NS question

Eighty problems in ten minutes. What's so hard about that?

NS question

Postby bradp » Sat Nov 11, 2006 10:21 am

What's the best way to find the product of the coefficients of (5a+7b)^2 ?`

take (ax+by) where a & b are coefficients

the product of all teh coefficients is then

a^2 * b^2 * 2 * a * b = 2*a^3*b^3
that seems too difficult though
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Postby hhhsmath06 » Sat Nov 11, 2006 6:24 pm

so there are 2 ways to look at it
1. a^2 * b^2 * 2 * a * b
2. 2*a^3*b^3
depends on the situation, use either 1 or 2...well, it's essentially the same...hm...i'd just look at it this way
a=5, b=7
2*5^3*7^3=10*25*343
write down the 0, so it's just 25*343= 8575
the answer is 85750...

q
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Postby AuSmith » Sat Nov 11, 2006 11:33 pm

If you want a general formula for the product of the coefficients of (ax+by)^n, take the product of the nth row in Pascal's triangle

1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
...

and multiply that by (ab)^T(n) where T(n) is the nth triangular number. Any other formulas or tricks are too complicated. Remember 2(ab)^3, like you found, and that it always involves (ab)^(1+2+...+n). The reason (5x+7y)^2 is hard is that not many people know 35^3 off the top of their heads. You should know 35^2 well, though. 2*35^3=2*35*1225=70*1225, which you should be able to do.

Don't expect n to get very big, since that would result in outrageous sizes. I think he asks for the product just because we're used to doing the sum of the coefficients. I know I often take the sum when it says product because I've grown accustomed to summing the coefficients. And, reading takes too much time for numbersense.
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Postby hhhsmath06 » Sun Nov 12, 2006 1:26 am

"reading takes too much time for numbersense"
true, true, true.
aaron's approach is good, even tho i'll never work it out that way. 1000 ways to approach a problem, just pick the one that works the best for you...
btw, LW started mixing up sum/products these yrs...on a UIL regional or state test expect something like products instead of sum...of course, always "try" to read the problem, even tho 80% of the time u can just get by w/out reading it...
aaron, do u still take ns tests? r u a junior @ missouri now? i heard u've been taken a lot of hrs...

man...i feel old...freshman @ TAMU returning to this forum...
nostalgic...those 6 am wake ups and 8 am ns tests...

good stuff, good stuff.

q
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Postby bradp » Sun Nov 12, 2006 8:15 am

When I'm trying to go fast I always just glance at the text and look at the numbers... sometimes that gets me with stuff like this, probability vs. odds, positive integral divisors vs. sum of positive integral divisors, etc.
On tests this year, I'm going to slow myself down and avoid dumb mistakes. I have the speed to finish but I make a lot of mistakes sometimes.

Speaking of cubes, how high did you two memorize your cubes? I've seen 16^3 and 15^3 on a test but never 18^3 or 17^3
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Postby hhhsmath06 » Sun Nov 12, 2006 12:29 pm

memorize the cubes up to 25...i've seen 22^3

q
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Postby AuSmith » Sun Nov 12, 2006 7:08 pm

I haven't taken a NS test since state two years ago. Unfortunately, I stopped enjoying all of it around 10th grade. So, I'll mention it to people to defend having memorized 7^5. 6^5, 2^25 (33554432=Dr Morgan's favorite power of 2), etc. Oh, and I find it funny that 13^3=2197 and 3^7=2187... anyways, Bryant Heath (two years my senior) and I have become better friends and were going to compete against eachother after years of not practicing. It still hasn't happened... but I think we will eventually.

I memorized cubes up to 25 and they promptly bacame vague past 16^3 (after rarely needing them). 17^3 stuck with me for some reason... and 20^3. Then, I'd develop familiar paths for calculating the missing cubes. Since n^2=1,-1(mod 5), it's convenient to round the square to the nearest 5, then multiply... like 18^3=18*(325-1)=650*9-18.

