Project Status

We have moved to www.infinitydots.org.

Welcome to the InfinityDots project website!

The project is still a work-in-progress.

• TST and October camp archives
• Problems of the Week (POTWs): POTWs 2 posted!
• Biweekly math blog and articles : soon!

Any feedbacks are welcome – just click here and write us something!

Episode 1: I HATED but LOVED math!

After thinking about it, I realized I really, really “hate” mathematics. In fact, right now, you can say that I hate it to the core. And yet, in the end I came to the point of becoming a representative of the mathematics olympiad. Many of you will probably be saying “Huh? If you hate math this much, why did you become a representative? and how could you endure attending the camps for so many years?” because I was attending them for the past 3 years and then finally becoming a representative in the fourth(4th) year. I’ll have to tell you now that, in the past, mathematics was my everything. If there was no mathematics, my life wouldn’t have been this successful.

Actually, I don’t really “hate” math that much now, I just feel that I cannot make any progress. Simply saying, I feel like I’m stuck, or even going backward (like a logarithm function). Perhaps it’s because I devoted my time to preparing for the scholarship examination lately, so much so that I have no time to practice my mathematical skills. So it’s natural that I cannot perform as well as when I was in my prime. Another reason, which is the most important, is because I stopped myself from learning something simply because I didn’t like or understand the definitions at first sight. I gave up reading the sheets, even when just reading a bit further will point me to examples to help me understand or further details, leaving me unable to study new techniques or topics especially college math. For I keep thinking “these topics won’t be appearing in the olympiad! Studying them right now is pointless.” I thought that I can study them later when I’m in university. This creates two huge disadvantages at the same time. One, I cannot understand the origin of some problems which was adapted or inspired from college-level math. This makes me feel awkward when I’m reading the solution, I feel like I understand, but not fully. Different from others who fully understand and are able to adapt it for other problems. Next, I am at a disadvantage compared to others who study in college because in the US, there are already college math courses for high school students, leaving me in an inferior position and at risk of failing to catch up if I can’t get rid of this bad habit. Because of these, I cannot say that I stopped liking mathematics, just that the wall, which was built from my lack of ability and efforts to understand, that prevents my love for mathematics at the moment.

But it is still not too late to change, both for me and my readers who may experienced this problem. Actually, I get rid of that ineffectual habit since I have been here at the land of opportunity, US.

EP 0: Fear of being Lost

I believe that everyone has fear inside their hearts. Some fear because of their past experiences; others fear meeting new and unfamiliar things. But if you ask me what I fear the most, I can tell you that I fear being forgotten…

Every time I wake up in the morning, I rejoice, because I did not want to think about what will happen if I did not. Every night when I let my body fall on my bed, I ponder, “What if I never wake up after sleeping tonight? What will happen? How will I feel? Where will I go? Will my memories be lost forever?” There is one more question that I often ask myself – “Will I still live on in the memories of other people?” – for we cannot deny that even if we are famous or well-known, someday, sometime, we will be replaced – lost in the sands of time. Every time this question surfaces in my mind, I keep trying to find the answer – when my body decays, what will I leave behind in this world?

I have always hoped that in my short life, I will be able to do something for this world. This seems unlikely to happen both in the present and in the uncertain future, but there is one thing that I can do now – seize the chance to write about my experiences, and tell my life’s story to all my readers. For at least, my memory would still live on – it is the best that I can manage in the present and this work will persist for all eternity. I hope that the following stories will inspire and give insights to every readers, whether a lot or little.

POTWs 4

Every week, we will select three notable or interesting problems, marked with E,M,H (“easy”, “medium”, “hard”) for the relative difficulty. Easy problems will be around TMO or easier than TMO; medium problems will be around Oct Camp / easy IMO, and hard problems will be around medium / hard IMO.

E4 [Classical]

Alice and Bob alternatively choose numbers from among , without replacement. The first to obtain numbers which sum to wins. Does Alice (the first to play) have a winning strategy?

