Problems

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 $k\ge 2$, 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 $x\ge 0$ and $|x|=-x$ if $x<0$. What are $|3|$, $|-4.3|$ and $|0|$?

Prove that $|x|\ge 0$.

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 $\frac{1}{10}$. What are they?