@wary bay
#bmo1
gritty viper
wary bay
this one is long lol
restive latch
I have no idea
graceful jackal
The answer is 4 I think
wary bay
so n = 1001k+1 = x^2
so x^2 - 1 = 1001k
(x-1)(x+1) = 1001k
Let k = ab
(x-1)(x+1) = 1001ab = 7 x 11 x 13 ab
Now, u kinda just guess how to split this
(x-1) = 7a and (x+1) = 143b
so 143b-7a = 2
now u solve this:
143u+7v=1
143= 20x7 + 3
7 = 2x3 + 1
so 1 = 1x7 - 2x3 = 1x7 - 2x(1x143 - 20x7) = 41x7 - 2x143
so u'=-2, v'=41
so for 143u+7v=2, u'=-4, v'=82
u = -4+7t, v=82-143t
so 143b-7a = 2
b=-4+7t, a=143t-82
Letting t=1 gives us two positive values of a and b
a=61, b=3
This gives k=183
so n=183184
restive latch
The answer is 4 I think
It has to be 6 digits
graceful jackal
It has to be 6 digits
6 then. That's what I meant
wary bay
this is a pretty bad question
gritty viper
how did you know what to do
wary bay
kinda just went with it
started by representing n
then i saw i could get a difference of two squares
now i needed to decide on two factors (this is just pure guess work), a bit of a problem is that the factors of k can be spread across both x-1 and x+1 so i let k=ab to solve this a bit
then i noticed they got a difference of 2 so i got a diophantine equation which i solved to get my answers
gritty viper
how did u know to do k = 'abc'
restive latch
Is this maths a level question or further maths a level
gritty viper
cuz i did n = 100000a + 10000b ....
then just got nowhere
gritty viper
no
wary bay
cuz i did n = 100000a + 10000b ....
then just got nowhere
ik that the first three is one less than the last 3
so i just grouped it up as 1
which reduces the number of variables by a lot and makes life a bit easier
wary bay
ye
gritty viper
also out of : numer theory, inequalities, geo and combi which one will prob take the longest
this geo book is huge
gritty viper
yeah
wary bay
hmm probs the geo or combi one
gritty viper
wary bay
ah
the triangle one is defo gonna be a lot more advanced then u gonna need
the other euclidean one is the more bmo1 focused
gritty viper
yeah i planned on asking you what i could leave out cuz it was so huge
whenever i get to that
gritty viper
||its 4 pages||
wary bay
bruh
gritty viper
ez stuff
wary bay
ye
gritty viper
and the primer book said the trig identities could be useful too
wary bay
most geo problems could solve be solved using trig
but often if u fuck that up, u gonna get less marks to if u did standard geo and fucked it up
gritty viper
im sure ill do a lot of that
oh well ill just do geo last
is combi hard
just im dreading it
i can barely do gcse combi
wary bay
is combi hard
some combi questions are pretty straight forward
others are pretty evil
like this yrs q4
is more on the straight forward side
for combi, all u need to know on top of gcse is the idea of double counting (expressing smth in 2 ways), colouring (basically, on a grid, or smth how can u colour it st u arrive at a certain conclusion), and then bijections (how can u represent the question in a new way)
wary bay
that is also more bmo2 ...
gritty viper
lmao
gritty viper
thats only for the past few years
wary bay
u can basically get all of them
sols vids go back to 2005/6
then https://www.dropbox.com/sh/w9mfy9qtjs68xzc/AADnnQKWONBsboMGVDiuS-kAa?dl=0 (misc -> BrMO ...) has two booklets with 1997-2004
then u can find the rest for 1975-1996 on https://libgen.is/book/index.php?md5=CE42BE019C9CB79C19268002D35ACF55
gritty viper
ty
calm phoenix
heeeeeeli
wary bay
so:
GCD(n-38,n+38) = GCD(76,n-38) so GCD(n-38,n+38) is a factor of 76
that means one of n-38 or n+38 is divisible by 5^3
also, they both must be divisible by 4 (otherwise, their product will not have a factor of 8)
thus that means one of n-38 or n+38 is divisible by 5^3x4 = 500
n-38 = 500k -> n=500k+38
n+38 = 500k -> n=500k-38
so n is of form n=500k +- 38
Lets say the GCD of n-38 and n+38 is h
inner shore
what does bmo stand for
wary bay
that means
n+38 = n-38 + 76
h divides the LHS and also (n-38)
so it must also divide 76
inner shore
british maths olympiad?
wary bay
so whatever the GCD of those n-38 and n+38 is will also divide 76
so it must be a factor of 76
from that we can deduce a couple things:
as the gcd does not have a factor of 5, if the product is divisble by 1000 = 2^3 x 5^3
that means exactly one of them is divisible by 5^3
additionally, from
n+38 = n-38 + 76
if 2 or 2^2 divides n-38, as it also divides 76, it will divide RHS so it also divides the LHS -> it divides n+38
and we repeat the argument in the other direction with n+38 - 76 = n-38
so, say that 2 divides n+38 but 4 doesnt, that be 2 divides n-38 but 4 doesnt, their product will be divisible by 4 but not 8
so that means 4 divides n+38 and n-38
if their product is divisible by 1000
from that, one of n+38 or n-38 is divisible by 500
thus we can represent it as n+38 = 500k or n-38=500k
n=500k + 38 and n=500k-38
so n=500k +- 38
and thats our answer
wary bay
british maths olympiad?
ye it does
quasi meadow
hi
wary bay
hello
eager scaffold
niolyan
gritty viper
and thats our answer
Tyty
wary bay
nice
gritty viper
nice
is it possible in bmo to get the variable in both the dividend and the divisor
but like not simple
where its all linear
ik that was in this year
when its quadratic and stuff
wary bay
ig it can but most likely smth u can deal with polynomial division
and probably will just be a q4-6
drowsy wave
so n = 1001k+1 = x^2
so x^2 - 1 = 1001k
(x-1)(x+1) = 1001k
Let k = ab
(x-1)(x+...
where did u and v come from?
also which year was this
wary bay
where did u and v come from?
it’s part of the way to solve diophantine equations