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 \(n\).
There are \(n\) balls labelled 1 to \(n\). If there are \(m\) boxes labelled 1 to \(m\) containing the \(n\) balls, a legal position is one in which the box containing the ball \(i\) has number at most the number on the box containing the ball \(i+1\), for every \(i\).
There are two types of legal moves: 1. Add a new empty box labelled \(m+1\) and pick a box from box 1 to \(m+1\), say the box \(j\). Move the balls in each box with (box) number at least \(j\) up by one box. 2. Pick a box \(j\), shift the balls in the boxes with (box) number strictly greater than \(j\) down by one box. Then remove the now empty box \(m\).
Prove it is possible to go from an initial position with \(n\) boxes with the ball \(i\) in the box \(i\) to any legal position with \(m\) boxes within \(n+m\) legal moves.
Given a natural number \(n\), find a formula for the number of \(k\) less than \(n\) such that \(k\) is coprime to \(n\). Prove that the formula works.
We can define the absolute value \(|x|\) of any real number \(x\) as follows. \(|x|=x\) if \(x\ge0\) and \(|x|=-x\) if \(x<0\). What are \(|3|\), \(|-4.3|\) and \(|0|\)?
Prove that \(|x|\ge0\).
Prove that \(|x|\ge x\). It may be helpful to compare each of \(|3|\), \(|-4.3|\) and \(|0|\) with \(3\), \(-4.3\) and \(0\) respectively.
Two fractions sum up to \(1\), but their difference is \(\frac1{10}\). What are they?
On her birthday, my grandma was asked how old she was. She said: "Start with the year I was born. Add the current year to it. Then, from the sum subtract the year I celebrated by \(20\)th birthday. From that, take away the year I was \(30\). The result will be \(16\)." How old is my grandma?