#Some bmo question

1 messages · Page 1 of 1 (latest)

candid gate
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q2 is 13 how do i do q3

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eager depot
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How did u do q2?

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Did u just solve the equation and then plugged it in?

candid gate
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so i wrote x + 1/x as a single fraction

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so x^2 + 1/x

eager depot
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Good

candid gate
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then x^2 = 13 x -1

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subbed into the single fraction and yh got 13

eager depot
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Ok

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I was gonna say if u know what x^2 is u can square it to find x^4

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But that might make it a bit harder

candid gate
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Lets try it anyway

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so would i swuare the x+x^-1

eager depot
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Ok so x^4 is 169x^2 -26x+1

candid gate
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oh alr yh

eager depot
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U cld combine the fraction to get x^8+1/x^4

candid gate
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wait how

eager depot
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Same way u did before

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Multiply x^4 by x^4

candid gate
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wait im lost

eager depot
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Ok remember how in q2 u made it 1 fraction?

candid gate
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yh

eager depot
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We’re gonna do the same for q3

candid gate
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alr

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and what we using

eager depot
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So we had x^4 + 1/x^4

candid gate
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yh

eager depot
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Which becomes x^8+1 all over x^4

candid gate
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ohhh

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yeah

eager depot
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Now we need to find x^8 which is x^4 squared

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We know x^4 is x^2 squared

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And we know x^2 is 13x-1

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So x^4 is 169x^2-26x+1

candid gate
eager depot
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Yes

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But we don’t know x^8

candid gate
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we do jnow x^4 tho

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right

eager depot
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Yes

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U need to square it but u get a massive number

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@stuck gyro is there a better method?

stuck gyro
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Which question I’ll do it later

candid gate
eager depot
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Wait I think I got it

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So x +1/x=13

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If u square it u get

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X^2 +1/x^2 +2 =169

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X^2+1/x^2=167

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If we square it again we get

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X^4+1/x^4+2=167^2

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So x^4 + 1/x^4 =167^2-2

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We need the last digit tho

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Anything that ends in a 7 will end in a 9 when squared so 167 squared ends in a 9

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Subtract 2 and the answer is 7

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@candid gate

stuck gyro
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Yes

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As long as x isn’t 0

eager depot
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X can’t be 0 as per the original equation

stuck gyro
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Yep

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So it works

stuck gyro
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are u doing the bmo question?

candid gate
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btw how did u square the x + x^-1

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like where did the +2 come from

stuck gyro
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just use (a+b)^2 = a^2 + 2ab + b^2

candid gate
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Oh

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i see now

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thanks

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q4 is a bmo question have a go at it if u want

stuck gyro
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done it before

candid gate
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ive got about 10 solutions so far

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at one point u end up with 200+ x / 12 + x init

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howd u get that to give an integer

stuck gyro
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i believe i did it slightly differently so gimmie a sec

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yeah you do get to that

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so you want that to be an integer

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so 12+x|200+x

stuck gyro
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||200+x is the 12+x times table so 12+x must also divide 200+x-(12+x) = 188||

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||12+x|188 so 12+x is a factor of 188||

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||we may find factors by looking at the prime factorisation of 188||

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||188 = 2^2 * 47||

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||factors of 188 are: 1,2,4,47,94,188||

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||most of these will not work as they will not give an n value in the interval [200, 300)||

candid gate
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huh

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why did you do 200 + x - 12+x

stuck gyro
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we know that 200+x is some multiple of 12+x so 200+x = k(12+x)
if we reduce that multiple by 1 then we have 200+x -(12+x) = k(12+x) - (12+x)
188 = (k-1)(12+x)

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from that we can see that 12+x divides 188

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note that this will only work if 12+x =! 200+x

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which is always true

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Or you could do it with modular arithmetic

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200+x = 0 (mod 12+x)
200+x -(12+x) = -12+x (mod 12+x)
188 = 0 (mod 12+x)

eager depot
stuck gyro
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Yep this years question 1

candid gate
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icl im still baffed

candid gate
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dont get why u subtract 12+x on both sides

candid gate
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@stuck gyro

stuck gyro
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watch the video solutions i think they explain

candid gate
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alr

cold geode
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Are you lads aiming for Oxbridge maths or is this for fun?

candid gate
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my school runs a program where they prepare u for step

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or other uni entrance exams

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this is the first hw task ive been given from it

livid kite
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ffs wheres my support

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i have to rely on 3rd party support like stem smart

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and amsp

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my skl got shit programmes

stuck gyro
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skill issue

candid gate
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this is the first year they've started it

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so ig im lucky

livid kite
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jk

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wish i went laet

livid kite
livid kite
stuck gyro
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Skill issue

livid kite
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@spice mica example of forum for bmo

spice mica
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This is british or balkan olympiad

stuck gyro
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British

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Only the last question