Hard Problems The Road to the World's Toughest Math Contest

[Music] his security instead of really carrying a blanket or a stuffed animal was was this number one in this puzzle I guess we had gotten him this puzzle because he did show an extreme interest in numbers from as I recall the age of nine months you can see it's very well not and chewed and all the papers worn off eventually I carry the number one around so much that it wore out and my parents had to buy a new puzzle for the new number one he just keep asking questions I keep asking for problems massive problems although he's very much interested in the maths and science but I think in my opinion he's not a nerd we don't believe we have the the kind of IQ that our three sons do and somehow her my combination created this little monsters [Music] [Applause] [Music] okay about a minute and a half everybody have everything they need as with yesterday the clock up here will be the official time good luck the Olympiad examination takes place over two days the students came yesterday morning and worked for four and a half hours we download the problems from the internet they're released all across the country at the same time there are three problems and they have four and a half hours to work on those problems and then they come back today and they do it again for another four and a half hours so that they have a total of nine hours of examination to do six very challenging and very difficult problems [Music] students that are able to perform at this level have the ability almost to sort of disappear into a zone they're able to focus and concentrate and exclude all other environmental considerations it's a good feeling when you finish the problem but also when you're working on it it might not look like you're entertained but I think I'm entertained this is what I think of as a good time I guess some kind of escapism when you're working on the problem that's the only problem my family is from China and I moved with them to Canada when I was three and we moved to the US when I was five I remember my parents teaching me how to write the numerals and always write five and seven sort of backwards so I guess it didn't really come so naturally to me the math in the states were a little bit too easy for him so we bought textbooks from China just teaching him at home for each grade my father he's a chemist he always taught me chemistry when I was young and it sort of laid the foundation of my interest in science in middle school I did a contest called math counts it really got me interested in learning more about math he has a remarkable ability to reason abstractly he's probably become more fluent with the language of mathematics than any other student that I've ever worked with there are five science Olympia in the world and that's biology chemistry physics math and computer science ISA had made it to the finalist of all five categories in the nation he's basically the guy everyone like who takes like math competitions like all their friends go like oh you son so good why aren't you more like him he's the person we're all expecting to him because he's really good at this sort of thing once I asked his mom I said well ISA is so right you think you're just pure lucky or is because of your gene and very humble his mother replied is because I'm lucky [Music] I always think the most difficulty in problem-solving or doing pure research for this age the problem bit so hard 1 months 2 months people can hang on with it but after a while they find it very discouraging and they give up if you have 10 or 12 students working on them together they don't feel that way can see deeper than I like the formality of it how there's a right answer but at the same time there's also nicer ways of getting to the right answer sometimes the search for the right answer but also sometimes to search for the right way to get to the right answer in fifth grade they let me start algebra when most people didn't start till seventh grade I heard about Exeter I decided to come here instead and here I met mr. Fung and he's helped me a lot basically it's a dottywot process mr. Fung is much more into the individual work not just giving people the answers and making people work through it if you guys are working hard actually you you might have remembered I actually give you this number ten in one of your what I want to get out of math is to learn how to think that's one I feel math is really useful and want to really work on he is a very tremendous original thinker so if the problem is about a combinatorics is about playing a game what's the best strategy a graph theory those kind of problem he can give you very original method so that's his story - three squared K minus 2h I like those problems where you need very good insights instead of technically doing brute-force equations for two hours when I get one of those insights that's when I feel really good about myself and doing mathematics Ryan is a very special talent he is extraordinary quick I was surprised by his speed he's really good at thinking of really nice and elegant ideas for math sometimes he'll just draw points or lines I've know where and they just immediately saw the problem and it's like how did you do that he's like it's obvious and it's not very obvious but it that's really cool he was six months I got a new calendar the calendar has baby pictures when he sold our calendar he had a big smile but he didn't like babies he liked the number he took part in math competition in Korea he got the first place among fourth graders nationwide and he was a seventh grader we moved to America Terry's heart for Americans to pronounce and read so my dad asked for a suitable English name for me and my brother so he got one each and that one that's right this is a completely new environment completely new culture and I felt like I didn't belong here at all and it was strange because in Korea I used to be on top of the things but when I came here I'm suddenly at the bottom of ladder I had little background in English and I had probably less than kindergarten level vocabulary well one thing I was still good at is math because I don't really need to talk in order to be good at math I just need to write stuff and numbers I think by doing some math competitions I got back my confidence and after that the transition to this world was very very easy for me he's a character Israel change it