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.
Category Archives: problem
2010-6 Identity on Binomial Coefficients
Prove that \(\displaystyle \sum_{m=0}^n \sum_{i=0}^m \binom{n}{m} \binom{m}{i}^3=\sum_{m=0}^n \binom{2m}{m} \binom{n}{m}^2\).
2010-5 Dependence over Q
Find the set of all rational numbers x such that 1, cos πx, and sin πx are linearly dependent over rational field Q.
2010-4 Power and gcd
Let n, k be positive integers. Prove that \(\sum_{i=1}^n k^{\gcd(i,n)}\) is divisible by n.
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.)
2010-2 Nonsingular matrix
Let A=(aij) be an n×n matrix of complex numbers such that \(\displaystyle\sum_{j=1}^n |a_{ij}|<1\) for each i. Prove that I-A is nonsingular.
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.
2009-23 Irrational number
Prove that \(\sqrt{2}+\sqrt[3]{5}\) is irrational.
2009-22 Integral and Limit
Evaluate the following limit:
\(\displaystyle \lim_{\varepsilon\to 0}\int_0^{2\varepsilon} \log\left(\frac{|\sin t-\varepsilon|}{\sin \varepsilon}\right) \frac{dt}{\sin t}\).
2009-21 Rank and Eigenvalues
Let A=(aij) be an n×n matrix such that aij=cos(i-j)θ and θ=2π/n. Determine the rank and eigenvalues of A.