And, I'm a senior now at U of Missouri-Rolla. It was possible for me to graduate next semester (2 years), but I've been wondering how ambitious that was. I got in a bad car wreck 6 weeks ago, was out of school for 3 weeks, and my parents got me to drop 6 hours. So, I'll graduate in 2.5 years, looks like. Not bad. I can now take the Putnam 3 times! lol. My RA just told me last night that a bunch of physics majors come in with over 100 credit hours, but it's from dual credit at an academy... kinda like TAMS, I presume.
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Postby Quelloquialism » Sun Nov 12, 2006 9:06 pm

hhhsmath06 wrote:memorize the cubes up to 25...i've seen 22^3

q

You wouldn't necessarily have to treat that particular one as memorization, though. If I encountered it, I would do 22^3 = (2*11)^3 = 8*1331 = 10648. I try to avoid memorizing things...my memory's too fallible.
Last edited by Quelloquialism on Sun Dec 03, 2006 2:48 pm, edited 1 time in total.
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Postby hhhsmath06 » Sun Nov 12, 2006 9:31 pm

"my memory's too fallible"
lol...
well...every single ns contestant has to find a balance between memorization and calculation. i've seen greatness from both ends of the spectrum....i've seen guys like aaron, who doesn't really memorize that much and still outplays everyone, and guys who memorize as much as they can and getting an edge in contests...

i am more a memorization type of guy...cuz when i get nervous my head doesn't function at it's peak...so i tend to make a lot of mistakes calculating big numbers in my head...

afterall...it's all about what is the best for a specific individual.

to aaron: yeah, graduating in 2.5 yrs is pretty amazing, i am guessing u r also sitting on a 4.0 too? wut r u gonna do after graduation tho? grad?

q
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Postby AuSmith » Mon Nov 13, 2006 1:43 pm

4.0? Ha... lol. Change a few things and yes I would have... I had a 4.0 my first semester. Then I took a "Writing and Research" English class which took all my time (10 hours/page not including research time) - I'm slow. Worked really hard and got a C. Though, if I would have taken the class with the prof I'm under next semester, it would have been an easy A.... oh well, that's life. And, I took some upper level physics where he'd go through a whole gob of boring algebra to explain wave functions - simple algebra turns me off, especially if your equations take 8 seconds apiece to write out. Ended up doing perfectly in everything but wave functions b/c those weren't in the book. Finally, I added a "Discrete Math for Comp Sci's" after about 7 weeks into the semester. By that time, I had missed half the homework, but he agreed to take that into account by the end of the year. I asked him 3 times in 3 different ways to change my homework grade around finals and he didn't. So, I got 0's on half the homework. And, discrete math is my forte... this is a watered down comp sci version. So, all that junk gave me a 3.05 GPA for that semester and a 3.48 cumulative. I think there's a good chance I'll make all A's except for last semester, though.

Yea, I'll go to grad school someplace. There's not much I would want to do with only a Bachelor's... and grad school's where the real math is anyway, right? Sounds like great fun. I'd like to be a TA.
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Postby hhhsmath06 » Mon Nov 13, 2006 2:09 pm

3.48 is not a bad gpa, plus i don't think u r the type of guy who is # over A's and B's...to me u seem to be more interested in math itself than the letter grades anyways...

were u required to take that english class or something? taking english class is like a waste of talent for u.

Ever thought about going back to azle and be a mathmatics coach? LOL

q
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Postby AuSmith » Mon Nov 13, 2006 7:59 pm

Yea, I was required to take that English class, and I'm about to take British Literature, which sounds much more enjoyable than writing about nothing. Seriously, what's with some of these essay prompts like, "Should you care about what your friends think about you? Why or why not?" Frankly, there's not much to say on that one - I don't worry about it. Whatever.

