Problems

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.

Let \(a,b,c\) be numbers. If \(a+b=c\), show that \(a=c-b\).

Prove that the sum of two consecutive numbers is always odd.

Show algebraically that the sum of two odd numbers is even.

Show that the sum of six consecutive numbers is a multiple of \(3\) but not a multiple of \(6\).

Alice sums \(n\) consecutive numbers, not necessarily starting from \(1\), where \(n\) is a multiple of four. An example of such a sum is \(5+6+7+8\). Can this sum ever be odd?