Category Archives: problem

2010-3 Sum

Evaluate the following sum

\(\displaystyle\sum_{m=1}^\infty \sum_{\substack{n\ge 1\\ (m,n)=1}} \frac{x^{m-1}y^{n-1}}{1-x^m y^n}\)

when |x|, |y|<1.

(We write (m,n) to denote the g.c.d of m and n.)

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2010-1 Covering the unit square by squares

Prove that finitely many squares on the plane with total area at least 3 can cover the unit square.

각각의 정사각형의 면적을 다 더했을 때 3 이상이 되는 유한개의 정사각형들이 있을 때, 이 정사각형들로 면적이 1인 단위정사각형을 완전히 덮을 수 있음을 증명하세요.

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2009-20 Expectation

Let en be the expect value of the product x1x2 …xn where x1 is chosen uniformly at random in (0,1) and xk is chosen uniformly at random in (xk-1,1) for k=2,3,…,n. Prove that \(\displaystyle \lim_{n\to \infty} e_n=\frac1e\).

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