Let n, k be positive integers. Prove that \(\sum_{i=1}^n k^{\gcd(i,n)}\) is divisible by n.
loading...
Let n, k be positive integers. Prove that \(\sum_{i=1}^n k^{\gcd(i,n)}\) is divisible by n.
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.)
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.
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.
Prove that \(\sqrt{2}+\sqrt[3]{5}\) is irrational.
Evaluate the following limit:
\(\displaystyle \lim_{\varepsilon\to 0}\int_0^{2\varepsilon} \log\left(\frac{|\sin t-\varepsilon|}{\sin \varepsilon}\right) \frac{dt}{\sin t}\).
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.
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 \(\displaystyle \lim_{n\to \infty} e_n=\frac1e\).
Let A and B be n×n matrices over the real field R. Prove that if A+B is invertible, then A(A+B)-1B=B(A+B)-1A.
Suppose y(x)≥0 for all real x. Find all solutions of the differential equation \(\frac{dy}{dx}=\sqrt{y}\), y(0)=0.