Square-free part of n!

Square-free part of n!

04:09:00  

Few days ago, while doing exercise with Indonesian IMO 2017 Team,  we came across this problem (which is from Tuymaada Olympiad 2015). Let $n!=ab^2$ where $a$ is a squarefree integer, prove that for any $\varepsilon>0$ the inequality $$2^{(1-\varepsilon)n} < a < 2^{(1+\varepsilon)n}$$ holds for sufficiently large $n$. This is...

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Perkuat soal USAMO 2014.

Perkuat soal USAMO 2014.

03:57:00  

Buktikan bahwa terdapat konstan $c>0$ yang memenuhi syarat berikut: Jika $a,b,n$ adalah bilangan bulat positif sedemikian sehingga $FPB(a+i, b+j)>1$ untuk setiap $i,j \in \{0,1,...,n\}$ maka berlaku $$\min\{a,b\} > c^{n} n^{n/2}$$ Kita bisa perkuat batas pada soal menjadi $(cn)^{1.09n}$. Motivasi pada pengerjaan soal ini adalah membatasi faktor-faktor dari $a$...

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A matrix problem with line-sums

A matrix problem with line-sums

03:07:00  

Last week, one of my junior gave me a Linear Algebra / Matrix problem, which smell like a combinatorial problem (although until now, I haven't found a combinatorial proof of it). Let $X$ and $Y$ be a $n \times n$  $(0,1)$-matrix , where the line-sums of $X$ (resp....

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