Thread: Math Q&A Game View Single Post
 P: 688 maze's question is classic... the answer is Spoiler N-floor (sqrt(N)) the nice animation makes it kinda obvious... the reason is only perfect squares have an odd number of distinct divisors. problem 2 above Spoiler it suffices to find # of ways to arrange coins such that each row has one coin. This is easily given as $$n^n$$, so answer is $$1-n^n/ \binom{n^2}{n}$$... if you want to find probability such that at least one row or one column has no silver coin, then it would be $$1-n!/ \binom{n^2}{n}$$ problem 3 above Spoiler LHS is less than or equal to $$a(a+b+c)+b(a+b+c)+c(a+b+c)$$ by weighted AM-GM inequality, which is clearly less than or equal to 1 Problem 1 seems too long and I am feeling kinda lazy... Now for my problem... hmmmm... after all these years.... let's see.....I'm not sure if this is appropriate: prove that all one dimensional compact connected Lie group (a manifold that is also a group with smooth multiplication) is diffeo-isomorphic to S1 (the circle). In case we wanna make things "elementary", evaluate: $$\int_{0}^\infty \frac{dx}{1+x^n}$$ In case you think a simple contour will bring this problem down... try to evaluate it for n>1, not just integers. edit: sorry, latex just doesn't work well with spoiler alerts...