Problems

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?

A numerical sequence is defined by the following conditions: $$a_1=1,a_{n+1}=a_n+\lfloor \sqrt{a_n}\rfloor .$$

Prove that among the terms of this sequence there are an infinite number of complete squares.

Is it possible to construct a 485 × 6 table with the integers from 1 to 2910 such that the sum of the 6 numbers in each row is constant, and the sum of the 485 numbers in each column is also constant?

What time is it going to be in $2025$ hours from now?

Prove that the product of five consecutive integers is divisible by $30$.

Prove that if $n$ is a composite number, then $n$ is divisible by some natural number $x$ such that $1<x\le \sqrt{n}$.

The natural numbers $a,b,c,d$ are such that $ab=cd$. Prove that the number $a^{2025}+b^{2025}+c^{2025}+d^{2025}$ is composite.

Prove that for an arbitrary odd $n=2m-1$ the sum $S=1^n+2^n+...+n^n$ is divisible by $1+2+...+n=nm$.

Multiply an odd number by the two numbers either side of it. Prove that the final product is divisible by $24$.

Mattia is thinking of a big positive integer. He tells you what this number to the power of $4$ is. Unfortunately it’s so large that you tune out, and only hear that the final digit is $4$. How do you know that he’s lying?