he learned how to get along with people so now he can make friend easily so I'm impressed you guys got it so that's very good right so you know that's why I never underestimate you guys every year the one way I can measure my team is how much better each one of them are better than me if they are much better than me I know I have a very strong team [Music] ie congratulations it's not often I get to deliver this kind of message in person this is elgin johnston belgians the chair of the Committee on the American math competitions we just finished the grading about what three four hours ago I'm glad we were able to make contact with you and you're one of the u.s. amo winners one of the top 12 congratulations we'll see you again in Washington in about what three or four weeks I guess maybe a little over a month two hundred and fifty thousand students in the United States take the AMC 10 and the AMC 12 will have 12,000 Oh take the Amy 140 take the US amo we can take the top 12 papers those become the US amo winners you've gone all the way from 250,000 kids who are the best kids in the in the high school anyway down to 12 through this intensely competitive process we now present the winners of the USA mathematical Olympiad medal yakov / chinko Kogan would you and your mother please come to the podium I think it's very good that he's good at something cancer he can meet other people it gives something interesting in his life of course he hopes that he didn't get into the team Jihan taken cold Bryan Lawrence get Rick Liang mess is all about exploring trying new methods and techniques Richard McCutcheon you son Arnab Tripathi how much should I carry gone I like everything about mathematics like I appreciate the elegance of proofs just the sheer power of it is also really beautiful actually for the for the problems she's doing like this kids doing many professors even cannot do it math competitions are great they introduce all these new ideas and in particular give students who are in school the first chance to see how you can be creative in solving a problem because you're asked to solve problems that are unlike anything you've ever seen before and so you have to develop a new way a new technique to you I mean you know of course may not be new to the the whole world but at that moment it has to be it has to be new to you because you've never seen anything like it and that's really exciting girls are still by far the minority at the IMO I mean I remember actually in Taiwan they tried to have some sort of dance as part of the IMO program and it was less than twenty percent girls and so this is a disaster being the first female and I feel like people kept looking at me and I had to represent all the females this is what women are this is what women who can make the eye emoji more like it's definitely worth it and I've been so excited to see in the past couple of years the US has had two more women on IMO teams the American mathematics competition started in 1950 with a competitive test in the New York section of the mathematical Association of America it has grown considerably since then I'd like to present the clay Olympiad scholar award - Bryan Lawrence of Kensington Maryland it honors the solution to the USA mathematical olympiad problem which is judged most original math is a beautiful creation in itself it's a human achievement it's an art in the sense it expresses an absolute truth that science doesn't I [Music] [Music] [Music] basically picked six medium problems maybe two of them on the easier side of medium four of them on a harder side of medium I told them the only scene I want to see on a TST is to really use these problems to magnify their difference I tell them the hardest problem on a team selection test will not be as hard as the hardest problem and I am all in a certain ways very brutal because there's no easy problems to lead them to make them feel comfortable on the test for enough power doing math it would kill most people that early I guess brain dead after that I just like that clicking part of math I could feel that my brain is actually working when those things happen if I do well on the TST or Sylvania then I'm pretty sure I can do well in the team so I guess it all depends of TST for the TST that's kind of special because you have to think really hard it decides whether you're on the team or not and you have a lot of pressure in there too I first got interested in contests math in sixth grade when my teacher introduced s-- me and my parents to math counts which is a mineral school competition I think it was the first time that I sort of discovered really the subject of math instead of being taught random stuff in school that's basically how I learned just by reading and working on problems that I thought were interesting I was born in Nanjing China I came to the US when I was two and a half to Pittsburgh while he was 67 year old he enjoyed doing mental calculation like addition subtraction multiplication things like that and he also enjoyed discover the patterns of the numbers like he found the difference between two consecutive numbers square is the addition of these two numbers he liked to think a lot of things like think deep but I think most kids at his age them to think that much as much as he thinks so in that way he's a little bit lonely Ross along with theory about continuous missions before you get there humans since people at my school don't really like I don't think anyone there really has the same interests in math so you can't really connect he always so in the mass world I thinking all the time so sometimes I just tried to get him out of a house and do something [Music] if you're working on a problem or satyr problems for a long time and you sort of start to figure out the structure and then like finally you get it then I think that's a very rewarding part of math in general timmi math is more of exploration than knowledge I think [Music] okay guys we're done this is not a line no that means in our control that means that means pbq is clear sorry I lost my diagram yeah everybody sort of has a sense of how well they've performed they know how the scoring goes they they have a good sense of I I really nailed that problem I should be able to get six or seven points out of it or wow I hardly made any progress at all I'll be lucky if I only get one or two points yesterday I totally secure though I think I well cover the