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. Each of us take turn 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 a trivial and a non-trivial combination 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 nonzero real numbers \(a,b,c\): \[\frac{b}{2a} + \frac{c^2 + ab}{4bc} - \left|{\frac{c^2 - ab}{4bc}} \right| - \left|{\frac{b}{2a} - \frac{c^2 + ab}{4bc} + \left|{\frac{c^2 - ab}{4bc}}\right|}\right| = \min\left\{{\frac{b}{a},\frac{c}{b},\frac{a}{c}\right\}.\]

On the picture below you can see graphs \(K_5\) a complete graph on \(5\) vertices and \(K_{3,3}\) a complete bipartite graph on \(3\) and \(3\) vertices. A theorem states that these graphs cannot be embedded into plane, namely one cannot draw graphs \(K_5\) and \(K_{3,3}\) on a plane in such a way that there would be no intersecting edges.
A converse statement is also true: if a graph \(G\) cannot be embedded into a plane (drawn on a plane without intersecting edges), then this graph contains either \(K_5\) or \(K_{3,3}\) as a subgraph.
The question is: can you draw the graph \(K_5\) without intersecting edges on a torus?

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?