Let M>0 be a real number. Prove that there exists N so that if n>N, then all the roots of \(f_n(z)=1+\frac{1}{z}+\frac1{{2!}z^2}+\cdots+\frac{1}{n!z^n}\) are in the disk |z|<M on the complex plane.

The best solution was submitted by Jeong, Seong Gu (정성구), 수리과학과 2007학번. Congratulations!

Here is his Solution of Problem 2010-9.

Alternative solutions were submitted by 최홍석(화학과 2006학번, +3), 김호진(2009학번, +3), 김치헌 (수리과학과 2006학번, +3).

Let S be the set of non-zero real numbers x such that there is exactly one 0-1 sequence {an} satisfying \(\displaystyle \sum_{n=1}^\infty a_n x^{-n}=1\). Prove that there is a one-to-one function from the set of all real numbers to S.

The best solution was submitted by Jeong, Seong Gu (정성구), 수리과학과 2007학번. Congratulations!

Here is his Solution of Problem 2010-7.

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

The best solution was submitted by Jeong, Seong-Gu (정성구), 수리과학과 2007학번. Congratulations!

Here is his Solution of Problem 2010-1.

Alternative solutions were submitted by 임재원 & 서기원 (2009학번, +3 -> +2, +2 each) and 권용찬 (2009학번, +2; almost correct). Thank you for participation.

Evaluate the following limit:
\(\displaystyle \lim_{\varepsilon\to 0}\int_0^{2\varepsilon} \log\left(\frac{|\sin t-\varepsilon|}{\sin \varepsilon}\right) \frac{dt}{\sin t}\).

The best solution was “again” submitted by Seong-Gu Jeong (정성구), 수리과학과 2007학번. Congratulations!

Here is his Solution of Problem 2009-22.

Let A=(a_ij) be an n×n matrix such that a_ij=cos(i-j)θ and θ=2π/n. Determine the rank and eigenvalues of A.

The best solution was submitted by Seong-Gu Jeong (정성구), 수리과학과 2007학번. Congratulations!

Here is his Solution of Problem 2009-21.

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

The best solution was submitted by Seong-Gu Jeong (정성구), 수리과학과 2007학번. Congratulations!

Here is his Solution of Problem 2009-20.

Suppose y(x)≥0 for all real x. Find all solutions of the differential equation \(\frac{dy}{dx}=\sqrt{y}\), y(0)=0.

The best solution was submitted by Seong-Gu Jeong (정성구), 수리과학과 2007학번. Congratulations!

Here is his Solution of Problem 2009-18.

There were 2 incorrect submissions.