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Showing posts from February, 2023

Definitely no geos [Feb 1-Feb 7]

Okay I did way more problems.. it's just I am too lazy to type up. And Ig no more posts before EGMO cause I am too much busy for boards :( bai people Problem 1.6: Prove that the number of answers for $|a_1|+|a_2|+...+|a_k|≤n$ is equal to the number of answers for $|b_1|+|b_2|+...+|b_n|≤k$, where $a_i,b_i$ are integers.  Proof:  Let $f(n,k)$ be the number of solutions to $|a_1|+\dots+|a_n|\le k$.  Note that $$f(n,k) = f(n-1,k)+f(n-1,k-1)+[f(n-1,k-1)+2f(n-1,k-2)+\dots+2f(n-1,0)]$$ $$=f(n-1,k)+f(n-1,k-1)+f(n,k-1).$$ Similarly, we get $$f(k,n)=f(k-1,n)+f(k-1,n-1)+f(k,n-1).$$  Now, we can simply induct on $a+b$ and show $f(a,b)=f(b,a)$.  Done!  EGMO 2012 P2 Let $n$ be a positive integer. Find the greatest possible integer $m$, in terms of $n$, with the following property: a table with $m$ rows and $n$ columns can be filled with real numbers in such a manner that for any two different rows $\left[ {{a_1},{a_2},\ldots,{a_n}}\right]$ and $\left[ {{b_1},{b_2},\ldots,{b_n}} \right]$ the fol