late I was very tired and I just wasn't doing very well even if I got only one problem that day I I would still be in pretty good position so I guess that might have that might help me think more clearly and that I finally got it miss reading a question when there's three questions it's not the best thing to do in the world that's exactly why we have this team selection exam is to train these students how to react after a bad day it's tough I mean it's an intensely objectively competitive field and your worth is measured by your success and so I think there's sometimes some real complex play of emotion it's a lot like the Olympics you can be a world-class competitor but to make the Olympic team is special same thing happens here I didn't really see myself not being on the team unless I just had a severe case of panic attack or something like that which actually did the night before the second half the TSD I couldn't sleep like I did not sleep at all which was rather perturbing the most interesting thing was Halloween candies he would collect the candies and the following year the candies are still at home because he is sorting them sorting them by colors sorting them by shapes all the time sorting I was actually born in Lansing Michigan so I'm an American by birth but I spend most of my life in North Carolina I moved there when I was three because my dad got a job at human see I guess by the time Italy there was nothing for him because all the high school courses added by the seventh grade he has finished all so from ninth grade onwards he started taking classes at even see math department he is doing his part we never have to tell him when to study do your homework never had to do that and I quit my job just to be able to take him back and forth from school to university and all that many parents we talked about oh now this the children are so competitive or we make it competitive for them I don't know and they do get stressed out but I don't see that in Arnab or his friends they recognize each other's strengths and really respect that I play classical music quite a lot and that's what I consider my other great love besides mathematics because I've played music since I was maybe five maybe younger but he's definitely been for a long time when I hear on a play piano then I tell him I don't think he's that good in math he touches your heart your soul when he plays the piano he should go to piano see the music is really much more emotional started intellectual for me at all whereas with math you can't really do math emotionally that doesn't make sense he uses maybe piano or music playing to unwind himself when he's doing some serious mathematics and he will come back and play the piano maybe it makes him it makes him feel good makes him feel relaxed and he comes back and plays a piece with more emotion and more strength more love and the funniest thing is that every time before an exam or something he will play the piano we probably have not seen the best of the best of him yet the rest of him is yet to come after the second day everyone went to watch a movie I guess it was a mindless action movie so pretty good for forgetting about the test and then by the time we got back there were only just two hours to wait each problem is given seven points and there's six of them and right now I believe downstairs all the graders are discussing whether this guy should merit six points or seven points for five points or two points or no points it's just essentially six problems even though there's possible 42 points my guess is that it's an indexing error so I really want I really want to give this a 6 you can decide whether the solution deserves to be graded from the bottom up or the top down from the bottom up you start giving points for things that they did that were right and that implies that they in general the problem was not particularly right and the top-down approach you're taking points away from seven you're saying that their solution was very close to a full solution and there were just some things wrong with it we spend a lot of time discussing even you know what is what is that word that they wrote there we can't work out their handwriting or do you think they really meant this instead of that he's off by n minus 4 over 2 it changed but you're not entirely sure right okay this sequence of computations certainly can yield you the right now yes yes and if you did them properly he would have gotten the right answer we had some discussions about particular papers that affected the few team members who were right on the border but in the end we had a division between the top six and everyone else and we announced them that night it's pretty much as soon as we know okay sounds good we have our team okay so we had a good TST this year I think you guys did well particularly on the second day we had lots of people solving two problems which is always good it was nice but that's not probably what you're here for the 2006 us IMO team in alphabetical order is a cable zebb Brady Ryan Co you son all right dr. pathy [Applause] and Alex Jack sitting there is kind of nerve-racking because Alex are sort of prolonging the announcement as long as possible he alternates the first alternate is Bo Hagen and the second alternate is tedric bling I was pretty sure I would not make the team because I knew they made many mistakes and if you go in there and thinking Cara you're not gonna make it it's not gonna hurt you as much because you prepare yourself for the fact that you're not gonna be there well I was traveling back to Exeter for my graduation so while we're in the airport I called someone back here and asked him and then he told me the results these things are very close often we have 12 problems to divide them and one might hope that that would be enough but a lot of the students cluster around the point where we have to draw the line for the top six to make the team I wasn't really disappointed with how I did because even looking at the problems afterwards I didn't think to myself oh I should have gotten those I had a good chance of making to you but you know it's always the competition so I missed it I'll try again next year and str2 years ago since eighth grade I was like okay I'm going to make the USIA mode some year and when it finally happened I mean I said that I knew I had a pretty good chance coming into this but I was really happy when it happened that