Is there a divisibility rule for \(2^n\), where \(n = 1\), \(2\), \(3\), . . .? If so, then explain why the rule works.
Can you find a formula relating \(1^3+2^3+\dots+n^3\) to \(1+2+\dots+n\)?
Prove the reverse triangle inequality: for every pair of real numbers \(x\), \(y\), we have \(\left| \left| x \right| - \left| y \right| \right| \leq \left| x - y \right|\).
Can you come up with a divisibility rule for \(5^n\), where \(n=1\), \(2\), \(3\), . . .? Prove that the rule works.
Show that for each \(n=1\), \(2\), \(3\), . . ., we have \(n<2^n\).
You and I are going to play a game. We have one million grains of sand in a bag. We take it in turns to remove \(2\), \(3\) or \(5\) grains of sand from the bag. The first person that cannot make a move loses.
Would you go first?
For every natural number \(k\ge2\), find two combinations of \(k\) real numbers such that their sum is twice their product.
Show that \(n^2+n+1\) is not divisible by \(5\) for any natural number \(n\).
Prove the following identity for any three non-zero real numbers \(a,b,c\): \[\frac{b}{2a} + \frac{c^2 + ab}{4bc} - \left|{\frac{c^2 - ab}{4bc}} \right| - \left|{\frac{b}{2a} - \frac{c^2 + ab}{4bc} + \left|{\frac{c^2 - ab}{4bc}}\right|}\right| = \min\{\frac{b}{a},\frac{c}{b},\frac{a}{c}\}.\]