Let \(A\) be an unbounded subset of the set \(\mathbb R\) of the real numbers. Let \(T\) be the set of all real numbers \(t\) such that \(\{tx-\lfloor tx\rfloor : x\in A\}\) is dense in \([0,1]\). Is \(T\) dense in \(\mathbb R\)?
Determine all \(n\times n\) matrices A such that \( \operatorname{tr}(AXY)=\operatorname{tr}(AYX)\) for all \(n\times n\) matrices \(X\) and \(Y\).
Let \( M=\begin{pmatrix} A & B \\ B^*& C \end{pmatrix}\) be a positive semidefinite Hermian matrix. Prove that \[ \operatorname{rank} M \le \operatorname{rank} A +\operatorname{rank} C.\] (Here, \(A\), \(B\), \(C\) are matrices.)
Let \(\{a_n\}\) be a sequence of non-negative reals such that \( \lim_{n\to \infty} a_n \sum_{i=1}^n a_i^5=1\). Prove that \[ \lim_{n\to \infty} a_n (6n)^{1/6} = 1.\]
Let \(T\) be a triangle. Prove that if every point of a plane is colored by Red, Blue, or Green, then there is a triangle similar to \(T\) such that all vertices of this triangle have the same color.
Let \( A\) be a set of \(n\ge 2\) odd integers. Prove that there exist two distinct subsets \(X\), \(Y\) of \(A\) such that \[ \sum_{x\in X} x\equiv\sum_{y\in Y}y \pmod{2^n}.\]
Remark (added March 3): n is an integer greater than or equal to 2 and A is a set of n odd integers.
Suppose that \(n\) points are chosen randomly on a sphere. What is the probability that all points are on some hemisphere?
(This is the last problem of 2014. Thank you!)
Let \(f:[0,1]\to \mathbb R\) be a differentiable function with \(f(0)=0\), \(f(1)=1\). Prove that for every positive integer \(n\), there exist \(n\) distinct numbers \(x_1,x_2,\ldots,x_n\in(0,1)\) such that \[ \frac{1}{n}\sum_{i=1}^n \frac{1}{f'(x_i)}=1.\]
For a nonnegative real number \(x\), let \[ f_n(x)=\frac{\prod_{k=1}^{n-1} ((x+k)(x+k+1))}{ (n!)^2}\] for a positive integer \(n\). Determine \(\lim_{n\to\infty}f_n(x)\).
Let \(\mathcal F\) be a non-empty collection of subsets of a finite set \(U\). Let \(D(\mathcal F)\) be the collection of subsets of \(U\) that are subsets of an odd number of members of \(\mathcal F\). Prove that \(D(D(\mathcal F))=\mathcal F\).