Q16 style problems thread (1 Viewer)

[vuhung]

Great.

Doing math on Word is a pain. That’s why.

You can upload your solution anywhere, Google Drive is a good choice.

This is a famous result and what I did was merely rewording it in HSC style.

 

[Burnt_Out]

Alr (sorry if my solution is unreadable. i was trying to put my thoughts on the word doc so its kind of like my working out page at the same time)

PS: looking at Apery's theorem, it looks interesting thats for sure.

 

[vuhung]

Hi Burnt Out,

ru in y 10 or 11?

 

[Burnt_Out]

11

 

[vuhung]

You should try IMO level math?
Probably ext 2 is too easy for you, it seems.

There are some hard problems in this sub-forum. I’d love to see your solutions and feedback.

 

[WeiWeiMan]

You should try IMO level math?
Probably ext 2 is too easy for you, it seems.

There are some hard problems in this sub-forum. I’d love to see your solutions and feedback.

IMO is a big step up from 4u

 

[vuhung]

Alr (sorry if my solution is unreadable. i was trying to put my thoughts on the word doc so its kind of like my working out page at the same time)

PS: looking at Apery's theorem, it looks interesting thats for sure.

Great work!

Total mark: 14/15

Some constructive feedback:

You should have gotten full marks if double-checked the minor issues pointed out below.

Part (a):
- Maintain the absolute value bars throughout the inequalities unless you have established the sign of the difference.
- Note that the inequality still holds for any integer p, q not just coprime ones.

Part (b):
- The solution is unnecessarily complicated, making it harder to follow.
- Should consider the boundary case p = 0.
- Should not use "p" as it is already defined as an integer in part (a).
- The max occurs in the interior (0, 1), not boundaries. (Why?)

Part (c):
- You got full marks here.
- C can be any constant and you can explicitly state C in term of (what?) integral.
- Considering the second derivative would have been a good idea to confirm the max (or min)

Part (d):
- Full marks here.
- Strong reasoning and logic.

 

[Burnt_Out]

IMO is a big step up from 4u

yeah. idk i wanna do IMO

 

[Burnt_Out]

This one was fun as I saw it as a vindication of my Task 1 results which i kinda bombed. This was a lot more structured though

(I attached the similar question from my school that i didn't get right in the same file)

 

[vuhung]

Great.

Solution for ii) should not take that long.
Apply I) to ii), as hinted.

iv) not an aha moment. Other (more direct) solutions available.

 

[vuhung]

This problem (Wallis Integrals) is a fantastic revision tool because it hits so many syllabus dot points at once:

- Complex Integration & Substitutions
- Lots of IBP, but not heavy lifting
- Inequalities & The Squeeze Theorem
- Reading the room (following "Do NOT prove" instructions)
- Deriving the Normal Distribution constant (Gaussian Integral)

This is unlikely to be a Q16 "Final Boss" anymore. The Wallis integrals are too famous. It’s now a standard that every top student needs to have in their toolkit.

The point is not just to be able to solve the problem, but aim for full marks.

 

[vuhung]

Came across this problem that ties together the Inequalities and Proofs topics really well. It’s a solid practice question for anyone looking to polish their calculus/proofs before trials.

It guides you through the whole derivation chain:

1. Part (i) & (ii): Using calculus (minimisation) to derive Young's Inequality.
2. Part (iii): The classic induction proof for AM-GM.
3. Part (iv): A satisfying application involving a cyclic sum.

Good luck! Let me know if you get stuck on the algebra in part (i).

 

[vuhung]

Here’s why this problem is a total legend and why you should give it a crack.

1. It’s Ancient Tech (4,000 Years Old!)

The question calls it "Newton’s Method," but the logic actually dates back to Babylonia, circa 1700 BC. Before calculators existed, ancient surveyors used this exact method to calculate square roots for building diagonal walls. It’s arguably the world’s first computer algorithm.

2. It’s the "Source Code" of Engineering

Planning on doing Computer Science or Engineering at uni? Pay attention.

Computers don’t magically know what sqrt{2} is. They can’t store infinite decimals. When you code Math.sqrt(2), the computer runs an iterative loop almost identical to this problem. It’s a sneak peek into Numerical Analysis, the maths that powers everything from game physics to rocket trajectories.

 

[f7eeting]

I wish I did extension 2 maths. Looks fun. Would have loved to do this problem if I could.

im sure theres still a way to solve it even without having ext 2 knowledge

 

[Burnt_Out]

I wish I did extension 2 maths. Looks fun. Would have loved to do this problem if I could.

well i'm doing ext 2 (and planning to do compsci) and it is kinda fun.

im sure theres still a way to solve it even without having ext 2 knowledge

I believe can be done by just doing extension 1 knowledge. I think recursive sequences is extension 2 knowledge but, its not really rocket science

 

[f7eeting]

well i'm doing ext 2 (and planning to do compsci) and it is kinda fun.


I believe can be done by just doing extension 1 knowledge. I think recursive sequences is extension 2 knowledge but, its not really rocket science

for some reason i can never see what you write when you send those documents

 

[Burnt_Out]

for some reason i can never see what you write when you send those documents

thats odd

 

[f7eeting]

I viewed it using Google Docs. Does it not work for you?

how do you do that

 

[coolcat6778]

what the fuck is "converge", this was never taught to us in the HSC syllabus

 

[vuhung]

what the fuck is "converge", this was never taught to us in the HSC syllabus

Thanks, spot on!

Btw, "Limiting sum" is so Aussie, should be replaced with an universal-understandable terminology with a broader context.