Prove that for every positive integer k, there exists a positive Fibonacci number divisible by k.
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Prove that for every positive integer k, there exists a positive Fibonacci number divisible by k.
Let n>1 be an integer and let x>1 be a real number. Prove that if
\(\sqrt[n]{x+\sqrt{x^2-1}}+\sqrt[n]{x-\sqrt{x^2-1}}\)
is a rational number, then x is rational.
Prove that each positive integer x can be written as a sum of at most 53 integers to the fourth power. In other words, for every positive integer x, there exist \(y_1,y_2,\ldots,y_k\) with \(k\le 53\) such that \(x=\sum_{i=1}^k y_i^4\).
If we choose n points on a circle randomly and independently with uniform distribution, what is the probability that the center of the circle is contained in the interior of the convex hull of these n points?
Let \(a_0=a\) and \(a_{n+1}=a_n (a_n^2-3)\). Find all real values \(a\) such that the sequence \(\{a_n\}\) converges.
Let \(\mathcal F\) be a collection of subsets (of size r) of a finite set E such that \(X\cap Y\neq\emptyset\) for all \(X, Y\in \mathcal F\). Prove that there exists a subset S of E such that \(|S|\le (2r-1)\binom{2r-3}{r-1}\) and \(X\cap Y\cap S\neq\emptyset\) for all \(X,Y\in\mathcal F\).
Let \(a_1<\cdots\) be a sequence of positive integers such that \(\log a_1, \log a_2,\log a_3,\cdots\) are linearly independent over the rational field \(\mathbb Q\). Prove that \(\lim_{k\to \infty} a_k/k=\infty\).
Let \(a_1\le a_2\le \cdots \le a_n\) be integers. Prove that
is an integer.
Let X and Y be independent and identically distributed random variables with real values. Prove that if E(X) is finite, then E(|X+Y|)≥E(|X-Y|).
Find all real numbers \(\lambda\) and the corresponding functions \(f\) such that the equation