I finally made it this was what I've been striving for for quite a long time I called everyone I knew I called my family and a few of my closest friends and they were all excited although I think my brother was most excited of all and possibly even more so than myself he was bouncing off the walls I think literally I heard a few crashes so yeah it was a nice surprise to make the team I've been told I can see diagrams in a way that most people can't in geometry that's very concrete you can literally look at a problem from many different angles I guess my interest in math started around seventh grade I was always good at math I have faint memories from like doing exponents in third grade and stuff from eighth or so on most of the math I've learned was not through school it was through reading books and just independently learning whatever I could so my dad is a psychologist self-employed my mom is a lawyer she works for Bank of America neither of my parents has that mathematical side of the brain Matthews a year older than Zach and they were in the same high school the Zack would sit there and just get award after award at these assemblies and Matt was his biggest fan he was his biggest supporter he would say I have a math tutor in the room next door to me there's nothing to complain about here some people think that people who are so good at math they're just nerds and they're whizzes and that's all they know how to do and Zack has never been into that category at all he's always played sports he won a conference championship and pole vaulting and you know he's always been completely outgoing and willing to do anything he just happens to love math he's always shown me a different way to look at math a different way it's opening up it doesn't have to be straight out of the book there's more ways to go around it there's more ways to look at it well he's a great brother we would have to pretty much ask him to stop doing math and finish his homework sometimes just because that's what he would prefer to do so yes you know his friends everyone would say he's a genius at math but they really didn't see how hard he worked at it and continues to work at it all the time that's just what he's passionate about I worry about him being a perfectionist he can be good he doesn't have to be great I worry more that he stresses himself out trying to be so perfect when it's okay to calm down and do well do your best and if it's not the best in the world it's okay [Music] for a really cool problem you can explain to anyone why it's cool without getting into the gritty details of the formulas and the calculations I think math does that it speaks for itself and it's very beautiful [Music] we've now got 50 kids roughly at each camp and there's the 12 kids in black group of the u.s. amo winners and high scores and from that we select the team and then the blue group is typically juniors and sophomores who have scored say honorable mention level on the US amo and then there's about 25 freshmen I'm Steve Dunbar I'm the director of American mathematics competitions welcome to the University of nebraska-lincoln the University is happy to host you here with that I want to introduce few people in order some of the instructors are advanced undergraduates or graduate students in mathematics they've all been through the IMO they've all had some of them two three even four years of experience at the IMO they'll be doing a combination of grading and instructing I was at mop 94 95 97 and I've been here twice officially other than those five times that I just mentioned rs1 jelka and I've also been here a couple of times yeah I think you'll see me mainly for geometry I am like I will be greater here I was a greater here last year and was on the russian IMO team in 2002 and 2003 and the US team 2004 deputy leader of the u.s. IMO team this year and I'll be one of the instructors teaching you you need to remember that your primary purpose for being here at the program is to do math and luckily this is another thing that you can do with your friends I started coming to mop in my eighth grade year and that year was the first year I qualified for the US amo I was pretty surprised when I got an invitation to come to the camp because I thought it was only for the top like 30 people I was definitely not really close to like the top that group this will be my fifth time in and this year I think I'm actually finally getting all the stuff in understanding it what we're doing right now is we have some of the graders the younger faculty members are working one-on-one with the students a pair of graders work with a pair of students and they're gonna find out what the students weaknesses some strengths are and talk to them try to convince them to do work on some problems that they maybe think they can't do they can a Evy you've lost me sir single single we have individual tests normally every other day we take a four or four and a half hour test we have team tests with different groups teams of about six and it really does help to have two or three minds bashing the same problem you get different perspectives on the same ideas in the first week for like the first three days we were working on problems at night like every day it's not that easy to be thinking hard all day long one of the best things about this camp is that you can sort of talk to other people other kids this is like one of the most fun parts of my year what I like about mom is that it is very collaborative that you know what's going on is that you've got a lot of individual people who come in here and they don't know a lot of math you come here and you're sort of dropped into the deep end and suddenly you're surrounded by people your age who know all these things these wonderful things beautiful magical amazing things that you never even imagined were possible and then you begin to learn you hunker down you've become a part of this community and it's a personal journey it's an individual journey and well for me at least it becomes a part of you for the rest of your life these kids discover that there are other kids out there who love mathematics who want to do mathematics more than anything who love to compete at mathematics who love solving problems and often these kids have been isolated in their high school and they get together with 50 other kids in this intense three week pressure cooker and they come out exhilarated [Music] and