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VCE Stuff => VCE Mathematics/Science/Technology => VCE Subjects + Help => VCE Mathematics => Topic started by: brightsky on August 10, 2010, 09:42:28 pm

Title: AIMO Past Papers
Post by: brightsky on August 10, 2010, 09:42:28 pm
Does anyone have any AIMO past papers (preferably with solutions) that they are willing to post up (excluding the practice paper on the AMT website)? Thanks. :)

Also, good luck to anyone doing the AIMO this Thursday.  :D
Title: Re: AIMO Past Papers
Post by: brightsky on August 11, 2010, 08:11:28 pm
is a perfect square where is an integer. Find the largest possible value of . Any help appreciated.
Title: Re: AIMO Past Papers
Post by: brightsky on August 11, 2010, 08:21:35 pm
Are you able to give me a scanned copy of just 2007-2008? I'm working on them now.  ::)
Title: Re: AIMO Past Papers
Post by: brightsky on August 11, 2010, 08:46:22 pm
is a perfect square where is an integer. Find the largest possible value of . Any help appreciated.
AIMO 2007 Q7.

Suppose x^2-19x+94 = m^2 where m is a non-neg integer. Then x^2-19x+94-m^2 = 0

Go from there.

Ah, solved. Thanks Azure!

Next:

5. Find where and are non-zerp solutions of the system of equations:




9. Find a prime, p, with the property that for some larger prime number, q, both 2q - p and 2q + p are prime numbers. Prove that there is only one such prime p.

Title: Re: AIMO Past Papers
Post by: kamil9876 on August 11, 2010, 09:59:55 pm
Probably not the best way, but the equations given are ugly anyway :P:

Subtract from the first equation the second times 5. You get:








So either x=0. y=5x, or y=17, lolz. Now you can plug these in to the original equation to see what u get in each case.
Title: Re: AIMO Past Papers
Post by: brightsky on August 11, 2010, 10:09:31 pm
Probably not the best way, but the equations given are ugly anyway :P:

Subtract from the first equation the second times 5. You get:








So either x=0. y=5x, or y=17, lolz. Now you can plug these in to the original equation to see what u get in each case.


Ooh, thanks kamil!

I got a related question:

Real numbers are linked by the two equations:





Determine the largest value for .
Title: Re: AIMO Past Papers
Post by: /0 on August 11, 2010, 10:35:16 pm
The Arithmetic-Quadratic mean inequality states that:



Using this,

So



Hmm well if this happens to be right so far then I guess you could solve for e.

EDIT: Oh wait, AM-QM only works for positive numbers
Title: Re: AIMO Past Papers
Post by: kamil9876 on August 11, 2010, 11:55:38 pm
Quote
9. Find a prime, p, with the property that for some larger prime number, q, both 2q - p and 2q + p are prime numbers. Prove that there is only one such prime p.

p=3, q=5 works.

Now suppose that there exists some other prime with this property, call it p. Thus (it is easy to see p cannot be 2)

Thus



Case 1:

Then one of p or -p is -1 mod 3. Thus one of 2q-p or 2q+p is a multiple of 3. Note also that 3<q<2q-p<2q+p  Since q>q.
It follows that one of these numbers must be a multiple of 3 greater than 3, hence not prime.

The other case is basically the same.
Title: Re: AIMO Past Papers
Post by: Ahmad on September 11, 2010, 02:46:50 am
Real numbers are linked by the two equations:





Determine the largest value for .

First I tried to reason by analogy, if instead of a, b, c, d, e we only had c, d, e then we can think about it geometrically in R^3. We can have e as the vertical axis, then we have a plane (first equation) and a sphere (second equation) intersecting in some sort of rotated circle in R^3 and we're trying to find the highest point on the circle, which we visually see occurs when c=d. By analogy we may expect that equality occurs when a=b=c=d, and generally when dealing with symmetric inequalities this is a good idea.

If a=b=c=d then the first equation gives us that a = b = c = d = 10 - e/2. We try to analyse the effect of deviation of the values of a, b, c, d away from 10 - e/2 (our suspected best point). So we let a = 10 - e/2 + a', and similarly for b,c,d, and we think of a' as representing a small deviation. Now we know that a' + b' + c' + d' = 0 from the first equation, this is very convenient. Plugging these into the second equation results in massive cancellation, and we're left with (assuming I did this right), which tells us our maximum occurs when 5e - 80 = 0, or e = 16.

It's interesting to think about why symmetry causes the massive cancellation, and why performing a change of coordinates (a,b,c,d) -> (a',b',c',d') to localize our variables at our suspected maximum is a good idea.
Title: Re: AIMO Past Papers
Post by: shrek27 on April 16, 2020, 11:30:47 pm
https://drive.google.com/open?id=1E3l2RGQvOg4C0V-LtzacIIKc--psVw2z

These has all the past AIMO papers with SOLUTIONS from 1998 to 2017.

Enjoy.