will be taught by Harold Stark, world-renowned number theoretician who did extensive work on the Gauss class number problem. It should be a fun quarter.
Interesting ideas from classes to come in many more blog posts!
SDMO posted under Home by George
The San Diego Math Circle held the SDMO today, a few of the problems were interesting. In particular, the last one, dealing with intervals.
Let I, I1, I2, ..., In be intervals such that I contains I1 through In. Show the the union of the left halves of intervals I1 through In contains at least half of I.
On first looking at this problem, it seemed to shout measure theory at me, and I was almost tempted to start thinking about it in terms of more general metric spaces and using Lebesgue measure. I don't think this should be too hard... some algebra should do the trick I think. Perhaps the intervals could also be generalized to be any desired set. A follow up may come if I get some time to work on this.
Math Team Heuristics Lesson posted under Home by George
I recently created a heuristics lesson which I taught to CCA's math team. It includes three basic principles: invariance, extremity, and pigeonhole, with explanations, examples, and problems for them. Additionally, there's a history section on pigeonhole and applications section on invariants. For the more advanced reader, there are more difficult things to think about which are included as sidenotes or extensions to problems.
Economics posted under Home by George
My current economics class just covered comparative advantage optimization problems, and naturally I was inclined to generalize the solution to the problem. I wrote an algorithm in javascript which allows for as many people/goods as one could want. Linkage.
Follow up on longest segment in polygon problem posted under Home by George
I did some searching on Google after renewed interest in my polygon problem (Given a region, find the longest segment fitting entirely within the region. That is, find an algorithm which will do it. Try for polygons first, then estimate curves with straight lines, predicting maximum error.) and came up with this: link. For other things I found, search "longest segment lying inside polygon" on Google. I haven't read the paper that is linked here, but it looks like it might have the solution to it...
Happy π (pi) day by the way.
Not more Pascal uggghhh posted under Home by George
Building off of the previous post, instead of using the Fibonacci sequence, use (-1)^n. I haven't investigated this much as of 1/15, but I plan on looking into it further when I get some spare time. If you find anything neat out about it, email me and I'll post it. Or you can make an account and comment.
1
-1 -1
1 -2 1
-1 -1 -1 -1
1 -2 -2 -2 1
-1 -1 -4 -4 -1 -1
Etc. We can also do this with (-2)^n, and it gets something else weird. Are there any results that hold for all (a)^n for negative integers a?
Pascal strikes back... posted under Home by George
I kept thinking about Pascal's triangle, and then I saw Fibonacci on my wall, so I thought about combining the two. I put the Fibonacci sequence on both sides of the triangle, and then let the normal recursion rules work like normal. Interestingly, a similar hockey stick identity holds. Traveling along any path downward (with exception of the continued middle path, shown in the picture on the right) all previous numbers will sum to the final, GIVEN THAT in terminating the path, you make one final turn, or else the sum will be one less. That is, following 1,1,2,3,7,15, (LLLRR) we see that 1+1+2+3+7=15 (-1). Also, if we follow the middle path, this will become a +1. There is only way of choosing the middle path though. For all other paths, the identity holds exactly.
Edit: Upon further inspection, these things seem to fail sometimes. Can we find certain identities that always work?
Pascal^3? posted under Home by George
Pascal's triangle is a commonly known recursive tree of numbers, where each row is determined by the row above by adding all pairs of two adjacent numbers. For an image and description, see the Wikipedia entry.
Now I pose the follow up question: what if we create a Pascal tetrahedron? With the top point 1, second slice (plane) a triangle of 1's, third slice a 3 in the middle with six 1's around it, etc. Do the numbers in this represent the coefficients obtained from the multinomial theorem? What about an nth dimensional triangle?
It might yield nothing... I was sitting around for two minutes thinking of something to think about, and looked at my mathematician wall, and Pascal stared out at me.
After looking around, it seems like some guys did a science fair project on it, linkage.
So the only remaining question: can we prove inductively on dimension n that the nth dimensional Pascal triangle has terms relating to the coefficients of an expanded multinomial with n terms? Might not be so bad...