I've thought about coming back to Azle to coach a team, or some other high school, given that I could get kids involved in AMC, USAMTS, and just math for math's sake ... but as far as UIL stuff goes, I would only know a few coaches, still have a hard time motivating kids to work, and not even get into the competitions. High school is really no fun to teach - not seeing it.
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Postby fht » Thu Nov 23, 2006 3:43 pm

[quote="AuSmith"]I haven't taken a NS test since state two years ago. Unfortunately, I stopped enjoying all of it around 10th grade. So, I'll mention it to people to defend having memorized 7^5. 6^5, 2^25 (33554432=Dr Morgan's favorite power of 2), etc. Oh, and I find it funny that 13^3=2197 and 3^7=2187...

Oh my goodness, what a coincidence
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Re: NS question

Postby Liking_math » Sat Mar 29, 2008 9:47 pm

Yeah, Sam, 1st in state Number Sense? nice...
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Re: NS question

Postby ZachB » Sun Mar 30, 2008 4:38 pm

What did you make on the Putnam this year, AuSmith?
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Re: NS question

Postby AuSmith » Sun Mar 30, 2008 7:02 pm

ZachB wrote:What did you make on the Putnam this year, AuSmith?


I don't know. I guess my coach got my score back, but I don't think it was good. I was much too ambitious. Whatever. I don't really care that much if they grade them so harshly. Every year I'm sure, a bunch of kids deserve good scores and get next to nothing.
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Re: NS question

Postby ZachB » Sun Mar 30, 2008 10:46 pm

AuSmith wrote:
ZachB wrote:What did you make on the Putnam this year, AuSmith?


I don't know. I guess my coach got my score back, but I don't think it was good. I was much too ambitious. Whatever. I don't really care that much if they grade them so harshly. Every year I'm sure, a bunch of kids deserve good scores and get next to nothing.


Thats definitely true. I thought I would have gotten around a 15, but all I managed to scrape out was a 6. Of course, I'm a freshman so hopefully that will get better next year. As a team, though, we did really well. One of our guys got top 50 individual and as a team I think we were 40th or something like that, so, by just getting a decent score, I guess it was a moral victory because it increased our team score dramatically. How many hours are you taking per semester to graduate so soon?
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Re: NS question

Postby AuSmith » Sun Mar 30, 2008 11:31 pm

ZachB wrote:
AuSmith wrote:
ZachB wrote:What did you make on the Putnam this year, AuSmith?


I don't know. I guess my coach got my score back, but I don't think it was good. I was much too ambitious. Whatever. I don't really care that much if they grade them so harshly. Every year I'm sure, a bunch of kids deserve good scores and get next to nothing.


Thats definitely true. I thought I would have gotten around a 15, but all I managed to scrape out was a 6. Of course, I'm a freshman so hopefully that will get better next year. As a team, though, we did really well. One of our guys got top 50 individual and as a team I think we were 40th or something like that, so, by just getting a decent score, I guess it was a moral victory because it increased our team score dramatically. How many hours are you taking per semester to graduate so soon?


That's pretty cool. Top 50 is awesome and positive scores aren't bad for freshmen. UMR had a fellow one year named Xi Chen, which sounds very crazy to me. Fellows just don't come from Rolla. I doubt we've had anyone place in the top 300 other than him. I even found him in a trivia question in the Kedlaya Putnam 90's book for being one of two fellows from the 90's to come from a public state school (the other was from Berkeley). Well, I find enough satisfaction in just solving the problems (which is a whole lot more than what I get out of solving these dumb graph theory problems!).

I'm at Texas A&M now in grad school, so I'm taking 9 credit hours because I'd have to pay for anything extra. I perhaps still pathetically wear my math team letter jacket, and every once in a while meet someone that actually knows what each of the competitions are! One of my friends told me he was talking about me in the lounge and was asked "who is this?" "Aaron Goldsmith" "Oh, I know him, but not from here. He was famous in high school." Ha, it's just a funky feeling.

Well ZachB, keep working Putnam problems if that's what you like. I don't think anyone comes in solving Putnam problems without some serious exposure to previous exams. It takes a whole gob of work to be able to write those proofs so quickly - more than I invested.
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