mop two years ago I probably gained more math skills in that three weeks than I did in the past two years that's because I was able to actually gear myself to work and in school or in other places really hard so I think the program really serves this purpose isn't to prepare people for the team so you realize that there are some people better at this and some people better and this and you are better and some other things you do very well tomorrow good you still have six more individual tests to go if you didn't do well it's okay you know you can't even call me and climb with me that's fine I have lots of teachers okay so are you passing is zarathustra LSR Brady and what people call me is my initials my parents thought of that and they were like it's cool [Music] my mom says like when I was little I learned multiplication quickly but I couldn't do the time to stay tall so that's where a sucked I actually skipped 2nd in 3rd grade one of the most important things that I noticed about Zeb is his commitment if you were to give them a book on any subject or any task and he would just elegantly work on it sometimes he would spend two hours and I would ask him what are you doing and he's like well I'm just trying to solve this problem he's just focused he doesn't give up he tries all kinds of different approaches he's gonna be very hard sometimes on himself I think I have a little bit of a distaste for schools just from my own experience so my philosophy was the fewer years he spends in school that better it will be for him he must have spent a total of three years in school by the time we moved out here to Los Angeles I was born in Philadelphia my sister was born in Georgia my other sister was born in California so we move a lot he always talks about math he doesn't do any work all he does is just do math all day sometimes he cooks us dinner if he's feeling nice and he eats a lot he teaches my little sister how to do math he's an okay brother I wouldn't trade him for a new one I'd like to upgrade him though Matt hates me I hate math I can't do it first of all it's boring second of all it takes so long for me to do math homework so do more like other classes like I can kind of just do them both math I have to actually study if I wanna do anything so Matt's just kind of sucks you have to allow a kid to recognize what they're interested in and you have to be able to not enforce what you think is important on them and that freedom will actually allow them to meet more of their potential his math teacher introduced me to him and when I was talking to him it became obvious like how much potential he had of smart he was he had some holes in his math education if it was to state forward or to obvious for him he kind of skipped those you know so we had to work on those to fill those holes and in coaching there's this thing that I liked so much especially students like zip when they understand they're like ice kind of shine and you see that they got the point I have this theory that the smarter a mathematician is the less they work or the less they like to work but I don't know if it's true see I haven't asked enough actual mathematicians how lazy they're I'm kind of amazed at his energy like he never seems to be tired or exhausted you come in here with all of that energy and all of that excitement and you feed into it all of the things that we're gonna teach here and some people are just like a sponge they take in what's given on the board and they say please can I have some more by the end of it they've mastered all of the techniques that we have to offer we don't want to do this as a factory you know if you go to mop for three years you will automatically win a gold medal or silver medal it's more about identify students ability train them to be more experienced but the ability is their own I see this as being a program that sets a tone or a standard that says that yeah we can meet this kind of international level of problem-solving and mathematical bill we can do it there's a sense that that these students are our students they're their American students and they represent something about our educational system and so we should all want them to do well just because it'll represent our schools well to the world we can be fans of them just like we're fans of a sports team or anything else we want them to do well I'll try to get a gold medal and I hope everybody gets it too well we'll come out with six gold medals hopefully and we'll have fun I don't know anything about Slovenia though so all I know is it's like in Europe somewhere [Music] that's not very much [Applause] [Music] dear students you have been selected as the best students in mathematics of your own country to come and compete at this 47th I am oh you can look forward to interesting mathematical problems and tough work in trying to solve them well they are most started in a year 1959 the first IMO was held in Romania they decided to invite teams from the neighboring mainly socialist countries of that time teams of eight students to participate in an international event it is a very important aspect of the IMO to really show examples of good mathematics problems this year's problems are quite beautiful the host country will have mathematicians meet and that they were pick the tops early problems they they liked the most and that's the IMO shortlist and they'll debate these problems and they'll they'll try and boil it down to six problems that they end up selecting there'll be one or two geometry problems one or two number theory problems one or two combinatorics problems one or two algebra problems and they'll try and make a few of them kind of encouraging so that just about everybody has a shot at it and they'll try and make one or two pretty hard after that I started getting nervous because now the test is there so you start to think about how well your team can do on this set of particular problems this year our team composition is very intriguing I think the two not experienced students are Zeb and Ryan but I think each one of them are very creative in their own way so they might give us very good surprised this is a very young team so I will not feel bad if they finish number five Number six number seven I will not also be surprised if they finish number three number two on even number one our expectations are always just that they'll do well we hope that no one will freak out on the exam