The reason I wasn't able to post a math trick on friday was because I was preparing for the AMT 2020 Math Competition (still going on). On saturday and sunday, we had the power round:http://attach.seedasdan.com/STEM/2020%20AMT%20PowerRound-problems.pdf, password=202008AMT. Now, my team had 4 members. However, one of them did not help, and another called Ross had Ross camp, so only had time to solve three problems(the hardest 3), and another called Daniel Chen also had Ross, and had to stay up very late in order to discuss the problems with me, and also only had time to solve three problems. But, Daniel helped us a lot in order to improve our LaTeX formatting and checking our solutions. In other, words, it was literally only the two of us and a bit of Ross's work.
Funniest moment: Daniel saying: "Imagine if I sent the pdf to submit through the wrong group chat"
What it meant: I was the team leader, so i had to submit the pdf solutions. Now, I do not have Adobe Acrobat, so Daniel helped us cut the pdf into 21 parts (we had to submit 1 pdf per problem), and so he had to send it to me. We were using WeChat. Now, there was a team group chat and the competition group chat(consists of all the teams). The 'wrong' group chat he referred to was the competition group chat.
Anyways, onto our math trick for this week... inspired by this year's power round... a proof technique called... INDUCTION! In the problems pdf, they explain principle of mathematical induction (PMI). In our solutions to 1c and 2a, we used PMI. 1a was some straight forward modular arithmetic, 1b was pigeonhole principle, 2b was Lifting the Exponent (we think that there should be a way easier solution), 2c was Strong Induction, and 2d was a combinatorial approach.
For PMI, just read the problems pdf, in fact their explanation is nearly the same as what I would say.
For STRONG INDUCTION, see below.
In strong induction, the base case is still the same. However, the assumption is everything below. In other words, strong induction is
The reason I wasn't able to post a math trick on friday was because I was preparing for the AMT 2020 Math Competition (still going on). On saturday and sunday, we had the power round:http://attach.seedasdan.com/STEM/2020%20AMT%20PowerRound-problems.pdf, password=202008AMT. Now, my team had 4 members. However, one of them did not help, and another called Ross had Ross camp, so only had time to solve three problems(the hardest 3), and another called Daniel Chen also had Ross, and had to stay up very late in order to discuss the problems with me, and also only had time to solve three problems. But, Daniel helped us a lot in order to improve our LaTeX formatting and checking our solutions. In other, words, it was literally only the two of us and a bit of Ross's work.
Here were our solutions: https://drive.google.com/file/d/1M-p_K2htOLwTPSLgb71K-4T42BVE3c6q/view?usp=sharing. Note that we used the ross formatting for LaTeX.
Funniest moment: Daniel saying: "Imagine if I sent the pdf to submit through the wrong group chat"
What it meant: I was the team leader, so i had to submit the pdf solutions. Now, I do not have Adobe Acrobat, so Daniel helped us cut the pdf into 21 parts (we had to submit 1 pdf per problem), and so he had to send it to me. We were using WeChat. Now, there was a team group chat and the competition group chat(consists of all the teams). The 'wrong' group chat he referred to was the competition group chat.
Anyways, onto our math trick for this week... inspired by this year's power round... a proof technique called... INDUCTION! In the problems pdf, they explain principle of mathematical induction (PMI). In our solutions to 1c and 2a, we used PMI. 1a was some straight forward modular arithmetic, 1b was pigeonhole principle, 2b was Lifting the Exponent (we think that there should be a way easier solution), 2c was Strong Induction, and 2d was a combinatorial approach.
For PMI, just read the problems pdf, in fact their explanation is nearly the same as what I would say.
For STRONG INDUCTION, see below.
In strong induction, the base case is still the same. However, the assumption is everything below. In other words, strong induction is
S(0) implies S(1)
S(0) AND S(1) implies S(2)
S(0), S(1), AND S(2) implies S(3),
and so on.
Compare this to usual induction, which is:
S(0) implies S(1)
S(1) implies S(2)
S(2) implies S(3)
For an example see our solution to Q2(c).