A whole number \(n\) has the property that when you multiply it by \(3\) and then add \(2\), the result is odd. Use proof by contrapositive to show that \(n\) itself must be odd.
Calculate the left side and the right side. \[2\times(12+3)\quad\quad 2\times 12 + 2\times 3\] \[3\times(0.8+1)\quad\quad 3\times 0.8 + 3\times 1\] \[(-2)\times (3-5)\quad\quad (-2)\times 3 + (-2)\times (-5)\] What do you notice?
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.
Show that the sum of any \(100\) consecutive numbers is a multiple of \(50\) but not a multiple of \(100\).