Problems

In the other room there are two doors. The statements on them say:

1. There is treasure behind at least one of the doors.

2. There is treasure behind the first door.

Your guide says: The first sign is true if there is treasure behind the first door, otherwise it is false. The second sign is false if there is treasure behind the second door, otherwise it is true. What would you do?

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 $x\le 0$. What are $\left|3\right|$, $|-4.3|$ and $\left|0\right|$?

Let $x$ and $y$ be real numbers. Prove that $x\le \left|x\right|$ and $0\le \left|x\right|$. Then prove that the following inequality holds $|x+y|\le \left|x\right|+\left|y\right|$.

Is there a divisibility rule for $2^n$, where $n=1$, $2$, $3$, . . .? If so, then explain why the rule works.

Find a general formula for the sum $1+3+\cdots +(2k+1)$.

Can you find a formula relating $1^3+2^3+\cdots +n^3$ to $1+2+\cdots +n$?

Prove the reverse triangle inequality: for every pair of real numbers $x$, $y$, we have $|\left|x\right|-\left|y\right||\le |x-y|$.

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.