Problems

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.

Show that \(n^2+n+1\) is not divisible by \(5\) for any natural number \(n\).

Prove the following identity for any three non-zero real numbers \(a,b,c\): \[\frac{b}{2a} + \frac{c^2 + ab}{4bc} - \left|{\frac{c^2 - ab}{4bc}} \right| - \left|{\frac{b}{2a} - \frac{c^2 + ab}{4bc} + \left|{\frac{c^2 - ab}{4bc}}\right|}\right| = \min\{\frac{b}{a},\frac{c}{b},\frac{a}{c}\}.\]

If we glue the opposite sides of the paper band in the same direction as on the picture, we will get a cylinder. What surface do we get, if we glue the circles of the cylinder in the same direction as well?

We start with a rectangular sheet of paper - preferably with proportions more than \(6:1\), so that it looks more like a band. For now assume that one can stretch or shrink the paper band as needed. Describe the surface we get if we start with a rectangular sheet of paper and then glue the opposite sides of the paper band in the opposite direction as in the picture.