# 2015-6 Dense sets

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 $$\mathbb R$$?

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# 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?

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# 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.

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# 2008-7 Find all real solutions

Find all real solutions of $$3^x + 5^{x^2} = 4^x + 4^{x^2}$$.

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# 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.

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