If a magician puts \(1\) dove into his hat, he pulls out \(2\) rabbits and \(2\) flowers from it. If the magician puts \(1\) rabbit in, he pulls out \(2\) flowers and \(2\) doves. If he puts \(1\) flower in, he pulls out \(1\) rabbit and \(3\) doves. The magician starts with \(1\) rabbit. Could he end up with the same number of rabbits, doves, and flowers after performing his hat trick several times?
For any real number \(x\), the absolute value of \(x\), written \(\left| x \right|\), is defined to be \(x\) if \(x>0\) and \(-x\) if \(x \leq 0\). What are \(\left| 3 \right|\), \(\left| -4.3 \right|\) and \(\left| 0 \right|\)?
Let \(x\) and \(y\) be real numbers. Prove that \(x \leq \left| x \right|\) and \(0 \leq \left| x \right|\). Then prove that the following 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. 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.