The evolution of an idea... posted under Home by George
I became interested with reflections over curves back when I first learned what calculus was and I got to thinking about what one could do with derivatives. I came up with the question of how to reflect points over the parabola y=x^2.
Phase 1: The very first stage of the idea involved finding the line normal to the parabola through the given point, and then using the distance formula to flip the point over the parabola. This was a pretty ugly process.
Phase 2: I learned about circular inversion. Tutorial here. I began trying to construct a parabolic inversion. This made me use, instead of distance formula, the intersection between the normal line, and the two points of tangency from the original point and the parabola. That is, the two points where the tangent lines to the parabola at those points go through the original point. This made everything somewhat nicer, but I was still dealing with a very nasty normal line to the parabola, which actually had more than one solution (the others were not real though).
Phase 3: Just today, it crossed my mind that I get rid of this normal line entirely. I decided to average the two points of tangency, and use that as the third point. In circular inversion, the inversion of a line is a circle. Now, inverting a line through my parabola like this gives another parabola! I believe the same approach can be used for all other conics (hyperbolae, ellipses). Also, something strangely interesting, is that if I use parabola of inversion y=x^2, then if I invert line y=k for constant k<0, then the area between the new parabola and the parabola of inversion is equal to the area between the parabola of inversion and y=k from any a to any b.
Notably, the reason it took so many steps to find the "right" way of doing it was because in circular inversion, all of these processes are the same, except for the first (which used distance). The first could be reformulated to using a proportion of the distance, in which case it too would be a method of circular inversion. In summary, this shows once again the unique "normalness" of circles, only in a new context.
Well, that was my longest post so far I think. One final note, this problem began somewhere around a year ago when I was doodling.
Four problems of my own posted under Home by George
1. Given a region, find the longest segment fitting entirely within the region. That is, find an algorithm which will do it. Try for polygons first, then estimate curves with straight lines, predicting maximum error.
2. Prove that for each pair of twin primes (p, p+2), there is another pair (q, q+2) such that p+2 < q+2 < p(p+2). That is, there is another pair of twin primes between the first and their product.
3. Prove that there are infinitely many prime pairs (p,q) such that |2p-3q|=1.
4. A pin is dropped on the coordinate plane with random orientation so that the center of the pin falls within a circle of radius 3. If the pin has length 2, what is the probability that the pin is completely within the circle?
These are just some problems I made up in the past week by thinking about math too much. (2) and (3) don't seem solvable, unless they aren't true, in which case a counterexample would suffice (for 2). (1) I'll have to get around to when I feel like programming. If anyone solves it for a polygon with a given set of vertices, email me the program/method and I'll post it with your name. Lastly, the problem that most interests me and seems most easy to solve is 4. However, it seems like it involves a gradient of probabilities, where each covers a circle, so it becomes difficult to sum them together when multiplied by their respective probability to find an area.
A coin game in one and two dimensions posted under Home by George
I recently came across this game in a puzzle book,
There is a row of 50 coins of different denominations. (There are two players.) A person takes a turn by choosing either the rightmost coin or the leftmost coin and taking it. At the end of the game, the winner is determined by who has the higher value in coins. Prove that the person who goes first can always win or tie.
And here's my follow up problem: what about the same game with a square grid of dimensions 2n by 2n and you can only take one of the outermost squares? Will the first player always win here? (Note: you could also do this in three dimensions, or even four, etc. I kept it at two, since I haven't solved that yet.)
Squares posted under Home by George
A problem from Naoki Sato's lecture at the SDMC today:
Say you have some arrangement of squares of uniform width on a table such that the sides are parallel to the sides of the table. Prove that you can put pins through the squares such that each square has exactly one pin going through it.
AP Calc POTW 3 posted under Home by George
Unfortunately, nobody answered last week's problem, but here's another one.
Let f(x)=xn for some real n. Then we have the point (a,an) for any a. Let y be the line perpendicular to the curve at a. Find the limit of the y-intercept of y as a approaches 0. (The answer for this will not be completely simple. Hint: think piecewise function.)
Have fun with this one, it's easier and nicer. Last week's was kind of ugly, but interesting nevertheless.
