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- Jun 7th 2010, 05:31 PM #1
## IMO 1960 problem 1

Determine all three-digit numbers N having the property that N is divisible by 11, and N/11 is equal to the sum of the squares of the digits of 11.

Anyone know how to solve this one. We write N as abc then...

Since N is divisible by 11 we know that a - b + c is divisible by 11. So a-b+c = 0 or 11.

Then we also know that

I can't get any sort of 'nice' solution from this and just end up with having to check 18 cases. (a+c=b, sub into above than a=0,1,2,... etc)

- Jun 7th 2010, 06:10 PM #2

- Jun 7th 2010, 07:44 PM #3

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Maybe there is another method which is better ...

the actual value of is

which leads to two cases and

Let's check the first one :

We have

Since substitute it we obtain

which is a quadratic equation of , check its discriminant and it should be a square ie:

We have

( Note we neglect the case because later we will find that must not be an integer . )

Therefore , the two possible ways to write in the form are :

and

and we have or the only solution is and ( corresponding to )

so or which gives

Therefore , the required number is

Use the similar method to see if there is another solution in the second case . It should be .

- Jun 8th 2010, 03:03 AM #4

- Jun 8th 2010, 04:11 AM #5

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