and they can't do any of the problems the students and the faculty hope that this might be the year we finished first but it's a hard row to hoe the United States fielded its first team in 74 we've had some real spectacular years 94 was the year that we had six gold medals scored first the Chinese certainly are strong the Russians are always strong the u.s. is pretty strong Eastern Europeans are pretty strong the Chinese are certainly the strongest now I remember last year which was the 2005 IMO in mérida Mexico and China won by a large margin not really competing at the same level as the rest of us they do much more training than we do it's builds into their school system in the year of 2005 they got a 5 code and won several with a drink on number one they've got a lot more people than we do you take you know a billion people and you picked the top six and they should be better than you know a quarter billion or what do we up to a third billion people now and take the top six and so that they even though they got us be pretty seriously on on numbers to the Odin uni away to Miami well wish you answer you know good had they expected to have six gold people in this country and in China and Russia and you know the other countries that do habitually very well like to spin it as a national contest a show of national pride or talent or what-have-you but that's not what it's supposed to be I think in the ideal the IMO is about individual achievement it's about one person's ability to supersede the vast majority of her or his colleagues it's good for them to see people that are really really good you know to see students that are better than them and to spur them on a little bit these students have their first exam tomorrow morning this is kind of a high point the whole thing that they've been working towards oh yeah so this is really big for them and I've just been trying to calm down today and I had to tell them to stop doing mathematics they've been doing mathematics all day every day for the last week so they've just gone and played some games played some football and chilled out this evening and hopefully they'll probably be getting ready for bed now and hopefully they'll get a good night's sleep but it's a big day for them tomorrow they're all quite nervous it's okay to be nervous as long as you're only nervous for like the first half hour and none of the problems will be I'm sorry this is the hardest problems you had a model so remember I mean each of you can get each of these problems yeah we don't expect you to necessarily to get all of them and remember the easiest way to get unstuck on a problem is to work on another problem you know one of the things that our team is always very good at is getting one points or two points on problems we don't know how to solve so keep up the tradition you know you found the problems you don't actually know how to do write down useful things okay good let's go [Music] for the first three hours I try to work on the problem that I choose to do instead of going back and forth you read all three poems first it goes there's also this rule that if you want to ask any questions it has to be in the first 30 minutes and then if I see a geometry problem I usually go after that one because I like geometry of course it's a huge responsibility the six of us representing the United States of America and that's a big burden but it'll be fun I'll do my best and what happens happens I start working on the problem and because you can't really concentrate and be nervous at the same time concentration sort takes over I guess sometimes you are know immediately what approach to take but sometimes it takes a while you know it takes a lot of stupid ideas to generate a good one let P be regular 2006 gun a diagonal of P is called good if it's endpoints divide the boundary of P into two parts each composed of an odd number of sides of P the sides of P are also called good suppose P has been dissected into triangles by two thousand three diagonals no two of which have a common point in the interior of P find the maximum number of isosceles triangles having two good sides that could appear in such a configuration that problem took me a while but I first considered small cases like where I replaced 2006 guns with squares and hexagons and other even cited polygons and so I figured out that the answer should be one thousand three by checking the small cases and then after that I just knew it had to be some kind of induction and I just worked I just tried a bunch of things and eventually something worked the way I approached it was inductively instead of looking at all 2006 edges at once I looked at three of them four of them five of them and inducted upwards and so I I initially found that if you have a small portion of the polygon at most half that the total number of good isosceles edges is at most n over two if you have n sides of the polygon and then I found a hole and I just kept fixing and patching and fixing holes by the time I finished writing it time was almost up [Music] they're always very excited when they get out of the test they want to compare to their their friends and they want to talk about how they did the problems because they're very proud of what they did in there also [Music] all we have right now is rumor innuendo it seemed like all the other schemes thought one was also very easy to was slightly more difficult not everyone got it three it seems like very few students solved I heard the rumor that five of the Chinese solved all three problems sum of squares greater equal to zero hopefully I mean I'm not there can't just be one geometry and it'd be really really easy right yes I'm a trip I mean there's no way there I'm looking forward to no comma networks looking for it the number five there definitely is an element of tension in taking on the tensions built up from taking the test and you still have the second day to go so it's just sort of eating away at you [Music] this is in a way a competition but in a way it's a feast of mathematics you really can feel the all impaired feeling that this is a smaller bird I am always not just a competition is something a place that people from different countries can gather together and get together and socialize and have fun [Music] so this is camera that's right Israelis took our rooms key when in our room rather moves and left a giant moose it's like a stuffed animal moose