M4 [The Guardian]

Given a grid with an arrow pointing to one of four direction (up/down or left/right) in each of cell. Initially you’re in a cell. The goal is to get out of the grid.  In each turn, you must move in the neighbour cell, if not out of the grid, that the arrow in your current cell pointed to, and once you leave the old cell, you must also turn the arrow in that cell by degree clockwise. Is it true that no matter which square you chose to begin, and no matter what directions the arrows are initially pointing at, you will eventually get out of the grid?

H4 [@gausskarl on AoPS]

Let  be a set of  points in the interior of a circle of radius  with be the center of the circle. For each , let  be the distance from  to the other point in that is closest to . Prove that

.

Solution will be available next week.

POTWs 3 Solutions

Here are the solutions to POTWs 3. Along with the solutions, we also give some comments and difficulty ratings in , where POSN Camp level is 1-2, TMO level is 1-3, October Camp is around 2-4, and IMO is around 3-5.

POTWs 3

Every week, we will select three notable or interesting problems, marked with E,M,H (“easy”, “medium”, “hard”) for the relative difficulty. Easy problems will be around TMO or easier than TMO; medium problems will be around Oct Camp / easy IMO, and hard problems will be around medium / hard IMO.

E3 [adapted from IMO 2006 P4]

Show that for all primes , is divisible by .

M3 [Bulgaria TST 2005]

Find the number of the subsets  of the set  such that the sum of the elements of  is congruent to  modulo .

H3 [reddit]

From any pair of positive integers , in each turn, you can choose to move to either or . Show that, starting from any pair of positive integers , you can reach a pair of two equal positive integers.

Solution will be available next week.

POTWs 2 Solutions

Here are the solutions to POTWs 2. Along with the solutions, we also give some comments and difficulty ratings in , where POSN Camp level is 1-2, TMO level is 1-3, October Camp is around 2-4, and IMO is around 3-5.

InfinityDots MO files now available!

The InfinityDots MOs are our Mock IMO contests on AoPS. We have held two InfinityDots MOs so far with lots of participants and great feedback.

Today we have added the InfinityDots MO and MO 2 problems and solutions to our archive. Here’s a direct link to the folder. Also, while this is the English site, we’d also like to note that we have translated the problems into Thai; the Thai versions are now available in our archives too.

We invite readers who are interested in IMO-like contests to try these problems. Read more about our contests on AoPS here: InfinityDots MO, InfinityDots MO 2.

TMO 15

Hi,

During the past several days, the Thailand Mathematics Olympiad was held in Nakhon Ratchasima. In this post, we are sharing our solutions, along with some comments, to the problems in the competition. But since it’s always good to try solving problems yourself before reading the solution, we’ve attached the solution file at the end of this post and give the non-spoiler version of comments to each problem first.

Note that we use the difficulty rates as in our POTWs, that is, POSN Camp level is 1-2, TMO level is 1-3, October Camp is around 2-4, and IMO is around 3-5.

POTWs 2

Every week, we will select three notable or interesting problems, marked with E,M,H (“easy”, “medium”, “hard”) for the relative difficulty. Easy problems will be around TMO or easier than TMO; medium problems will be around Oct Camp / easy IMO, and hard problems will be around medium / hard IMO.

E2 [@Konigsberg on AoPS]
In a convex pentagon, show that we can choose diagonals such that their lengths can form a triangle.

M2 [China TST 2007 Quiz]
Let be the incenter of triangle Let be the midpoints of respectively. Points lie on respectively such that The line perpendicular to through intersects the line perpendicular to through at Prove that

H2 [Google CodeJam 2011]
Goro wants to sort a list of distinct numbers in an increasing order. In each round, Goro can fix some elements of the list. All non-fixed elements of the list will then be permuted randomly (with each permutation having equal probability.) Given and the initial list, determine the expected number of rounds Goro will need to sort the list, under Goro’s best strategy.

Solutions will be available next week.

POTWs 1 Solutions

Here are the solutions to POTWs 1. Along with the solutions, we also give some comments and difficulty ratings in , where POSN Camp level is 1-2, TMO level is 1-3, October Camp is around 2-4, and IMO is around 3-5.