Problems

Convert 2000 seconds into minutes and seconds.

There are $n$ integers. Prove that among them either there are several numbers whose sum is divisible by $n$ or there is one number divisible by $n$ itself.

Prove that there is a power of 3 that ends in 001.

Prove that any $n$ numbers $x_1,\dots ,x_n$ that are not pairwise congruent modulo $n$, represent a complete system of residues, modulo $n$.

Prove that for any natural number there is a multiple of it, the decimal notation of which consists of only 0 and 1.

All integers from 1 to $2n$ are written in a row. Then, to each number, the number of its place in the row is added, that is, to the first number 1 is added, to the second – 2, and so on.

Prove that among the sums obtained there are at least two that give the same remainder when divided by $2n$.

a) We are given two cogs, each with 14 teeth. They are placed on top of one another, so that their teeth are in line with one another and their projection looks like a single cog. After this 4 teeth are removed from each cog, the same 4 teeth on each one. Is it always then possible to rotate one of the cogs with respect to the other so that the projection of the two partially toothless cogs appears as a single complete cog? The cogs can be rotated in the same plane, but cannot be flipped over.

b) The same question, but this time two cogs of 13 teeth each from which 4 are again removed?