AP Calc POTW 2 posted under Home by George
(Use the notation int(a)(b)(f(x)) mean the integral of f(x) from a to b.) We define +int to be the POSITIVE area from a to b (similar to a normal integral, only don't include the negative area). Now let g(x)=2x and h(x)=3x2. Now we know +int(0)(1)(g(x))=+int(0)(1)(h(x)). Find some functions a(t) and b(t) that are both increasing on (0,1) such that +int(0)(1)(g(x)-a(t))=+int(0)(1)(h(x)-b(t)) for all tε[0,1].
AP Calc POTW 1 posted under Home by George
Solutions are due Friday, 11/3.
Let lim_(x->infinity) f(x) = a for some real a. Find, with proof, lim_(x->infinity) f'(x).
Calculus problem of the week posted under Home by George
Schoolwork has gotten really crazy, consuming my life graaagh. Posts will start coming more regularly as I'll be hosting the AP Calculus problems of the week here for Shay's classes. Here are the rules of the problem of the week "contests":
1. Don't directly cheat off of each other, collaboration is okay, but it shouldn't be one person doing all of the work
2. No commenting on this site until after they have been turned in.
3. Don't post the question on a forum for someone else to answer for you.
Each of these problems will be worth some extra credit, and they are meant to be at least somewhat challenging (so don't give up after five minutes!). Mr. Shay and I will pick the best solution(s) and put them up on the board (and I'll put them up here).
Math Team POW 1 posted under Home by George
Prove that x5+x4+x3+x2+x+1 divides x55+x44+x33+x22+x11+1. Good luck with this!
UCSD Lecture Notes posted under Academic by George
I'll be posting the lecture notes from the algebra class I'm in at UCSD under the notes section (problem sets as well perhaps). PDFs are on their way, word docs are there for the meantime though.
Problem of the week for math team posted under Academic by George
I'll be writing a problem of the week for our team each week. This is optional, but if you like doing challenging problems, then check the posts on ElNerdo home where I will put it up (with the title Math Team POW 1 or whatever number it is).
Problem set 9/12 and 9/13 posted under Academic by George
Member solutions will be posted in a PDF document around Friday (or on the weekend).
Distance stuff... posted under Home by George
Take a regular polyhedra with n sides (n=4,6,8,12,20). Pick a point on each of the n sides such that the distance between any two points on adjacent sides is the same for any pair of points. Prove or disprove: This can occur if and only if the points picked are the centers of the faces.
Pizza problems... posted under Academic by George
Solutions will be posted on the weekend. Look for more problem sets to come! Written by myself and others...
aa=2 posted under Home by George
Then a is irrational.
Proof: assume a is rational to proceed by contradiction. a=x/y for some positive integers x and y such that (x,y)=1. Then (x/y)x/y=2 gives (x/y)x=2y thus (x/y)x is an integer giving (x,y)!=1 or y=1. Thus y=1. This means a is an integer, but 1^1=1 and 2^2=4 so a cannot be an integer. Contradiction.
Late night spontaneous problem idea posted under Home by George
How many solutions does aa+2b=cc have (in integers, of course)? Can we generalize with kb for integers k? What about working in rationals? Or strictly rational+nth root numbers, without transcendentals? Are there even any solutions at all?
Also, I've been pondering about making some kind of notation to denote the solution to aa=k. Also how calculation of the value of a would work out. Possible patterns of aa also interest me, but I can't see them yet. Lastly, can we prove transcendence of the value of a when aa=k is not rational?
Some problems of my own... posted under Home by George
A few interesting problems that I've thought of:
1. How do you reflect points over a parabola? I believe I have solved this, and I am currently working on a program that will do it. So far this is my output: image 1 and image 2.
2. An open question: Take a cube and decompose it into 27 smaller cubes in the typical way. Put a base 3 digit where each little cube is (0, 1, or 2). Let us then read the sequences of the cube as follows: we pick any of the smaller cubes that show on the front, left, or top face. We then go straight through the cube from that cube. So for example, if it was the very top left front cube that we chose, relative to the top face, we would count first that cube, then the one directly below it, then the one below that. Thus we have 27 total possible strings. Now the question: can we put our base-3 digits in this such that every base-3 string from 000 to 222 occurs exactly once?
Can we generalize to an n dimensional cube with digits in base n with strings ranging from 000 to (n^n)-1?
