- #1

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its an exercise of the math olympiad of my city... i know i should have posted at least a bit of my work, but i think there is a trick to solve this category of problems that i dnt know...where should i start???

- Thread starter Born2Perform
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- #1

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its an exercise of the math olympiad of my city... i know i should have posted at least a bit of my work, but i think there is a trick to solve this category of problems that i dnt know...where should i start???

- #2

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- #3

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i think it was meant as

- #4

uart

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The following works for me. First let **a=0** and consider what integers you can construct with just the difference of two sqaures [tex]b^2 - c^2[/tex]. By considering the expansion you should easily be able to show that any odd number can be composed from the difference of the two squares alone.

Actually lots of even numbers can be made from the difference of the two sqaures as well, though not all. For example it's also easy to show that all integers that are a multiple of 4 can also be contructed, though you dont even need that result here. Once you've establish that b^2-c^2 can make any odd number then just leave**a=0** for odd numbers and set **a=1** to make all the evens.

Actually lots of even numbers can be made from the difference of the two sqaures as well, though not all. For example it's also easy to show that all integers that are a multiple of 4 can also be contructed, though you dont even need that result here. Once you've establish that b^2-c^2 can make any odd number then just leave

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- #5

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n must be even or odd or 0.

The case of n=0 is trivial, a=b=c=0

n is even: n=2k, k=1,2,...

2k= 1^2 + k^2 - (k-1)^2

n is odd: n=2k-1, k=1,2,...

(2k-1)= 0^2 + k^2 - (k-1)^2

The case of n=0 is trivial, a=b=c=0

n is even: n=2k, k=1,2,...

2k= 1^2 + k^2 - (k-1)^2

n is odd: n=2k-1, k=1,2,...

(2k-1)= 0^2 + k^2 - (k-1)^2

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- #6

uart

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Actually zeron must be even or odd or 0

- #7

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Yes, you are right. But, here as the case of n=0 gives rise to a trivial solution of all 0's (a=b=c=0), I separately mentioned the case notwithstanding it was a repetition.Actually zeroisan even number.

As a matter of fact, the values of 'k' I mentioned, should be k=...-2,-1,0,1,2,...

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- #8

uart

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The result that I was hoping that the OP would come to by himself (by looking at the expansion of the difference of the two sqaures) was that any integer that can be expressed as a product of two factors can be expressed as a difference of two squares

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