Q16 style problems thread (1 Viewer)

[vuhung]

I've composed a problem so student can practise with problem 16s.
Let me know what do you think.

 

[WeiWeiMan]

I've composed a problem so student can practise with problem 16s.
Let me know what do you think.

View attachment 51038

(i)

Spoiler: Solution

Let f(x) = ln(1+x) - x + x^2/2
f(0) =ln(1) - 0 + 0 =0
f’(x) = 1/(x+1) -1 + x
= x+1 + 1/(x+1) -2
> 2 -2 {by AM/GM for x>0, no equality since x =/= 0 => x+1 =/= 1/(x+1)
= 0
f’(x) > 0 for x > 0 and f(0) = 0
Thus, f(x) > 0 for x > 0
=> 0 < ln(1+x) -x + x^2/2
=> x - x^2/2 < ln(1+x) QED

(ii)

Spoiler: Solution

Let x = k/n^2 for k = 1,2,3,…, n

Via x-x^2/2 < ln(1+x) < x: [given]
1/n^2 - 1/2n^4 < ln(1+1/n^2) < 1/n^2
2/n^2 - 2^2/2n^4 < ln(1+2/n^2) < 2/n^2
…
n/n^2 - n^2/2n^2 < ln(1+n/n^2) < n/n^2

Adding these:
(1/n^2 + 2/n^2 + … + n/n^2) - (1/2n^4 + 2^2/2n^4 + … + n^2/2n^2) < ln[(1+1/n^2)(1+2/n^2)…(1+n/n^2)] < 1/n^2 + 2/n^2 + … + n/n^2

Factorising and simplifying a bit:

1/n^2 (1 + 2 + … + n) -1/2n^4 (1 + 2^2 + … + n^2) < ln(Pn) < 1/n^2 (1+2+… + n)

I am far too lazy to prove 1+2^2 + … + n^2 = n(n+1)(2n+1)/6 so I’ll just accept it as is

Using this alongside 1+2+…+n = n(n+1)/2 (Via AP):

n(n+1)/2n^2 - n(n+1)(2n+1)/12n^2 < ln(Pn) < n(n+1)/2n^2

Exponentiating all 3:
e^[n(n+1)/2n^2 - n(n+1)(2n+1)/12n^2] < Pn < e^[n(n+1)/2n^2]

I’ll do (iii) and (iv) later since I’m doing these on phone and it’s a genuine pain to type it out

My rough ideas are:

Spoiler: Idea for (iii)

Sandwich/Squeeze, looks like sqrt(e)

Spoiler: Idea for (iv)

Question literally tells you what to do

 

[vuhung]

Nice work Wei Wei.

You'll find some interesting findings working with part (iv), linking to part (i), Taylor expansion and the squeeze theorem.

 

[WeiWeiMan]

Nice work Wei Wei.

You'll find some interesting findings working with part (iv), linking to part (i), Taylor expansion and the squeeze theorem.

Yeah I’ll get it done later when I’m bothered

 

[justletmespeak123]

is this even in the syllbus

 

[WeiWeiMan]

is this even in the syllbus

Almost certainly
(i), (ii) and (iii) can be very reasonably asked in 4U, however, (ii) is a bit annoying to do without the realistic ability to quote without proof that (1+2^2+3^2+...+n^2) = n(n+1)(2n+1)/6.

(iv) is nice to think about, however, not super HSC style I'm pretty sure. I think it's still a reasonable question since it tells you how you should arrive at the proof though.

 

[vuhung]

I think it is an easy one.

 

[vuhung]

That sigma is in syllabus.

 

[vuhung]

Not a hard problem.

 

[ivanradoszyce]

Is this problem from a text book? If so, can you supply the title and author(s).

Thanks

 

[vuhung]

I am the author of the booklet “The Last Resorts” which the problem is taken from.
You can download the booklet from my profile page.

 

[ivanradoszyce]

Excellent material.

Thank-you so much for the time and effort in creating this resource.

 

[vuhung]

A nice problem links Mechanics, Integrals and Sequences.

 

[vuhung]

With the trials/HSC season approaching (in one year), I've composed a problem that I think is a perfect "final boss" challenge for those aiming for the top of the state. It’s a beautiful walkthrough of the proof that ζ(3) is irrational.

Why you should try this:

• Problem 16 Vibes: This is classic "Question 16" material - it's non-routine, deeply conceptual, and honestly a bit harder than the Q16s we’ve seen in the last few years.
• The "Top 0.001%" Test: If you can navigate the reasoning behind these four parts without getting lost in the algebra, you’re in a very strong position for a U4.
• Focus on Logic: The "heavy lifting" (the massive integral evaluations and LCM bounds) is already provided as hints. Your job is to connect the dots and provide the rigorous reasoning that bridges these complex results.

The Challenge:
The problem guides you through an irrationality criterion, a 3-variable optimization, and an integral construction that eventually forces a contradiction.

Good luck to everyone grinding through Extension 2! Post your solutions or where you’re getting stuck below. Let’s see who can crack the logic for the final contradiction first.

 

[Burnt_Out]

Hi, i'm doing b right now and all i'm feeling is just pain. Good question, just pain

 

[vuhung]

g is symmetric.

 

[Burnt_Out]

well, i already did b but thanks. (also i have no idea if my answer for a would be sufficient)

 

[vuhung]

Yes. Part a) is important.

but I mentioned, focus on the core reasoning.

I designed the problem so that it can be solved 20 30 mins or so, including everything, not to give you pain.

Q16s style, as I said.

 

[WeiWeiMan]

With the trials/HSC season approaching (in one year), I've composed a problem that I think is a perfect "final boss" challenge for those aiming for the top of the state. It’s a beautiful walkthrough of the proof that ζ(3) is irrational.

Why you should try this:

• Problem 16 Vibes: This is classic "Question 16" material - it's non-routine, deeply conceptual, and honestly a bit harder than the Q16s we’ve seen in the last few years.
• The "Top 0.001%" Test: If you can navigate the reasoning behind these four parts without getting lost in the algebra, you’re in a very strong position for a U4.
• Focus on Logic: The "heavy lifting" (the massive integral evaluations and LCM bounds) is already provided as hints. Your job is to connect the dots and provide the rigorous reasoning that bridges these complex results.

The Challenge:
The problem guides you through an irrationality criterion, a 3-variable optimization, and an integral construction that eventually forces a contradiction.

Good luck to everyone grinding through Extension 2! Post your solutions or where you’re getting stuck below. Let’s see who can crack the logic for the final contradiction first.


View attachment 51228

I shall do this when I am bothered

 

[Burnt_Out]

I solved it (at least I think i did)! My solutions are probably longer than it what would be needed but oh well (plus i'm doing this on word).

How did you come up with this problem anyways and should i send it (and i'm going to bed)?