The Student Room Group
Competition - post a photo you've taken of nature >>
Competition - post a photo you've taken of nature >>
Competition - post a photo you've taken of nature >>

The Proof is Trivial!

Scroll to see replies

Original post by Renzhi10122
Someone's been doing some BMO1 :smile:


haha you know the papers too well :smile: Thats where I first saw it, but for some reason I had a dream about doing this question the other night so thought I should share :tongue:. I have been doing some BMO recently though- I saw a comment (perhaps by you) about people who say olympiad stuff isnt proper maths, mainly because they cant do it themselves, and I dont want to be one of them!
Original post by EnglishMuon
haha you know the papers too well :smile: Thats where I first saw it, but for some reason I had a dream about doing this question the other night so thought I should share :tongue:. I have been doing some BMO recently though- I saw a comment (perhaps by you) about people who say olympiad stuff isnt proper maths, mainly because they cant do it themselves, and I dont want to be one of them!


Lol that was me.
BMO is wonderful I think great problem solving.



Posted from TSR Mobile
Original post by 16Characters....
Problem 581*

You select a real number entirely at random from the interval [0,2π][0, 2\pi]. What is the probability that sin⁡x+cos⁡x\sin x + \cos x is greater than 1 in magnitude?

Disclaimer: I made this up myself. I think my logic is correct, but my probability is **** so if this is an impossible question (or significantly harder than the * I gave it) then my apologies.


Am I being dozy or is this just a half?
Original post by physicsmaths
Lol that was me.
BMO is wonderful I think great problem solving.



Posted from TSR Mobile


I agree :smile: Although the final result for some questions I might not find amazing, the techniques involved are always quite nice. Did you notice a significant improvement in all your maths problem solving skills (including STEP) after BMO 2 revision or does it just help your number theory/ geometry stuff?
Original post by joostan
Am I being dozy or is this just a half?


Not being dozy. I just write easy questions haha.
Original post by EnglishMuon
I agree :smile: Although the final result for some questions I might not find amazing, the techniques involved are always quite nice. Did you notice a significant improvement in all your maths problem solving skills (including STEP) after BMO 2 revision or does it just help your number theory/ geometry stuff?


It helps alot with general spots aswell alot of step are off theorems simplified with stepsI seen a serious improvement after doing BMO2 properly to the extent where most of step is fairly straightforward and obvious. Ofcourse only will actually know until the 18th august but I should be ok for S,1 this year. I seen a serious significant improvement.
I only answered a few questions in III 2014 and 2015 so when I redo those papers not including the questions i already did then I will know how much I improved.


Posted from TSR Mobile
Problem 582

Let b and c be uniformly distributed with −2<b<2 -2 < b < 2 and −1<c<1 -1 < c < 1.

Find the probability that x2−bx+c=0x^2 - bx + c = 0 has real roots less than 1 in magnitude.
Original post by physicsmaths
It helps alot with general spots aswell alot of step are off theorems simplified with stepsI seen a serious improvement after doing BMO2 properly to the extent where most of step is fairly straightforward and obvious. Ofcourse only will actually know until the 18th august but I should be ok for S,1 this year. I seen a serious significant improvement.
I only answered a few questions in III 2014 and 2015 so when I redo those papers not including the questions i already did then I will know how much I improved.


Posted from TSR Mobile

Thanks, Ill have another look at BMO then. Im getting to the stage where most step qs are pretty straight forwards and BMO feels like how I used to find step when I was just getting into it. I still make dumb mistakes though so hopefully it will help to phase those out :tongue:. Im pretty sure you'll get those grade atleast though, my personal goal is Ss so if you cant get them I most certainly wont!
Original post by 16Characters....
Problem 581*

You select a real number entirely at random from the interval [0,2π][0, 2\pi]. What is the probability that sin⁡x+cos⁡x\sin x + \cos x is greater than 1 in magnitude?

Disclaimer: I made this up myself. I think my logic is correct, but my probability is **** so if this is an impossible question (or significantly harder than the * I gave it) then my apologies.


