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DeanK22
question

answer :p:
The Muon
answer :p:


Prove that 2+3 \sqrt{2} + \sqrt{3} is algebraic

Spoiler

A quaint paper 2 question;

4) Prove there are infinitely many integers [distinct] that are positive such that x2+y3 x^2 + y^3 is divisible by x3+y2x^3 + y^2
correct
I do
JohnnySPal
I know I'm being hick here, but why doesn't the solution set (x,x):x∈N{(x,x) : x \in \mathbb{N}}


Oh. Yeah I made that mistake at the first time I read it.

(x,y) are such that x≠y x \ne y - distinct.
It would considereably give the game away as the hint is the solution.

Spoiler

Solution;

Spoiler

Reply 1268
Avatar for Kevin_B
Kevin_B
1
Unparseable latex formula:

\displaystyle \frac{1}{1} + \frac{1}{3} = \frac{4}{3} , \hspace5 4^2 + 3^2 = 5^2 \\ \frac{1}{3} + \frac{1}{5} = \frac{8}{15}, \hspace5 8^2 + 15^2 = 17^2 \\ \frac{1}{5} + \frac{1}{7} = \frac{12}{35}, \hspace5 12^2 + 35^2 = 37^2



State and prove a generalization of this
Kevin_B
Unparseable latex formula:

\displaystyle \frac{1}{1} + \frac{1}{3} = \frac{4}{3} , \hspace5 4^2 + 3^2 = 5^2 \\ \frac{1}{3} + \frac{1}{5} = \frac{8}{15}, \hspace5 8^2 + 15^2 = 17^2 \\ \frac{1}{5} + \frac{1}{7} = \frac{12}{35}, 12^2 + 35^2 = 37^2



State and prove a generalization of this


erm. Not quite getting what you want us to do. Elaboration
Reply 1270
Avatar for Kevin_B
Kevin_B
1
Notice the pattern, find a generalization for it and prove it
Reply 1271
Avatar for SimonM
SimonM
18

Spoiler

Spoiler



I'm not sure this is what you meant?

edit: Sorry didn't see Simon's post:o:
Reply 1273
Avatar for Swayum
Swayum
18
Something I just discovered while playing on my calculator (as you do):

tan(99) = tan(999) = tan(9999) = tan(99999) = tan(999999) (in degrees)

Prove/disprove that it holds in general (i.e. beyond 999999). Sounds easy to be honest.
Swayum
Something I just discovered while playing on my calculator (as you do):

tan(999) = tan(9999) = tan(99999) = tan(999999) (in degrees)

Prove/disprove that it holds in general (i.e. beyond 999999). Sounds easy to be honest.


All of these numbers = tan 999

(All of them can be written as {a multiple of 180} + 999)
Reply 1275
Avatar for Swayum
Swayum
18
thomaskurian89
All of these numbers = tan 9

(All of them can be written as {a multiple of 90} + 9)


Nope.

*Edit*

And in response to your edit, they're not -tan9 either.
Swayum
Nope.

*Edit*

And in response to your edit, they're not -tan9 either.


Re-edited. Sorry.
Reply 1277
Avatar for SimonM
SimonM
18
Swayum
Nope.

*Edit*

And in response to your edit, they're not -tan9 either.


I think he's got the right idea?

tan⁡(9∘+90∘(2k+1))=1tan⁡9∘\tan ( 9^{\circ} + 90^{\circ} (2k+1)) = \frac{1}{\tan 9^{\circ}}
Reply 1278
Avatar for Swayum
Swayum
18
thomaskurian89
Re-edited. Sorry.


Yup, that works :smile: (although I've edited the question to include tan99, but obviously you just add 99 instead of 999).

SimonM
I think he's got the right idea?

tan⁡(9∘+90∘(2k+1))=1tan⁡9∘\tan ( 9^{\circ} + 90^{\circ} (2k+1)) = \frac{1}{\tan 9^{\circ}}


Ah, I hadn't noticed that.
Reply 1279
Avatar for Swayum
Swayum
18
Yup :frown:. It's a known bug though so they ought to fix it soon (but they may not make it high priority since it affects just 1 forum mainly).

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