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More math illiteracy August 20, 2009

Posted by lumixedia in General, math education, number theory.
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Just for fun, here’s a rather pointless anecdote.

My third-grade teacher decided to have a fun, hands-on activity to teach our class about primes. Now I have a low opinion of all fun, hands-on activities (give me a good, proper whiteboard lecture any day, and if you’re incapable of doing so you should really just work on improving your teaching skills before making me pay attention to you, and yes, this was my opinion even when I was very, very young) but that’s not the point of this post. (more…)

Another math illiteracy moment August 15, 2009

Posted by lumixedia in General, history of mathematics, math education, number theory.
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I was recently informed that the Goldbach conjecture is popularly known in China as the “1+1=2” conjecture. As in, “every positive even number can be written as the sum of two primes. For example, 1+1=2.” [Edit–I was told this by a Chinese person who might nevertheless not be representative of how this nickname is understood–see comments.]

When I mentioned that this nickname is not in fact accurate, the person who so informed me got rather annoyed with my pointless pedantry. Why shouldn’t 1 be prime? Why not define a “prime” to be a positive integer with at most two distinct divisors, rather than a positive integer with exactly two distinct divisors? Clearly the “1+1=2” conjecture sounds way cooler than the “2+2=4” conjecture to a layman, and we are talking about popular mathematics here, so why not?

Okay, I guess it might not be immediately obvious why current notation is preferable. Maybe. From a certain perspective. It is also admittedly true, according to Wikipedia, that 1 was indeed widely considered to be prime by mathematicians up to a few hundred years ago. Fine. So let’s temporarily redefine “prime” to mean a positive integer with at most two distinct divisors, and see if it’s acceptable today. (more…)

USAMO 1973 #2 August 11, 2009

Posted by lumixedia in Problem-solving.
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USAMO 1973 #2. Let {\{X_n\}} and {\{Y_n\}} denote two sequences of integers defined as follows:

\displaystyle X_0=1,\hspace{0.1cm}X_1=1,\hspace{0.1cm}X_{n+1}=X_n+2X_{n-1}\hspace{0.1cm}(n=1,2,3,...)

\displaystyle Y_0=1,\hspace{0.1cm}Y_1=7,\hspace{0.1cm}Y_{n+1}=2Y_n+3Y_{n-1}\hspace{0.1cm}(n=1,2,3,...)

Thus, the first few terms of the sequence are:

\displaystyle X:\hspace{0.1cm}1,1,3,5,11,21,...

\displaystyle Y:\hspace{0.1cm}1,7,17,55,161,487,...

Prove that, except for “1”, there is no term which occurs in both sequences. (more…)

IMO 2009 #1 July 18, 2009

Posted by Martin Camacho in Problem-solving, Uncategorized.
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The 2009 IMO was a few days ago – in this post I tackle what I think is one of the easier IMO problems, IMO 2009 #1.

The question is as follows:

Let {n} be a positive integer and let {a_1,a_2,a_3,\cdots,a_k} ({k\ge 2}) be distinct integers in the set {1,2,\cdots,n} such that {n} divides {a_i(a_{i+1}-1)} for {i=1,2,\cdots,k-1}. Prove that {n} does not divide {a_k(a_1-1)}.


USAMO 1972 #1 July 18, 2009

Posted by lumixedia in Problem-solving.
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My first post was going to be an introduction to combinatorial game theory, but putting that together would have been rather more complicated than grabbing some USAMO problem and putting up my solution, so of course I chose the path of less resistance. The intro to game theory will come eventually, but in the meantime, here’s the first USAMO problem ever:

USAMO 1972 # 1. The symbols {(a,b,...,g)} and {[a,b,...,g]} denote the greatest common divisor and the least common multiple, respectively, of the positive integers {a,b,...,g}. For example, {(3,6,18)=3} and {[6,15]=30}. Prove that

\displaystyle \frac{[a,b,c]^2}{[a,b][b,c][c,a]}=\frac{(a,b,c)^2}{(a,b)(b,c)(c,a)}.

Here is, based on my first instinct when seeing this problem… (more…)