Let M=(mi,j)1≤i,j≤n be an n×n matrix such that mi,j=i(i+1)(i+2)…(i+j-2). (Note that m1,1=1.) What is the determinant of M?
loading...
Let M=(mi,j)1≤i,j≤n be an n×n matrix such that mi,j=i(i+1)(i+2)…(i+j-2). (Note that m1,1=1.) What is the determinant of M?
Prove that for every skew-symmetric matrix A, there are symmetric matrices B and C such that A=BC-CB.
Let \(t_1,t_2,\ldots,t_n\) be positive integers. Let \(p(x_1,x_2,\dots,x_n)\) be a polynomial with n variables such that \(\deg(p)\le t_1+t_2+\cdots+t_n\). Prove that \(\left(\frac{\partial}{\partial x_1}\right)^{t_1} \left(\frac{\partial}{\partial x_2}\right)^{t_2}\cdots \left(\frac{\partial}{\partial x_n}\right)^{t_n} p\) is equal to \[\sum_{a_1=0}^{t_1} \sum_{a_2=0}^{t_2}\cdots \sum_{a_n=0}^{t_n} (-1)^{t_1+t_2+\cdots+t_n+a_1+a_2+\cdots+a_n}\left( \prod_{i=1}^n \binom{t_i}{a_i} \right)p(a_1,a_2,\ldots,a_n).\]
Prove that there is a constant c>1 such that if \(n>c^k\) for positive integers n and k, then the number of distinct prime factors of \(n \choose k\) is at least k.
Let f be a continuous function on [0,1]. Prove that \[ \lim_{n\to \infty}\int_0^1 \cdots \int_0^1 f(\sqrt[n]{x_1 x_2 \cdots x_n } ) dx_1 dx_2 \cdots dx_n = f(1/e).\]
Let f(n) be the largest integer k such that n! is divisible by \(n^k\). Prove that \[ \lim_{n\to \infty} \frac{(\log n)\cdot \max_{2\le i\le n} f(i)}{n \log\log n}=1.\]
Let \(a_1\le a_2\le \cdots \le a_k\) and \(b_1\le b_2\le \cdots \le b_l\) be sequences of positive integers at most M. Prove that if \[ \sum_{i=1}^{k} a_i^n = \sum_{j=1}^l b_j^n\] for all \(1\le n\le M\), then \(k=l\) and \(a_i=b_i\) for all \(1\le i\le k\).
Find all linear functions f on the set of n×n matrices such that f(XY)=f(YX) for every pair of n×n matrices X and Y.
Added: The value f(X) is a scalar.
Let n>2. Let f (x) be a degree-n polynomial with real coefficients. If f (x) has n distinct real zeros r1<r2<…<rn, then Rolle’s theorem implies that the largest real zero q of f´(x) is between rn-1 and rn. Prove that q>(rn-1+rn)/2.
Let us write \([n]=\{1,2,\ldots,n\}\). Let \(a_n\) be the number of all functions \(f:[n]\to [n]\) such that \(f([n])=[k]\) for some positive integer \(k\). Prove that \[a_n=\sum_{k=0}^{\infty} \frac{k^n}{2^{k+1}}.\]