Calendar Up posted under Academic by George
Math Team competition calendar is up at the official site. For problem sets and notes upload your stuff here. We'll eventually have some mock CAML tests and possibly mock AMC 10s, depending on when I get the time to construct such tests...
CCA Math Team is ready to get serious... posted under Math by George
We now have an official website, thanks to Irina Kufareva. Everyone should check out this site and fill out registration, as well as fill out this questionnaire to turn in at the first meeting. Otherwise you can choose to fill it out during the first meeting and hand it in at the end. I'm hoping to possibly get a math team separate site up under ElNerdo, similar to the academic portion (or to just convert the academic portion entirely to math team).
Polynomial problems... posted under Home by George
Here are some fun polynomial problems I've come across during my study of polynomials in math comps.
1) If P(x) = x4 - bx - 3 has roots a, b, c, d, then find the polynomial with roots (a+b+c)/(d2) cycled.
2) If (1+x+x2)n = a0 + a1x + ... + a2nx2n is an identity in x, find a0 + a1 + ... + a2n
3) If x1, ..., xn are the roots of 1+X+X2+...+Xn then prove that 1/(1-x1) + 1/(1-x2) + ... + 1/(1-xn) = n/2
Back posted under Home by George
I'm back from SUMaC, and I got bored so here's a 2-D Rubik's cube I just programmed. (I'm pretty sure it's a correct representation.) Link!
Stanford Math Camp: Day 1 posted under Home by George
People here are interesting. Fun to laugh with. Been sharing some of the fun NT problems that I know. Working on problems with people, cubing (which surprisingly few people besides me can do, so I guess I'll have to teach them), and the like. Problem of today: prove that k consecutive positive integers cannot be the kth power of an integer. (The head counsellor solved this in a few minutes.)
Grades in... posted under Home by George
Finally received all of my grades. A's in high school for a GPA of 5.0, which I'm very pleased about. As for UCSD, I got an A+ in number theory and an A- in statistics.
Update: AP exam grades were 4, 4, 4, 5 in Chemistry, World History, Psychology, and Calculus, respectively.
Getting there posted under Academic by George
Been working on submit, upload, accept forms and actions. They're starting to finally work out, good clean administrator end. Need to make it clean so that if this ever takes off like I hope then other people can start reviewing and accepting papers etc.
Slow going through algebra posted under Home by George
Been reading Abstract Algebra by Dummit and Foote, it's very slow going. I'm barely through chapter 1 and I still don't understand all of it completely. Isomorphisms and homomorphisms as well as group actions are the topics that have me stuck at the moment, but I should understand them eventually. This book is not written for the casual reader.
Another problem for you posted under Home by George
Label the primes in order (p1=2, p2=3, etc.). Prove that pi<=22i-1.
And the marquee thing is broken right now because AoPS is redoing their site and moving to a new server.
Number theory final posted under Home by George
So I took my NT final yesterday from 3-6. Yes, a three hour long final. I thought I did pretty well, the only question I didn't answer was the last one (worth 4 points):
Prove that for any odd prime p, there exist x and y such that
x2+y2=-1 (mod p)
It's really not that hard when you see the trick, but I didn't unfortunately.
Finals week and AIME preparation posted under Home by George
Yeah, the AIME is a long time away from now, but I'm starting to get in preparation mode because I want to qualify for USAMO. Finals week though. NT final Monday, stats final Friday. Then guess what? Summer! No more school...
I see the problem posted under Home by George
I now understand why the Goldbach conjecture is so hard to prove. If it were proven, it would say a lot about the distribution of primes, and very little about that is known. It's an interesting problem to think about though. See if you can find what implications its proof would have for the distribution of primes.
Goldbach posted under Home by George
Recently I've been trying to focus myself on a very specific problem which will give me something to struggle with. I've selected one which I found very nice: the Goldbach conjecture.
Prove that any even number can be written as the sum of two primes.
I'm making a little progress, restating it and rephrasing it, turning it into a more tangible form. Anyways, that's my goal in the next few months, to solve that. I don't really expect to, as it hasn't been solved in over 100 years, but I'll have fun trying.
