F=ma/CMIMC 2019
So a week ago, I went to CMIMC at Carnegie Mellon University in Pittsburgh University. It was a nice experience, I went with some friends and we didn't expect to do well. Which was good, because we didn't :D.
We left on Friday after school. Conveniently, I took the F=ma right before leaving. I won't make an entire blog post about it because I simply don't focus on physics that much. Maybe I will focus on it later, but not really a concern of mine right now. I haven't even finished the entire physics curriculum, so there's 0% chance I can do any angular momentum or harmonic motion problems X_X. And I've also barely done any prep for it, so I didn't really expect to do well. I think I got at least a few questions correct, so that's good :D.
So we left after that, and it was a grueling 6 hour car drive to Pittsburgh. It was literally all on a highway, so it was a straight and uneventful drive. On the way there, I did some programming, looked at some CMIMC problems, and mostly slept. When we got there, it was around 11:00, so we went to our hotel, took a shower, and slept.
The competition began at around 7:00, and I didn't really know what to expect. As we were assigned our team, I noticed a common theme among large math competitions like this: they don't really have a lot of security. Obviously, cheating is totally morally wrong, but it would be so easy to take your phone out, do a quick calculation, and then put it away. Especially because it's such a large auditorium and there were no proctors in the back paying attention.
As for the actual testing environment, it was pretty bad. We were each sitting in auditorium chairs with 2 sheets of scrap paper for 10 problems and around a clipboard's size area to work with. To make matters worse, we got one sheet with 10 problems, one for putting answers on, and then a few for scrap paper. By the time we got to the final round, I had so many old papers that I kept dropping them and having to run after them to get them back.
So, the actual competition:
Problems/Solutions: http://www.cmimc.org/archive
Round 1: Algebra and NT - I really didn't know what to expect. I'm not really a specialist, more of a, can do all problem types, not super well, sort of person.
#1 was pretty easy. You can just write out the terms, factor, and find that you just need to find r. Since consecutive terms are given in the problem statement, the problem didn't take too much time. However, I actually trolled and wrote down the wrong value of r, meaning I put 1/9 at first. Luckily, I caught this mistake later and fixed it to put 8/9
#2 was interesting, but not hard. You can just findout that you need to find the maximum number less than or equal to 3000 that is a multiple of the first n consecutive numbers. Then the answer is just n + 1. So you can list out the prime factorizations of the first 15 or so consecutive numbers and then find the lcm of the first n to find the answer. You can find easily that n=10 works, so the answer is 11.
#3 was not that interesting. You can simply generate a system of equations with 3 variables by equating coefficients and givens, and then solve. The answer is then easily evaluated to be 7/5.
#4 was bad. I think a lot of people got this wrong because of random edge cases. Only one person on my team got this correct, and multiple people got the same wrong answer. Essentially, with some intuition/basic number theory, you can get that n must be a multiple of 6. I stopped here, as did many others. Some other people however, realized that n could not be a multiple of 18 either. But there are more edge cases that can be considered with more cases, and it turns out that n cannot be a multiple of 18, 42, or 60. So therefore, after a bit of careful counting and summing, the answer was 360
#5 was an interesting problem, but also really easily cheesed. You can easily look for matters mod 7^n and then use those to quickly and efficiently generate values for x1, x2, and x3 to get 121 as the answer.
#6 was just a bashy as heck problem. Using Vieta's formulas, you can do a lot of bashing and fancy manipulations to arrive at a clever answer with cancellations. However, the method I was trying to do was to somehow find a way to manipulate the polynomial to get the sum I wanted. I had no clue how to do it, and apparently, it's possible if you use imaginary partial fraction decomposition :P. Anyways, the answer sucked, and was bleh to derive by myself later.
I didn't even look at #7-10. However, I looked at #10 solution because why the heck not, and that is just hell. It's literally 2 pages long and references other solutions X__X. It's a pain and I never want to see it again...
Round 2: Geometry - Ugh, why do they have a section just for geo...
#1 was easy. Drawing the iscoseles triangle through the points in the center of the sides yields the angle to be 170 deg.
#2 was interesting only in the proof that the shortest possible length was 2. Otherwise, the problem was relatively straight forward in that you can assume that YW is perpendicular to XYZ.
#3 was a rip from me X_X. I somehow drew my diagram wrong and accidentally did a bad assumption, which led me to getting a totally wrong answer. So I got this wrong by a lot :(.
#4 was really bashy, but not that bad. Essentially, coordinate geo kills the problem as you can make each point a coordinate and calculate the equation of the plane. Then, by seeing the vertex has to be (s, s, s), you can plug in and solve.
#5 was another problem where you just have to draw a big diagram, stare at it, draw a few lines, and realize some random things XD. Also coordinate bash works well on this problem, giving the answer easily.
#6 was doable, but I drew my diagram really badly and couldn't really see anything, so I couldn't get it.
