Problems

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?

In the long addition above, each letter corresponds to a different digit. What is the sum \(D + O +G + C +A +T\)?

Let \(ABCDE\) be a regular pentagon. The point \(G\) is the midpoint of \(CD\), the point \(F\) is the midpoint of \(AE\). The lines \(EG\) and \(BF\) intersect at the point \(H\). Find the angle \(EHF\).

A paper band of constant width is tied into a simple knot and tightened. Prove that the knot has the shape of a regular polygon.

Is it possible to draw the graph \(K_{3,3}\) without intersecting edges on a Moebius band?

Is it possible to link three rings together in such a way that they cannot be separate from each other, but if you remove any ring, then the other two will fall apart?

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?