Theorems posted under Home by George
I'll post number theory theorems in this post for reference.
Really popular ones:
Fermat's little theorem: (a,p)=1 and p is prime then ap-1 =1 (mod p)
Euler's totient theorem (generalized Fermat): (a,n)=1 then aφ(n)=1 (mod n)
Wilson's theorem: if p is prime then (p-1)!=-1 (mod p)
Chinese remainder theorem: click the link
Was kinda bored this Saturday afternoon... posted under Home by George
So I decided to made a regular icosahedron. It's pretty awesome. Now I'm going to write numbers on it and make it a sweet big paper D20.
Another SDMO problem posted under Home by George
Here's another SDMO problem which I found to be much easier after I thought about it a little more, but sadly I didn't think of a solution on the test.
Find all integer solutions (x,y,z) to the following system of equations.
I. x2z+y2z+4xy=40
II. x2+y2+xyz=20
A cooler one posted under Home by George
Find the remainder when 7243 is divided by 317.
A few questions from number theory homework 4 posted under Home by George
1. Prove that 49 | (5*34m+2 + 53*25m) for m>=0.
2. Find the remainder of 7243 when divided by 11.
Both pretty easy.
Here are a few hints
Number Theory homework posted under Home by George
Well, number theory homework has gotten harder. It's more fun that way though. Here's one problem that I especially liked.
Define μ(x) to be the Mobius function.
| {1 if x=1 | |
| μ(x)= | {(-1^k) if x=p1p2...pk (is square-free) |
| {0 if x is not square free |
1. Prove that μ(x) is multiplicative. That is, μ(mn)=μ(m)μ(n) for any m and n such that (m,n)=1
2. Find Fμ(x). (That is, Σd | x μ(d).)
I thought this problem was pretty interesting. Not hard really, I give it a difficulty 4/10, but it was neat.
SDMO posted under Home by George
I just went and competed in San Diego Math Olympiad (SDMO). I think I got around 7-15 out of 35. There were 5 questions, 7 points each. I definitely got #1, but none of the others, so I'm hoping for some partial credit. Anyways, it was tough, and I'm tired. Relax time. Here's question #1 in case you're interested:
Find, with proof, all triples of primes (p,q,r) such that p^q+1=r.
First midterm in number theory class posted under Home by George
Well, I took my first midterm today. The average was 18/30, with a standard deviation of 5. I got a 25, and was told that'd probably be an A/A- so I'm happy enough about it. Finished in half of the time given, so even more so.
Anyways, long day for me because I stay at UCSD until 8:00 since NT section is 7-8. Sleeptime.
I'm happy at college :)
Gil's NT problem from AoPS 1 posted under Home by George
Here's a problem that a friend of mine brought me (I solved half of it as of now, but haven't bothered making a rigorous proof of the second half, although I could do it using the same method).
Prove that if p can be represented as a sum of two squares, then 2p and 5p can be represented as sums of two squares.
Reveal Hidden Content
Easy NT posted under Home by George
Here are a few excercises from my new NT book (by Harold M. Stark)
1. Solve 1841r+3647s=1 in the integers.
2. Prove that there are infinitely many primes of the form 4x+3.
3. Show that if p does not divide n for all p less than the cube root of n (n^(1/3)) then n is either prime or a product of two primes.
4. Show that the sum of two consecutive odd primes can be factored into at least three (not necessarily distinct) prime factors.
By the way, I solved any of these so if you're curious about a solution, feel free to contact me.
UCSD comp is coming posted under Math by George
The UCSD honors math competition is coming up the 22nd, so everyone get ready (because we got owned last time) and maybe we can do decently this time. I'm hoping to compete with Tung, but he might be too good :/
Get ready over spring break. See you all at school Monday. (A week from tomorrow.)
Some geo for you... posted under Home by George
Here are a few geometry problems I just did (not very hard, but fun enough).
1. If ABC is a right triangle with right angle B, then median BD=AD=CD.
2. In square ABCD, M is the midpoint of AB. A line perpendicular to MC at M meets AD at K. Prove that <BCM~<KCM. Prove that triangle KDC is a 3-4-5 triangle.
