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 \(\mathbb R\)?
Tag Archives: real
2008-9 Integer-valued function
Let \(\mathbb{R}\) be the set of real numbers and let \(\mathbb{N}\) be the set of positive integers. Does there exist a function \(f:\mathbb{R}^3\to \mathbb{N}\) such that f(x,y,z)=f(y,z,w) implies x=y=z=w?
2008-8 Positive eigenvalues
Let A be a 0-1 square matrix. If all eigenvalues of A are real positive, then those eigenvalues are all equal to 1.
2008-7 Find all real solutions
Find all real solutions of \(3^x + 5^{x^2} = 4^x + 4^{x^2}\).
2008-2 Strange representation (9/11)
Prove that if x is a real number such that \(0<x\le \frac12\), then x can be represented as an infinite sum
\(\displaystyle x=\sum_{k=1}^\infty \frac{1}{n_k}\),
where each \(n_k\) is an integer such that \(\frac{n_{k+1}}{n_k}\in \{3,4,5,6,8,9\}\).
x가 \(0<x\le \frac12\)을 만족하는 실수일때, x는 아래와 같은 무한급수로 표현할 수 있음을 보여라.
\(\displaystyle x=\sum_{k=1}^\infty \frac{1}{n_k}\).
여기서 각 \(n_k\)는 정수이며 \(\frac{n_{k+1}}{n_k}\in \{3,4,5,6,8,9\}\)을 만족한다.