Prove that for all positive integers m and n, there is a positive integer k such that \[ (\sqrt{m}+\sqrt{m-1})^n = \sqrt{k}+\sqrt{k-1}.\]
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Prove that for all positive integers m and n, there is a positive integer k such that \[ (\sqrt{m}+\sqrt{m-1})^n = \sqrt{k}+\sqrt{k-1}.\]
Evaluate the sum \[ \sum_{n=1}^{\infty} \frac{n \sin n}{1+n^2}. \]
(UPDATED: 2011.2.18) I have fixed a typo in the formula. Initially the following formula \[ \sum_{n=1}^{\infty} \frac{\sin n}{1+n^2}\] was posted but it does not seem to have a closed form answer. I’m sincerely sorry!
Let \(a_1=0\), \(a_{2n+1}=a_{2n}=n-a_n\). Prove that there exists k such that \(\lvert a_k- \frac{k}{3}\rvert >2010\) and yet \(\lim_{n\to \infty} \frac{a_n}{n}=\frac13\).
Let X be a finite set of points on the plane such that each point in X is colored with red or blue and there is no line having all points in X. Prove that there is a line L having at least two points of X such that all points in L∩X have the same color.
Suppose that \(V\) is a vector space of dimension \(n>0\) over a field of characterstic \(p\neq 0\). Let \(A: V\to V\) be an affine transformation. Prove that there exist \(u\in V\) and \(1\le k\le np\) such that \[A^k u = u.\]
Let f be a differentiable function. Prove that if \(\lim_{x\to\infty} (f(x)+f'(x))=1\), then \(\lim_{x\to\infty} f(x)=1\).
Let A, B be Hermitian matrices. Prove that tr(A2B2) ≥ tr((AB)2).
Let n be a positive integer. Let D(n,k) be the number of divisors x of n such that x≡k (mod 3). Prove that D(n,1)≥D(n,2).
Let A, B be 2n×2n skew-symmetric matrices and let f be the characteristic polynomial of AB. Prove that the multiplicity of each root of f is at least 2.
Let n be a positive integer. Prove that
\(\displaystyle \sum_{k=0}^n (-1)^k \binom{2n+2k}{n+k} \binom{n+k}{2k}=(-4)^n\).