Hello friends, try these problems and post solutions :
\((1)\) For each positive integer \(n\), show that there exists a positive integer \(k\) such that
for some polynomials , with integer coefficients, and find the smallest such as a function of .
( Prove that there exists a polynomial with real coefficients such that for all real numbers and , which cannot be written as the sum of squares of polynomials with real coefficients.
Let and for .
Find, with proof, all nonzero polynomials such that
Find the least real number such that for each triangle with side lengths , , ,
Let , , be positive real numbers. Prove that