Creative Problem Solving

[Kilgamayan]

I'm taking this course right now and it seems like the kind of thing a lot of people here would enjoy. It's basically a course full of "make four liters using only a three-liter jug and a five-liter jug"-type problems, although those problems are, of course, the easiest to solve.

Try some:

- Let a, b and c be positive numbers. Prove that (a/b) + (b/c) + (c/a) >= 3.
- For which real numbers c is there a straight line that intersects the curve y = (x^4) + 9(x^3) + c(x^2) + 9x + 4 at four distinct points?
- How many arithmetic problems of the following form are possible? You must use each of the digits 1 through 9 in numerical order from left to right, and you can use any combination of the + and * symbols as you like provided your syntax is still correct. Proper examples include 123+45*6789, 1+2+3+4+5*6+7+89 and 123456789, while 12+3*+456++78*9 does not count.

Now obviously I'm not going to ask any questions where I don't already know the answer, so don't bother accusing me of not doing my own homework. >_>

[T0rajir0u]

Re: Creative Problem Solving

first one is AM-GM

second one "straight line" = ignore the linear and constant terms

(x - a)(x - b)(x - c)(x - d) = x^{4} - (a + b + c + d) x^{3} + (ab + ac + ad + bc + bd + cd) x^{2} + useless ****

basically a + b + c + d = -9 and we want to know the range of C = (ab + ac + ad + bc + bd + cd) given that a, b, c, d are distinct

well (a + b + c + d)^{2} = 81 = (a^2 + b^2 + c^2 + d^2) + 2C

basic rearrangement we have 3(a^2 + b^2 + c^2 + d^2) \ge 2C

so 81 \ge (8/3) C

or C \le 243/8

the equality case is a = b = c = d so we ignore it, giving C < 243/8

set a = 1, b = 0 and we have

C = ab + ac + ad + bc + bd + cd = (c + d) + cd

where c + d = -10

obviously we just let c be a huge negative number and d be a huge positive number and we don't have a lower bound

so C < 243/8 unless i'm totally ****ed up right now

third one edit: oh we fix the order okay so there are eight slots between them that can be +, *, or nothing for 3^8 possibilities



if you want to learn how to do **** like this go to http://www.artofproblemsolving.com



ps this is the best idea for a course i've ever heard where the **** do you go

pps this should be a ****ing middle school or high school class not a college class

ppps if you're stuck on your homework ask questions in the forum in the link

pppps seriously kilga you would probably like this site it has the least retards out of basically anywhere on the internet

[Reach]

Re: Creative Problem Solving

Second one can't be right.

However, I just got home from work and there's no way I'm figuring out the correct answer right now XDXD



Here are two good math problems.

1)

Several straight line segments are drawn on a plane surface in such a way that their intersecting lines form 1,597 areas that are not further subdivided. What is the minimum number of line segments that must be drawn to form the described pattern?

2) 1 + 10^1,234,567,890 triangles are drawn on a plane surface. What is the maximum number of areas, not further subdivided, that can be formed as these triangles intersect each other?

[T0rajir0u]

Re: Creative Problem Solving

first one number of areas in which m lines divide the plane is { m \choose 2 } + { m \choose 1} + { m \choose 0 } = m(m+1)/2 + 1

gay calculation gives m = 56 exactly o tricky

second one too lazy to do whatever



ps hey guys itt we post problems

- find all positive integers n such that 5^5 - 5^4 + 5^n is square
- find the maximum and minimum value of 3 sin x + 4 cos x
- find the largest even number that can't be expressed as the sum of two positive odd composite numbers
- find all positive integers n such that 2^n - 7 is a square

[Reach]

Re: Creative Problem Solving

Interesting method XD

I just derived .5x^2+.5x+1 = 1597 from a sequence of the first 4 lines = 56 lines, correct.

Second one is tricky

[Kilgamayan]

Re: Creative Problem Solving

number 2 is rite but there's a much easier way to do it

number 3 is rite obviously

[Patashu]

Re: Creative Problem Solving

sup riddles site

http://www.ocf.berkeley.edu/~wwu/riddles/intro.shtml

'putnam' is the pure maths section, 'hard' would probably also be worth checking out.

[T0rajir0u]

Re: Creative Problem Solving

second problem that kilga posted would probably have been a lot easier with calculus is that what you were thinking of

hey putnam i'll be doing that in college lol

[Kilgamayan]

Re: Creative Problem Solving

yeah basically

take the double derivative, set it equal to zero, find all c that makes the determinant greater than zero

[T0rajir0u]

Re: Creative Problem Solving

yeah thats what i thought but screw calculus basically none of the competitions i do allow it lol

[Reach]

Re: Creative Problem Solving

o I read the question incorrectly.


See, I forget almost everything I've learned in calculus like 2 months of being outside the class and not using it. XD

[Moogy]

Re: Creative Problem Solving

mathmania

[T0rajir0u]

Re: Creative Problem Solving

AAAAing is hard

[cetaka]

Re: Creative Problem Solving

yeah I want a class like this too wut da fux =(

ps how many sets S=({A},{B},{C}) can you have such that A U B U C = {1,2,3,4,5,6,7,8,9,10} and A, B, and C are disjoint

pps artofproblemsolving.com more like if your post doesn't solve a millenium problem you get reported for spamming

[Kilgamayan]

Re: Creative Problem Solving

Here's a couple more we went over today. Two of them I figured out last week but the third got me.

1. Prove that if p is prime and (p^2)+2 is prime, then (p^3)+2 is prime and (p^4)+2 is prime.

2. If you divide a prime number by 30, is the remainder necessarily prime?

3. Without doing calculations, determine (and prove) which of 31^11 and 17^14 is greater.