Everything else I barely looked at/couldn't do. IMO, #8 and #9 have really dumb solutions. #8 literally says: observe that this situation has to look like this, which is not a great solution. #9 literally uses calculus to find the area under a curve .__. IMO, competitive math problems just shouldn't use calculus.
Round 3: Combo/CS - Why are these combined? I guess a lot of CS is counting, but to anyone who just came for the math, CS is weird...
#1 was not bad and is just complementary counting.
I'll talk about #2 later.
#3 was funny for me. The first time I saw it, I was literally dead and just bashed the entire thing out. My entire paper was filled up and it was a pain. After 30 minutes, I went back to the problem and realized I could just do casework on a and find out how many values of b worked to produce a number c that was less than 60. So you can get through those much faster and I found out that my original bash was off by one :D.
#4 was a wordy, but nice, problem. You can do a bit of math to simplify the expression and use a table. I don't remember the exact math I used, but it was spot on and worked perfectly. Except for the tiny fact that I literally got the answer 808 on my paper and then wrote 708 on the answer sheet. Ughhhhhhhhh.
#5 was interesting, but I didn't have enough time to go through it (Thanks #2-4)
Everything else was really wordy and I couldn't do it/didn't have time.
Back to #2. Wow, this problem sucked. I first looked at it and spent a solid 3 minutes trying to find a bijection to simplify counting. However, I couldn't find one and then I resorted to the most random casework in the world with just making random cases as I went. And when I finished the comp, I asked my entire team, and no one got the same answer as anyone else. When we went home after the competition, I still had no clue what the answer was, and then I went on Wolfram MathWorld and found this incredibly elegant and simple formula to solve the problem: (x-1)(x)(x^6-11x^5+55x^4-159x^3+282x^2-290x+133). Such simpleness ._____. Yeah no. Anyway, I plugged in the number 3 for x, which is the amount of colors to use, and bam, I got 114, the same number I got on the competition through my random casework. CMIMC's solution for the problem used similar casework to mine, but also simpler (I think?) I honestly can barely follow through any casework solutions of problems like these. The rest of the test was nigh undoable.
Recap
So in the end, I got a 4 on A/NT, 4 on Geo, and 3 on Combo/CS. Honestly not my best competition, as I could've made it 6, 5, 4. but whatever :\. It didn't matter much and I didn't expect to win much. I'm not going to talk too much about team/power b/c I don't know what my teammates were thinking. As of posting this, AMC 10A is in 2 days. I've bene doing a lot of practice and hope that I can make JMO. But it's still robotics build season and there's a lot of work to be done tomorrow >_<. I'll try to do better!
We left on Friday after school. Conveniently, I took the F=ma right before leaving. I won't make an entire blog post about it because I simply don't focus on physics that much. Maybe I will focus on it later, but not really a concern of mine right now. I haven't even finished the entire physics curriculum, so there's 0% chance I can do any angular momentum or harmonic motion problems X_X. And I've also barely done any prep for it, so I didn't really expect to do well. I think I got at least a few questions correct, so that's good :D.
So we left after that, and it was a grueling 6 hour car drive to Pittsburgh. It was literally all on a highway, so it was a straight and uneventful drive. On the way there, I did some programming, looked at some CMIMC problems, and mostly slept. When we got there, it was around 11:00, so we went to our hotel, took a shower, and slept.
The competition began at around 7:00, and I didn't really know what to expect. As we were assigned our team, I noticed a common theme among large math competitions like this: they don't really have a lot of security. Obviously, cheating is totally morally wrong, but it would be so easy to take your phone out, do a quick calculation, and then put it away. Especially because it's such a large auditorium and there were no proctors in the back paying attention.
As for the actual testing environment, it was pretty bad. We were each sitting in auditorium chairs with 2 sheets of scrap paper for 10 problems and around a clipboard's size area to work with. To make matters worse, we got one sheet with 10 problems, one for putting answers on, and then a few for scrap paper. By the time we got to the final round, I had so many old papers that I kept dropping them and having to run after them to get them back.
So, the actual competition:
Problems/Solutions: http://www.cmimc.org/archive
Round 1: Algebra and NT - I really didn't know what to expect. I'm not really a specialist, more of a, can do all problem types, not super well, sort of person.
#1 was pretty easy. You can just write out the terms, factor, and find that you just need to find r. Since consecutive terms are given in the problem statement, the problem didn't take too much time. However, I actually trolled and wrote down the wrong value of r, meaning I put 1/9 at first. Luckily, I caught this mistake later and fixed it to put 8/9
#2 was interesting, but not hard. You can just findout that you need to find the maximum number less than or equal to 3000 that is a multiple of the first n consecutive numbers. Then the answer is just n + 1. So you can list out the prime factorizations of the first 15 or so consecutive numbers and then find the lcm of the first n to find the answer. You can find easily that n=10 works, so the answer is 11.
