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Official TSR Mathematical Society

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JohnnySPal

Not that I want to butt in on the problem Dean set, but I remember finding this puzzle really rather interesting:

http://xkcd.com/blue_eyes.html

The answer given isn't particularly short and there isn't a simple trick involved to get to the solution, but the solution is none-the-less rather elegant.

Is this it:

Spoiler

Reply 1201

Zhen Lin

Here's another examples sheet problem:

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Reply 1202

SimonM

DeanK2

Given;

x^2 + y^2 = 6x + 8y

What is the greatest possible value of

x + 7y ?

Consider tangents to the circle

Reply 1203

Aurel-Aqua

Swayum

Hmm, I haven't posed any problems since the society was reformed. Most of the ones so far have been Olympiad style, so I'll post a few A-levely ones:

1) Find the equation of the tangent/s from the origin to the circle x^2 + y^2 - 10x - 6y + 25 = 0

2) Find

\int \frac{1}{4\cos^2(x) - 9\sin^2(x)}\ \mathrm{d}x

3) Given that

(1 + x)^n = 1 + c_1x + c_2x^2 + c_3x^3 + ... + c_rx^r + ...

c_{s-1}

c_s

and

c_{s+1}

are in arithmetic progression, find the possible values of s when n = 62.

That last one is quite good I think - the hardest AP question I've done.

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Please check, I'm interested. Seems I got most of it wrong :sigh:

Reply 1204

Glutamic Acid

I get

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Reply 1205

Aurel-Aqua

Glutamic Acid

I get

Spoiler

Quote me next time.

I see where I went wrong, that's so obvious :sigh:

Reply 1206

Aurel-Aqua

Glutamic Acid

I get

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Reply 1207

Dadeyemi

Swayum

2) Find

\int \frac{1}{4\cos^2(x) - 9\sin^2(x)}\ \mathrm{d}x

Spoiler

Reply 1208

Glutamic Acid

Aurel-Aqua

Spoiler

My method was to

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Reply 1209

DeanK2

Given;

x^2 + y^2 = 6x + 8y

What is the greatest possible value of

x + 7y ?

Attempted this yet anybody?

Hint

Spoiler

Reply 1210

SimonM

DeanK2

Attempted this yet anybody?

I already gave a similar hint.

Reply 1211

DeanK2

SimonM

I already gave a similar hint.

It didn't come up at the quote tool because you were not within the five most recent people - did not see it - soz.

Reply 1212

The Muon

DeanK2

Attempted this yet anybody?

Hint

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I'm not sure if my answer is correct because my final answer isnt a nice number:

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Reply 1213

DeanK2

The Muon

I'm not sure if my answer is correct because my final answer isnt a nice number:

Spoiler

How did wyou go about the problem

My anwer;

31 + 25\sqrt2

Approach;

[spoiler] A circle which has centre 3,4 and radius 5. Also you need to find the point where the line 7y+x=c touches the circle as this is the maximum [or minimum] value for this. The gradient of the line

y = \frac{c-x}{7}

Unparseable latex formula:

\frac{-1}{7} [\latex]. The normal of this tangent will then have the negative reciprocal of this value. The gradient of the normal is [latex] \frac{dy}{dx} [/latex] which is [latex] \frac{7a}{a} [/latex]. This means the point at which the line [latex] y = \frac{c-x}{7} [/latex] meets the circle has coordinate ([3+a] , [4+7a]). Substituing into [br][br][latex] (x-3)^2 + (y-4)^2 = 25 [/latex] gives a = [late] \frac{\sqrt2}{2} [/late]

Reply 1214

Dadeyemi

DeanK2

Given;

x^2 + y^2 = 6x + 8y

What is the greatest possible value of

x + 7y ?

Do it with Lagrange multipliers just for lulz :p:

Reply 1215

Dadeyemi

[QUOTE="DeanK2"]How did wyou go about the problem

My anwer;

31 + 25\sqrt2

Approach;

Spoiler

same answer as me

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Reply 1216

The Muon

[QUOTE="DeanK2"]How did wyou go about the problem

My anwer;

31 + 25\sqrt2

Approach;

Spoiler

Reply 1217

NCIShippo

can someone help with this question please:

Based on Factor theroy

Find the factors for the equation: 3x^3 - 2X^2 + 7x - 15

is it meant to have a rem?

Reply 1218

Swayum

Glutamic Acid

I get

Spoiler

Yup.

Dadeyemi

Spoiler

You're forgetting the + c :p:

. But, more importantly, arctanh(tan) isn't very nice - I think a t-substitution is the way to go. Creative method though :yep:

Aurel-Aqua

Spoiler

Please check, I'm interested. Seems I got most of it wrong :sigh:

Acid already pointed out the mistake in the first one (although I used a more geometric argument than evil discriminant method). Third one has the right answer (can't be bothered to check the working). The second one I don't know about actually (haven't done it recently), but it seems plausible.