For any real number \(x\), the absolute value of \(x\), written \(\left| x \right|\), is define to be \(x\) if \(x>0\) and \(-x\) if \(x \leq 0\). What is \(\left| 3 \right|\), \(\left| -4.3 \right|\) and \(\left| 0 \right|\)?
Prove that for any real number \(x\), \(x \leq \left| x \right|\) and \(0 \leq \left| x \right|\). Then prove that for any real numbers \(x,y\), the triangle inequality holds: \(\left| x+y \right| \leq \left| x \right|+\left| y \right|\).
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. Each of us take turn 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 a trivial and a non-trivial combination 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 \(n\).