#3 was not that interesting. You can simply generate a system of equations with 3 variables by equating coefficients and givens, and then solve. The answer is then easily evaluated to be 7/5.
#4 was bad. I think a lot of people got this wrong because of random edge cases. Only one person on my team got this correct, and multiple people got the same wrong answer. Essentially, with some intuition/basic number theory, you can get that n must be a multiple of 6. I stopped here, as did many others. Some other people however, realized that n could not be a multiple of 18 either. But there are more edge cases that can be considered with more cases, and it turns out that n cannot be a multiple of 18, 42, or 60. So therefore, after a bit of careful counting and summing, the answer was 360
#5 was an interesting problem, but also really easily cheesed. You can easily look for matters mod 7^n and then use those to quickly and efficiently generate values for x1, x2, and x3 to get 121 as the answer.
#6 was just a bashy as heck problem. Using Vieta's formulas, you can do a lot of bashing and fancy manipulations to arrive at a clever answer with cancellations. However, the method I was trying to do was to somehow find a way to manipulate the polynomial to get the sum I wanted. I had no clue how to do it, and apparently, it's possible if you use imaginary partial fraction decomposition :P. Anyways, the answer sucked, and was bleh to derive by myself later.
I didn't even look at #7-10. However, I looked at #10 solution because why the heck not, and that is just hell. It's literally 2 pages long and references other solutions X__X. It's a pain and I never want to see it again...
Round 2: Geometry - Ugh, why do they have a section just for geo...
#1 was easy. Drawing the iscoseles triangle through the points in the center of the sides yields the angle to be 170 deg.
#2 was interesting only in the proof that the shortest possible length was 2. Otherwise, the problem was relatively straight forward in that you can assume that YW is perpendicular to XYZ.
#3 was a rip from me X_X. I somehow drew my diagram wrong and accidentally did a bad assumption, which led me to getting a totally wrong answer. So I got this wrong by a lot :(.
#4 was really bashy, but not that bad. Essentially, coordinate geo kills the problem as you can make each point a coordinate and calculate the equation of the plane. Then, by seeing the vertex has to be (s, s, s), you can plug in and solve.
#5 was another problem where you just have to draw a big diagram, stare at it, draw a few lines, and realize some random things XD. Also coordinate bash works well on this problem, giving the answer easily.
#6 was doable, but I drew my diagram really badly and couldn't really see anything, so I couldn't get it.
Everything else I barely looked at/couldn't do. IMO, #8 and #9 have really dumb solutions. #8 literally says: observe that this situation has to look like this, which is not a great solution. #9 literally uses calculus to find the area under a curve .__. IMO, competitive math problems just shouldn't use calculus.
Round 3: Combo/CS - Why are these combined? I guess a lot of CS is counting, but to anyone who just came for the math, CS is weird...
#1 was not bad and is just complementary counting.
I'll talk about #2 later.
#3 was funny for me. The first time I saw it, I was literally dead and just bashed the entire thing out. My entire paper was filled up and it was a pain. After 30 minutes, I went back to the problem and realized I could just do casework on a and find out how many values of b worked to produce a number c that was less than 60. So you can get through those much faster and I found out that my original bash was off by one :D.
#4 was a wordy, but nice, problem. You can do a bit of math to simplify the expression and use a table. I don't remember the exact math I used, but it was spot on and worked perfectly. Except for the tiny fact that I literally got the answer 808 on my paper and then wrote 708 on the answer sheet. Ughhhhhhhhh.
#5 was interesting, but I didn't have enough time to go through it (Thanks #2-4)
Everything else was really wordy and I couldn't do it/didn't have time.
Back to #2. Wow, this problem sucked. I first looked at it and spent a solid 3 minutes trying to find a bijection to simplify counting. However, I couldn't find one and then I resorted to the most random casework in the world with just making random cases as I went. And when I finished the comp, I asked my entire team, and no one got the same answer as anyone else. When we went home after the competition, I still had no clue what the answer was, and then I went on Wolfram MathWorld and found this incredibly elegant and simple formula to solve the problem: (x-1)(x)(x^6-11x^5+55x^4-159x^3+282x^2-290x+133). Such simpleness ._____. Yeah no. Anyway, I plugged in the number 3 for x, which is the amount of colors to use, and bam, I got 114, the same number I got on the competition through my random casework. CMIMC's solution for the problem used similar casework to mine, but also simpler (I think?) I honestly can barely follow through any casework solutions of problems like these. The rest of the test was nigh undoable.
Recap
So in the end, I got a 4 on A/NT, 4 on Geo, and 3 on Combo/CS. Honestly not my best competition, as I could've made it 6, 5, 4. but whatever :\. It didn't matter much and I didn't expect to win much. I'm not going to talk too much about team/power b/c I don't know what my teammates were thinking. As of posting this, AMC 10A is in 2 days. I've bene doing a lot of practice and hope that I can make JMO. But it's still robotics build season and there's a lot of work to be done tomorrow >_<. I'll try to do better!
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