3. See diagram below. CD is an altitude, CE is an angle bisector (of C) and CF is a median. Prove <DCE~<ECF.
Image for problem 3.
UCSD and tired posted under Home by George
Went to the first classes today. Not too hard. I'll be taking 3 AP's and 2 college math classes now, but it shouldn't be too tough. Lots of free time over spring break. What to do. Better start mathing.
Tired. Sleep. Even though it's ~6 ish.
Finally UCSD classes posted under Home by George
Okay, so I'm finally enrolled in some higher level math classes at UCSD. I attend the first ones tomorrow (one at 2-2:50 and the other 3-3:50) and then every Monday, Wednesday, and Friday. Woohoo. I hope Shay is okay with me leaving a few minutes early every other day.
Oh, and I dropped digital photography in favor of math.
Bell numbers posted under Home by George
Here's the output of the first 20 Bell numbers. If you don't know what a Bell number is, look it up on Wikipedia or click the link provided.
Prove that it's a square! posted under Home by George
Given parallelogram ABCD and the squares made off of it's sides, prove that quadrilateral WXYZ is a square if W, X, Y, and Z are the midpoints of the squares. Here's a diagram:
UCSD NT update posted under Home by George
Went in at lunch and Mr. Napora called UCSD and we got instructions about how to register for a class. Selected the class in digital photography, and all that I need to do now is get the registrar's signature and go to the UCSD admission building and work it out etc. If all goes well, I'll be in an "upper level" university course in which I hope to be challenged. Also, it's Monday, Wednesday, and Friday from 2-2:50, so I'll be able to skip three days of digital photography a week for math!
Geometrical circle area posted under Home by George
Here are some neat problems I came up with while drawing diagrams for another problem. Find the red shaded region in each picture (the boxes are squares with side length 1, and the curves and quarter circles). (Click for larger image.)
UCSD math classes posted under Home by George
So I finally went and asked my counsellor about getting some neat outside math teachings done and suggested UCSD, and she talked to Napora, and then I saw him and talked to him, and so now I'm going in at lunch and we're calling up UCSD and getting me in some class(es). Hooray! About time I did some neat math (that wasn't on my own).
AoPS NT chapter posted under Home by George
AoPS volume 2 has a chapter on number theory that I found very helpful. I'm almost finished with it, and once I do, and finish all of the problems, I'm going to finally pick up The Higher Arithmetic by Davenport once again. About time I learned some number theory. I also need to start doing some more geometry...
NT Tutor posted under Home by George
My dad is asking around (contacting UCSD professors, I think) about who might want to tutor number theory. It would be pretty neat to start learning some advanced NT stuff. Anyways, homework to do and stuff to do.
New goals and things to do posted under Home by George
My newest goal is to make it to IMO. This is pretty ambitious, and I'm nowhere near ready, so I've resolved to start working much harder on my extracurricular math by reading, foruming, and the like. Right now I just can't seem to do anything, so I'm just lazing around waiting for tomorrow to come. It seems I can get more work done with motivation from school, but I need to motivate myself to get to IMO. Anyways, SDMO and UCSD are soon, and I should start practicing some more for those.
Am I right? posted under Home by George
I just did question 3 on the SDMO in less than two minutes, but I'm not sure whether or not I'm right. I came to the conclusion that the only solution is (0,0,0) by infinite descent (using mod5). Anyways. Have fun on April Fool's day people reading this.
SDMO comp link posted under Home by George
Since I qualified for SDMO, I've begun to try to get a feel for what is going to be on it, and the best way to do that is relatively simple: go over old competitions. I thought I'd provide a link so people can see what kind of difficulty I'm preparing for, it's definitely going to be challenging.
PDF of the 2005 SDMO
If you come up with a neat solution to any of them, post it here!
Average day at school, fun in the math room posted under Home by George
Today in math class, while Shay was lecturing the class, I worked (and finally managed to solve!) this really cool log problem.
Given that p,q > 0, and log9p=log12q=log16(p+q), find q/p.
Answer
Blog submitted to AoPS posted under Home by George
I just submitted this site to AoPS as a blog. For the reviewers: there's a link to AoPS on the right (at the top of the list nonetheless!) which was there even before I read that you guys wanted for me to have one on my site.
