Find all triangles whose sides are consecutive integer and areas are alse integers.

Prove that there are infinitely many triples of consecutive  integers each of which is a sum of two squares

Detemine (with proof ) whether is a subset X of the integers with the following property : for any intefer n there is exactyly one solution of a +2b = n with a,b \(\in\) X

Determine which binominal coefficients \(_{C^n_r=\dfrac{n!}{r!\left(n-r\right)!}}\)are odd.

Let p an odd prime number. For any positive integer k, show that there exists a positive integer m such that the rightmost k digits of \(m^2\), when expressed in the base p, are all 1's.

Show that the sequence {\(a_n\)} difined by \(a_n\) = [n\(\sqrt{2}\)] for n = 1,2,3,... (where the breackets the greatest integer function) contains an infinite number ò integral powers of 2.