Category Archives: problem

2009-6 Sum of integers of the fourth power

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 \(x=\sum_{i=1}^k y_i^4\).

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2009-3 Intersecting family

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 \(X,Y\in\mathcal F\).

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2009-2 Sequence of Log

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 \(\lim_{k\to \infty} a_k/k=\infty\).

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