Getting ready for UCSD posted under Home by George
The UCSD competition is coming up on April 22nd! Everyone prepare for it and then we might have a chance of winning. I've been doing old comps and I'm pretty comfortable with the various types of questions, but I'm going to prepare a little more just to be sure. I'm hoping to do moderately well this year. See you all (those 8-12 people) there!
Cool system 1 posted under Home by George
Solve the system of equations: (AoPS Volume 2)
(x+y)(x+y+z)=66
(z+y)(x+y+z)=99
(x+z)(x+y+z)=77
Hint
Easy system #1 posted under Home by George
Here's a fun problem out of AoPS volume 2. (I liked it because of the incredibly elegant algebra that can solve it.)
In a rectangular solid, the area of the top face is 135, the area of the front face is 30, and the area of the right face is 50. Find the volume of the solid.
My solution
Interesting probability posted under Home by George
Here are a few probability problems from a book I have that are pretty interesting (in order of increasing difficulty).
1) The amount 2.5 is split into two nonegative real numbers uniformly at random. Then each number is rounded to the nearest integer. What is the probability that the two integers sum to 3?
2) Three points A, B, and C are selected at random on the circumference of a circle. Find the probability that the points lie on a semicircle.
3) Three numbers are choisen at random between 0 and 1. What is the probability that the difference between the greatest and the least is less than 1/3?
I've done the first two, but the last one is definitely tough. Post any ideas/answers as comments! Oh, and problem credits go to MAθ, MAθ, and Mandelbrot #3, respectively. (Questions compiled in AoPS Vol. 2)
Number theory problem 1 posted under Home by George
Here's a neat problem my dad's old number theory professor sent me:
Prove that p is a prime iff 1/p = 1/m - 1/n for only one m and only one n.
Saturday 3/25 posted under Home by George
I went to the fifth round of the San Diego Math League to compete yesterday (Saturday). I ended up getting 3rd place, qualifying for SDMO (San Diego Math Olympiad) in which I hope to do well.
Cheers to UCSD and SDMO in a few more weeks!
Results can be seen here. They haven't put round 5 up just yet.
George wins programming competition posted under Math by George
Well, it's been about a week, and it just occurred to me to post this here. Anyways, we went to the annual Cuyamaca field day competition, and I got first place in programming. Woohoo. Afterwards I didn't like my program though, and made a better one instead in java once I got home. Anyways, congrats to Parima and Matthew also for getting second place in engineering, and go CCA for placing 4th overall.
Oh yeah, and Xiaozhe and I got honorable mention for technology tandem.
George fails AMC posted under Math by George
Got a 90.5 on the AMC 10. Kinda over it now, but it was incredibly bad news when I found out. Needed a 120 to qualify for AIME, but I made careless mistakes on 6 problems. Had I not made those mistakes, I would have gotten a 126.5, but take what you can get eh? Anyways, preparing for the 5th round of the San Diego Math League and if I get in the top 5 (I got 6th place in round 4) then I go to the San Diego Math Olympiad. That'd be cool. Anyways, just been mathing around on yet another 4-day weekend, but I have a psych test tomorrow and a chem lab to complete. Too bad I can't drop out of school and just study math...
Math Circle at UCSD posted under Math by George
I just got back from this math circle thing at UCSD. It was fun. There were some good problems and stuff. The topic was geometry. The teacher was good. I learned some stuff. Also spent some time looking through my calculus book. It looks like it'll be a fun class. Spent some time learning about Taylor series and some other junk. That's about it for math this break. Oh, and browsing the AoPS forum of course.
Forum for math posted under Math by George
Ok, here's how the math team portion of this site is laid out. News will come in here. You may comment/ask questions by clicking on the comment link and posting what you will, but avoid spamming, as this site is a privelege. For math team practice, we will begin to use the forum as the place to go with questions and ideas. So you get it? Math Team tab for news, forum for practice. I hope that the whole team begins to use the forum, it will definitely help our skills and will get us ready for competitions.
Math team! posted under Math by George
Math team is now under the realm of the almighty Hand of Blog. Officially this portion isn't up yet, but I aim for it to be up by tonight. This is such a better site than the old crummy one.