Maths Olympiad Daily Problems

2^m sheets of paper

There are 2m sheets of paper, with 1 written on each initially. In each step choose two distinct sheets with numbers a,b, and replace both numbers by a+b. Prove that after m^(2m−1) steps, the sum of all numbers on the sheets is at least 4^m. Can anyone solve this?

There are 2m sheets of paper, with 1 written on each initially. In each step choose two distinct sheets with numbers a,b, and replace both numbers by a+b. Prove that after m^(2m−1) steps, the sum of all numbers on the sheets is at least 4^m. Can anyone solve this?

Guys solve these Don't do baikati Do hard study IOQM is comming

Let $P$ be a regular $99$-gon. Assign integers between $1$ and $99$ to the vertices of $P$ such that each integer appears exactly once. (If two assignments coincide under rotation, treat them as the same. ) An \textit{operation} is a swap of the integers assigned to a pair of adjacent vertices of $P$. Find the smallest integer $n$ such that one can achieve every other assignment from a given one with no more than $n$ operations. try

Good

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Please solve Q3

Question 3

standard problem must do. has been used as lemma in many other problems