My probability is far crappier than anybody, but this looks very/quite nice, so:

Solution 581

Unparseable latex formula:

\displaystyle [br]\begin{equation*}\sin x + \cos x = \sqrt{2}\sin \left(x - \frac{\pi}{4}\right)\end{equation*}



So for that to be greater than one in magnitude we need ∣sin⁡(x+π4)∣≥12\left|\sin \left(x + \frac{\pi}{4}\right)\right| \geq \frac{1}{\sqrt{2}}

So that gives us
Unparseable latex formula:

x + \frac{\pi}{4} \in \left[\frac{\pi}{4}, \frac{3\pi}{4}\right] \union \left[\frac{5\pi}{4}, \frac{7\pi}{4}\right]



That's a total width of π\pi in the total interval length of 2π2\pi - so the probability is π2π\frac{\pi}{2\pi}. I quite liked this!! :biggrin:
Original post by EnglishMuon
Thanks, Ill have another look at BMO then. Im getting to the stage where most step qs are pretty straight forwards and BMO feels like how I used to find step when I was just getting into it. I still make dumb mistakes though so hopefully it will help to phase those out :tongue:. Im pretty sure you'll get those grade atleast though, my personal goal is Ss so if you cant get them I most certainly wont!


Lol my goal is sss aswell. 100+ in each paper is the goal!


Posted from TSR Mobile
Original post by 16Characters....
Problem 582

Let b and c be uniformly distributed with −2<b<2 -2 < b < 2 and −1<c<1 -1 < c < 1.

Find the probability that x2−bx+c=0x^2 - bx + c = 0 has real roots less than 1 in magnitude.


Just to check the obvious, is that all roots <1 or sum of roots less than 1? (I guess the former)
Original post by Zacken
My probability is far crappier than anybody, but this looks very/quite nice, so:

Solution 581

Unparseable latex formula:

\displaystyle [br]\begin{equation*}\sin x + \cos x = \sqrt{2}\sin \left(x - \frac{\pi}{4}\right)\end{equation*}



So for that to be greater than one in magnitude we need ∣sin⁡(x+π4)∣≥12\left|\sin \left(x + \frac{\pi}{4}\right)\right| \geq \frac{1}{\sqrt{2}}

So that gives us
Unparseable latex formula:

x + \frac{\pi}{4} \in \left[\frac{\pi}{4}, \frac{3\pi}{4}\right] \union \left[\frac{5\pi}{4}, \frac{7\pi}{4}\right]



That's a total width of π\pi in the total interval length of 2π2\pi - so the probability is π2π\frac{\pi}{2\pi}. I quite liked this!! :biggrin:


Pi/2Pi= 1/2(pi/pi)=1/2


Posted from TSR Mobile
Original post by EnglishMuon
Just to check the obvious, is that all roots <1 or sum of roots less than 1? (I guess the former)


-1< aswell


Posted from TSR Mobile
Original post by physicsmaths


ah yep. Hurray for skim reading :wink:
Original post by physicsmaths
Pi/2Pi= 1/2(pi/pi)=1/2


Original post by EnglishMuon
Just to check the obvious, is that all roots <1 or sum of roots less than 1? (I guess the former)


Yes the former, both roots are < 1 in magnitude.

Original post by Zacken

That's a total width of π\pi in the total interval length of 2π2\pi - so the probability is π2π\frac{\pi}{2\pi}. I quite liked this!! :biggrin:


Yup, now try 582 above :-)
Original post by 16Characters....

Yup, now try 582 above :-)


Did you get this idea off a STEP paper? I remember something similar. (referring to the roots of a quadratic, not the sin x + cos x)
Original post by Zacken
Did you get this idea off a STEP paper? I remember something similar. (referring to the roots of a quadratic, not the sin x + cos x)


I have been discovered.
Original post by Zacken
Did you get this idea off a STEP paper? I remember something similar. (referring to the roots of a quadratic, not the sin x + cos x)


Yeah I think this was in a Siklos booklet (or similar)
Problem 583

xi=±1 x_i=\pm 1 and x1x2x3x4+x2x3x4x5+...+xnx1x2x3=0 x_1x_2x_3x_4+x_2x_3x_4x_5+...+x_nx_1x_2x_3=0
Prove that n n is divisible by 4.

Quick Reply

Related discussions

Latest

Trending

Trending

Articles for you