Show that for any number \(a,b,c,d\), we have \((a+b)(c+d) = ac + ad + bc + bd\).
Expand \((x_1+\dots + x_n)^2\) where \(x_1,\dots,x_n\) are real numbers.
Prove the Cauchy-Schwarz inequality \[(a_1b_1+\dots+a_nb_n)^2\leq (a_1^2+\dots+a_n^2)(b_1^2+\dots+b_n^2)\] where \(a_1,\dots,a_n,b_1,\dots,b_n\) are real numbers. If you already know a proof (or more!), find a new one.
Prove that there exist infinitely many natural numbers \(a\) with the following property: the number \(z = n^4+a\) is not prime for any natural number \(n\).
Let \(a,b,c>0\) be the length of sides of a triangle. Show that the triangle is right-angled if and only if \((a^4+b^4+c^4)^2 = 2(a^8+b^8+c^8)\). Note that this is a symmetric characterization of right-angled triangles by its side lengths.