TT's Maths Thread

[TrueTears]

sketch:

we need to know what the values of f(n) are first of all.

iff:





How many values of satisfy this will depend on , notice that the difference is an indication of how many values of solve this. In fact it is the number of values of n that solve this, since the difference is itself an integer(since it's a difference of two numbers with the same decimal part). After doing the messy algebra, you get an expression for the number of times appears, now you can figure out which numbers and how many of them you are exactly summing, and then see if you can try to find the sum.

Ah okay I get how you came up with the inequality, but how did you get the difference?

[QuantumJG]

This stuff is much more advanced than what you will see in first year uni maths. I think you may see it if you do accelerated maths. My friends who did accelerated maths in first year were trying to solve a problem similar to one of the problems you posted.

I think I may understand what this stuff is by the end of next year (during hopefully). MUMS always post problems like that in the maths building. If you are this interested in maths you may want to pursue something like a PhB in maths at ANU or some course with a lot of maths. With commerce the only area of study with a lot of maths is actuarial studies.

The only way you will see this maths is with a pure maths degree or if you decide to undertake studies to become a theoretical physicist. Yep you are not only a lover of maths, but you love pure maths.

[kamil9876]

It's often best to picture numbers on a number line. So we want to know how many numbers are in between a and b:

a----|--(a+1)----|--(a+2)----|--(a+3)----|--(a+4)----|--(a+5)....

we see that there are k integers in between a and a+k. Hence letting b=a+k, there are b-a integers in between b and a.

[Over9000]

This stuff is much more advanced than what you will see in first year uni maths. I think you may see it if you do accelerated maths. My friends who did accelerated maths in first year were trying to solve a problem similar to one of the problems you posted.

I think I may understand what this stuff is by the end of next year (during hopefully). MUMS always post problems like that in the maths building. If you are this interested in maths you may want to pursue something like a PhB in maths at ANU or some course with a lot of maths. With commerce the only area of study with a lot of maths is actuarial studies.

The only way you will see this maths is with a pure maths degree or if you decide to undertake studies to become a theoretical physicist. Yep you are not only a lover of maths, but you love pure maths.

Some of it is olympiad level stuff

[TrueTears]

This stuff is much more advanced than what you will see in first year uni maths. I think you may see it if you do accelerated maths. My friends who did accelerated maths in first year were trying to solve a problem similar to one of the problems you posted.

I think I may understand what this stuff is by the end of next year (during hopefully). MUMS always post problems like that in the maths building. If you are this interested in maths you may want to pursue something like a PhB in maths at ANU or some course with a lot of maths. With commerce the only area of study with a lot of maths is actuarial studies.

The only way you will see this maths is with a pure maths degree or if you decide to undertake studies to become a theoretical physicist. Yep you are not only a lover of maths, but you love pure maths.

Yeah this stuff is pretty cool, I'm not finding it too hard, it's actually really enjoyable =)

It's often best to picture numbers on a number line. So we want to know how many numbers are in between a and b:

a----|--(a+1)----|--(a+2)----|--(a+3)----|--(a+4)----|--(a+5)....

we see that there are k integers in between a and a+k. Hence letting b=a+k, there are b-a integers in between b and a.

Thanks I think I got the idea now, gonna try to solve now.

[kamil9876]

Yeah it is, I prefer those more olympiad problems that use tricks, this stuff contains a lot of algebra and infinite series so it sort of is closer to uni maths, but the previous problems we did like the combinatorial ones or number theoretical ones like

1. How many of the first 1000 positive integers can be expressed in the form:
?

are more the olympiad stuff I like and you won't find it in first year uni.

[TrueTears]

Yeah it is, I prefer those more olympiad problems that use tricks, this stuff contains a lot of algebra and infinite series so it sort of is closer to uni maths, but the previous problems we did like the combinatorial ones or number theoretical ones like

1. How many of the first 1000 positive integers can be expressed in the form:
?

are more the olympiad stuff I like and you won't find it in first year uni.

Yeah these are the problems that I find really fun.

[dcc]

In Accelerated Maths (1&2) @ uom, we studied linear algebra (vector spaces / inner product spaces) / real analysis / calculus, so there isn't much relation between the types of questions posted here and the stuff you will study at uni. 

Uni mathematics is set up so that you are able to tackle other subjects later on (i.e. complex analysis follows from real analysis, group theory follows from linear algebra and so on), whereas Olympiad maths doesn't really 'lead' anywhere (in terms of learning more about different topics in the future).

At least, that's my take on it.

[enwiabe]

I think Olympiad Maths teaches the much more important skills of critical thinking and problem solving.

[kamil9876]

True, I think it is best to be selective about which problems to care about since some may be more "natural" and some more "artificial"(contrived problems requiring to find some needle that has been cleverly placed in a haystack, ie not something you would yourself ask/invent before knowing the answer). e.g the latest problem is quite shit and not natural in my opinion so I wouldn't bother with it. But other problems can be quite natural and complement more serious stuff, they're alright at stimulating some creative thought every now and then. I don't neccesarily get into the whole competitive nature of IMO, but just pick and choose whatever I think is natural to think about, stress free, and try out problems not neccearily ones you would find on an IMO but still ones that provide good ideas that may build generic skills.

[TrueTears]

Let be the integer closest to . Find .

Okay I think I finally got it, god this method is pretty primitive but it is the only one I can think of. There must be some other method.

So first of all we know that for some positive integer m.

Thus the number of n that satisfies this inequality is determined by:

Now let's list all the possibilities we have:





 







 











 



Therefore we require






So yeah, what if the question was like

I don't wanna list forever Need a more general method!

[dcc]

1. How many of the first 1000 positive integers can be expressed in the form:
?

601 ?

[TrueTears]

1. How many of the first 1000 positive integers can be expressed in the form:
?

601 ?

Hmm I got 600 a few days ago: http://vcenotes.com/forum/index.php/topic,19896.msg201890.html#msg201890

Maybe you included 0

[kamil9876]

expand that difference and you get:



so this is the number of times appears in our sum. Therefore it contributes to the sum.

So now we have to add over m=1 to to m=6. We add to m=6 since 7^4>1995. Then we add on the remaining numbers like you did for f(n=7).

[dcc]

Ah yes 'positive integers'.  600 it is.