I know that the Canadian team brought some stuffed animal doll then the Americans came saw Tom Lussier I actually know nothing about it I think the Canadian came as a moose sent something this little blanket mug the moose put it there and that's the most not to buy us so how we found it and then we just decided to take it and nobody goes where did them our nerve went into the Canadiens room grabbed it and ran out with the Canadian in the room and though they'd be quite a juvenile thing to do really like stealing some other teams moose [Music] this morning they were a little quieter but they're definitely more relaxed than yesterday morning before the test I think they're a little bit tired I'm a little worried about that but I think they'll make it through once they start looking at the problems it's a slight caffeine to them well I'm not a morning person I can't get up so I just made sure I was awake long before the test itself so I had time to actually wake up maybe eat some breakfast I fell off my bed I just fell onto the bed below and I think I hit Zeb's away I just sit there quietly I just tried to keep myself calm and I think I was okay at that [Music] you can't really know what the problems are gonna look like you can't know how many you're gonna be able to solve each day but I mean you just do what you can you work on the problems if it's hard for you it's probably hard for a lot of other people [Music] we certainly feel a little useless right now this afternoon we'll meet the leaders and we'll start to actually do our job the fishing village if AF sufficiently large mean on most condition evaluates waiting to see how we've done nervously now they'll be fine they'll be up see fine had a good day yesterday - easy number theories and geometry this is like just what we wanted I got no geometry I got the case cake is - that's that was on a previous I ever do you - a equals plus minus C - B but it can't be like negative I don't know like my thing almost works there's four like hexagons with parallel sides and I was trying to do it with Bell's equation it took me forever I was like no outside III do something interesting but sort of I basically mean small mistakes on two and four I definitely thought about them during the contest I just forgot to write it down there's not really much I can do now number three was an inequality and I had essentially the right method if I had done the algebra correctly I would definitely have been able to finish the problem I said I didn't yes the testers would be hard and rather unconventional but I'm still not really sure how how it would come out so we'll see three pong definitely it was it was a tough interesting test that a reasonably so um looking forward to results like I got there would be infinitely many we might have done pretty well what's pretty funny though is like I didn't well like comparatively because there is like number theory problems like - number three - problems on day two and there were things like a combinatorial problem that not everyone got it and you know a really easy job she probably those are problems for me they still have to argue for points that's what's gonna be happening during the next three days trying to see if they could get like one extra point for that one thing I put in my scratch paper somewhere that hinted towards the solution it was alright it was certainly hard you know our students you know didn't do as well as they probably hopes and but you know from we only talked to a couple of other people but it looks like they also didn't do as well as they would hope so we think it's hard and now we're gonna go on like excursions like on the program it says excursion that's all it says although someone said something about a castle [Music] it's when the test is over that the real work for us begins because then we have to read over our students papers find the best way of presenting the students solutions to the coordinators and try to get as many marks as possible we heard that the Canadians went through coordination on number one and had 42 points on number one most countries the top country whoever yeah Alex and I will start to make our own grid this problem we feel like this guy definitely 7 points this guy we have a wrench maybe 2 to 4 how do we are things they cannot shara shara combination you know good triangles I don't know but in if you only but it's stronger I mean you only put it in the negative and you've got about 20 minutes maybe 30 minutes to go in and argue your six papers if you're a leader the advocates trying to argue it up to about a six or a seven the coordinators are trying to keep the scores low to keep everything separated and so it's the middle cases that are tough although the power is sort of in the hands of the coordinators and all you can do is sort of make the best case for people who tell them solve the form you give them a point many students have had original ideas and it can be quite tricky to determine whether they're flawed or not so an intricate argument which involves cutting up a 2006 gone into one thousand and three triangles can go wrong on a very minor detail and it might be hard to spot that in a solution that is written in a completely unknown language if you don't have this mistake then you have seven right he made a proof where one of the steps one of the steps is false mistake there are so many parents in parallel science is to basically approach this problem to parallelogram for such a proof you have to do perform some number of steps and some progress so this was a very hard problem and in general this irony is a very hard one and each point can be very important teams and they would like four four instead of three and they would like one instead of a zero it's a bad luck to them that they don't have a complete solution this is the key right I mean I mean this this idea is the key and I I mean I'm not claiming that he can do it or that he spent more than ten minutes thinking about it but so now you are interpreting he didn't do it but you tell me I cannot say oh he did it that's not very fair I said over here he has some ideas you say no you cannot assume this on the paper you have to only assume he didn't know but and over here you said okay even not it's not he didn't know you said he didn't know anything I can understand that they want more point and I can say I would not like to change my mind so probably those score