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    (Original post by Renzhi10122)
    Someone's been doing some BMO1
    haha you know the papers too well 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 . 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!
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    (Original post by EnglishMuon)
    haha you know the papers too well 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 . 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.



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    (Original post by 16Characters....)
    Problem 581*

    You select a real number entirely at random from the interval [0, 2\pi]. What is the probability that \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?
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    (Original post by physicsmaths)
    Lol that was me.
    BMO is wonderful I think great problem solving.



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    I agree 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?
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    (Original post by joostan)
    Am I being dozy or is this just a half?
    Not being dozy. I just write easy questions haha.
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    (Original post by EnglishMuon)
    I agree 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.


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    Problem 582

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

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


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    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 . 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!
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    (Original post by 16Characters....)
    Problem 581*

    You select a real number entirely at random from the interval [0, 2\pi]. What is the probability that \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

    \displaystyle 

\begin{equation*}\sin x + \cos x = \sqrt{2}\sin \left(x - \frac{\pi}{4}\right)\end{equatio  n*}

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

    So that gives us 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 \pi in the total interval length of 2\pi - so the probability is \frac{\pi}{2\pi}. I quite liked this!!
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    (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 . 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!


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    (Original post by 16Characters....)
    Problem 582

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

    Find the probability that x^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)
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    (Original post by Zacken)
    My probability is far crappier than anybody, but this looks very/quite nice, so:

    Solution 581

    \displaystyle 

\begin{equation*}\sin x + \cos x = \sqrt{2}\sin \left(x - \frac{\pi}{4}\right)\end{equatio  n*}

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

    So that gives us 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 \pi in the total interval length of 2\pi - so the probability is \frac{\pi}{2\pi}. I quite liked this!!
    Pi/2Pi= 1/2(pi/pi)=1/2


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    (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


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    (Original post by physicsmaths)
    -1< aswell


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    ah yep. Hurray for skim reading
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    (Original post by physicsmaths)
    Pi/2Pi= 1/2(pi/pi)=1/2
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    (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 \pi in the total interval length of 2\pi - so the probability is \frac{\pi}{2\pi}. I quite liked this!!
    Yup, now try 582 above :-)
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    (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)
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    (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.
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    (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)
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    Problem 583

     x_i=\pm 1 and  x_1x_2x_3x_4+x_2x_3x_4x_5+...+x_  nx_1x_2x_3=0
    Prove that  n is divisible by 4.
 
 
 
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