the three I suspect that that they also know that but but they tried it seems to me that maybe we need to speak to the coordinator and clear up this mathematical matter and if we clear up the mathematical matter then the coordinator is supposed to assess a solution without necessarily understanding the language when the team leader comes to him and they coordinate they have to agree on a score now in principle if in a perfect world they would agree immediately maybe people will expect that this has to be so in mathematics but when you look at student solutions that are scribbled on several pages and sometimes they are a bit confused or illegible then things are not so clear anymore everything is done in the spirit of fairness the representatives of the country are not supposed to root for as high up score as possible and my coordinators are not supposed to try to get score that is as low as possible but really it is in fact done by cooperation that's why this is called coordination and not grading [Music] we know about that that he understands but we need to prove this inequality which is obvious obvious is it obvious for you I think it's really obvious but you need to do to do some work yes I think this one is like this one if you if you accept this one and this one and this one why don't you accept this one I hope that our students would win three gold medals and three silver medals I hope that for China we had a serious language problem we were not able evaluate any of their papers yet so we have asked written translations for all the six papers and we will continue today every hour or so they were updated a score to be posted on a big wall that's pretty interesting very intense so they have constantly updated boards in the cafeteria online everyone was checking them constantly in fact we had seven a few other kids go into Excel and run some simulations what's the highest possible cutoff what's the lowest possible cutoff what's the cutoff if we like populate them with linear and and values exponential random values this year it seems that the two difficult problems proved a bit too difficult so the coordination is still going on we don't know exactly how many student solved each of those problems but it seems that below 30 students solve problem three one of the heart problems and only about ten student solved perfectly problem number six right he has three things and you said yesterday he has no intention of doing anything with this we're taking seriously they want to know those same score and afford another one right it's really about math it's not about you know do you wanna that's fine we were sorry we also so the final score for the u.s. on six was three points from Alex we were trying to get an extra point for Zeb an extra point for Alex but neither worked so just three points for Alex the forty-sevens IMO is coming to an end in the final count we had 498 students from 90 countries participating one of the biggest IMO was ever [Music] [Applause] [Music] it does increase your self-confidence a lot to get a gold medal it I am out like when I get home I'll be like hey I got a gold medal I won't keep it a secret [Music] this so hasn't really sunk in that I met the IMO which I mean is still pretty exalted in the US Paulie go home and then one day it'll hit me that I've actually been to the IMO and got a gold medal it is like pretty incredible the biggest accomplishment for me was to qualify to come to IMO I got a better understanding of the worldwide mathematical community how math works in other countries in other parts of the world and it's not too different from where I'm from [Music] I am about to make known the names of three skilled thinkers whose performance could be rated as superior since they gathered the entirety of points URI Boreyko gu u Alexander magazine oh no surprise I think I deserve it tell me why - um well the interest in mathematics and I study mathematics using my hat we always seem to contest is only a very little stuff in their life or the other career contest is only a little so how do we guide our students all those students they bring back that experience about how to deal with contests and how to deal with success and how to move up I think that's very very important it was so great to be able to make so many friends with people who were similarly interested and talented at math and they also gave me a picture of what the math world was like the people that I met through the math Olympiad program introduced me to the world of being a professional research mathematician the IMO is the culmination of all the training that you've done before it I feel I have a lot more confidence and I've also learned how to sort of reflect on problems that I've done no matter what your result is on the test it doesn't erase what you've learned from your training and your experience I think the IMO inspired me to study math as deeply as I did if I didn't have the competitive aspect of that especially my earlier years honestly I might not have been spurred on as I was to propel myself forward in mathematics the IMO definitely does serve to introduce this whole generation of young mathematicians to what they might do in their future life [Music] my goal is just to uh publish some really good paper my current goal is to publish an insightful paper by the time I turn 18 whenever I actually got something by my own effort I think that's when the whole thing got a bit exciting I do think doing the competitions has helped me to think better mathematically and understand things better they've taught me how to write rigorous proofs that's something that will be used in doing research [Music] I don't do mathematics because it's mathematics I do it because of the astounding connections and interpretations and general creativity that it allows [Music] this must be the vehicle cross Madhab champaign-urbana Christmas tree [Music] plus 30 March is no spirit can name haha because three gospel a partial screen tanana b2c [Music] big ugly bug poisoning a safe home for Vanya speak with her grip on poison impossible [Music] oh this our grilled cheese Tucson the support versus love voltage on grated cheese no sound plus y 4 o'clock bus Islam [Music] you [Music]

Channel: Ning Shan

Views: 267,987

Rating: 4.9454374 out of 5

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Id: VboXBHVaCFU

Length: 81min 20sec (4880 seconds)

Published: Sat Oct 20 2018