qid	term	tex	More tex formulas follow ...	
A.201	invertable, matrix, division ring, inverse, elementary			
A.202	rings theory, fields	F \subset D \subset E		
A.203		-(-x) = x		
A.204	morphisms, agree, residue fields, schemes, locally closed subscheme	Y \rightarrow Y \times_Z Y		
A.205		x^n = n^x		
A.206	proof, prove, show	\lim_{n \rightarrow \infty} \left(1+\frac{1}{n}\right)^n=e		
A.207	find function asymptotically equivalent	f(N) \sim \sum_{k \geq 0}\frac{k!}{N^k}		
A.208	approximation, asymptotics, harmonic number, derived, proof, prove	\sum_{i=1}^n \frac{1}{i^k}	H_n^{(k)}=n^{-k}    \left(-\frac{n}{k-1}+\frac{1}{2}-\frac{k}{12    n}+O\left(\frac{1}{n^3}\right)\right)    +\zeta (k)	
A.209	Evaluate, calculate, solve, integral	\int_0^\infty e^{-hx^2}\;\mathrm{d}x	\frac{\sqrt{\pi}}{2\sqrt{h}}	
A.210	extremas, max, show, prove	x(1-x) \leq \frac14	f(x) = x(1-x)	
A.211	MIT integration bee	\int\sqrt{x^2\sqrt{x^3\sqrt{x^4\sqrt{x^5\sqrt{x^6\sqrt{x^7\sqrt{x^8\ldots}}}}}}}\,dx		
A.212		f(x)=\sum_{n=0}^{\infty}(2n+1)(2x)^{2n}		
A.213		\sum_{x=1}^{\infty} \frac{(x-1)}{2^{x}}		
A.214		\int\limits_{-\infty}^\infty e^{-\pi x^2}dx = 1		
A.215	everywhere differentiable, set of derivative discontinuities, first category, 1st, prove, proof, union, nowhere dense, interval			
A.216		\lim\limits_{t \to + \infty} \int_0^{+ \infty} \frac{ \mathrm d x}{e^x+ \sin tx}		
A.217	maximum max area, isosceles triangle, inscribed in ellipse			
A.218	Ramsey, identity	(m, n) \leq \binom{m+n-2}{m-1}	R(m, n) \leq R(m-1, n)+R(m, n-1)	
A.219		\dbinom{n+r+1}{r}=\sum_{k=0}^{r}\dbinom{n+k}{k}		
A.220		\sum_{k=0}^{n} \binom{x+k}{k}=\binom{x+n+1}{n}		
A.221		g(n) \le \sum_{k=1}^{\ell(n)} \left( g(k) + 1 \right)		
A.222		(x+x^2/2!+x^3/3!+...)^4	(e^x-1)^4	
A.223		\sum_{i=0}^n {n+i\choose i}\frac{1}{2^i} = 2^n		
A.224		|f(x) - f(x_0)|&lt;\varepsilon	\epsilon - \delta	
A.225		\alpha^3-3\alpha^2+5\alpha-17=0\tag{1}	\beta^3-3\beta^2+5\beta+11=0\tag{2}	
A.226		x^{x^y}=y		
A.227		1+2+3...=-\dfrac{1}{12}	\frac{x(x+1)}{2}	
A.228		\int_{-\infty}^{+\infty} f(t+x) f(t+y) d t		
A.229	function intergral, fractional part	\int_{0}^{1}\big\lbrace\frac{1}{x}\big\rbrace \big\lbrace\frac{1}{1-x}\big\rbrace \big\lbrace1-\frac{1}{x}\big\rbrace dx		
A.230	definition, Ramsey number, colour, color, permutation, symmetricity, multi			
A.231	1 divided by aleph null; undefined; reciprocal of transfinite (cardinal) number	\frac{1}{\aleph_0}		
A.232	Induced Matrix Norm	\left\lVert A \right\rVert =\max_{\Vert \mathbf w\Vert = 1}\Vert A \mathbf w\Vert.		
A.233	tangent spaces, Smooth Manifolds, directional derivative, why action, smooth functions, vector			
A.234	Sards theorem, polynomial, measure, zero	Z = \{a \in X : f'(a) = 0\}		
A.235	solve	ax^2 + bx + c = y^2		
A.236	normalizing, Dirichlet distribution, beta, Gamma, area, triangle, dimensions			
A.237	rewritepropositions without implication	p \land \lnot q \land r		
A.238	Definition of Equivalence Relation, set, reflexive, symmetric, transitive			
A.239	prove reflexive, symmetric transitive			
A.240	derive discriminant	\Delta=b^2-4ac		
A.241	deivides, divisors, divisible	n^3 +6n^2-7n		
A.242	deivides, divisors, divisible, prime greater than 3 24, evenly	p^2-1=(p+1)(p-1)		
A.243	Pell equations, divisors, divisible 	2^{2k}-x^2 \mid 2^{2k}-1		
A.244	Cardinality, quotient set, Equivalence relation			
A.245	find	f(f(x))=\sin x		
A.246		\frac1{A_1A_2}=\frac1{A_1A_3}+\frac1{A_1A_4}		
A.247	endpoints, maximal arc, circle, eye, circular arc, screen. co-ordinate, coordinates, visible			
A.248	abelian group, even order, product, is not identity element, iff, if and only if, Wilson criterion	M=\{g\in G:g^2=e\}		
A.249	abelian, group, commutative	(G, * , e)		
A.250	proof, prove, show, GM-AM inequality	\frac{x_1+ \ldots + x_n}{n} \geq \sqrt[n]{x_1 \cdots x_n} 		
A.251	induction	(k+1)^{\frac{1}{k+1}} < k^{\frac{1}{k}}		
A.252	redundant, unexpected, include, appearance	\zeta(2)=\frac1{1^2}+\frac1{2^2}+\frac1{3^2}+\cdots = \frac{\pi^2}{6}	\frac 16\pi^2	\frac 16 \pi^2
A.253	contour integration	\int_{-\infty}^{\infty} \frac{\cos (bx) - \cos (ax)}{x^2} dx = \pi (a-b)		
A.254	solve	\int_0^\pi \frac{x }{1- \sin x \cos x}\ dx		
A.255	Integral	 \int_{-\infty}^\infty e^{-u^2} du		
A.256	factorize the function before taking limit			
A.257	taking a limit a function, ternary operation, group, homomorphism	\lim_{x \rightarrow k} (fg)(x) = \lim_{x \rightarrow k} f(x)g(x),	\lim_{x \rightarrow k}\ c(f(x)+g(x)) = c \lim_{x \rightarrow k} f(x) + c\ \lim_{x \rightarrow k} g(x),	
A.258	proof, prove, show, other approach	\lim_{x \to \infty} \frac{x^k}{e^x}=0		
A.259	faster, grows, slower	\lim_{n \rightarrow \infty} \frac{2^n}{n^c} = 0		
A.260	polar coordinates	\lim_{(x,y)\rightarrow (0,0)}\frac{xy^{2}}{x^{4}+y^{2}}		
A.261	matrix, diagonal, field	\det(T_n)=\sum_{k=0}^{n}\alpha^{n-k}\beta^k		
A.262	positive and negative parts, real number	a = a^+ - a^-		
A.263	formal definition, rule inference,  logicalform, concepts, notations, same, different			
A.264	Modulo Power Arithmetic	a^b \pmod x = a^{b\pmod x} \pmod x		
A.265	cannot be integer, prime, 2	(a/b)^2		
A.266	"mixed number" notation, improper fractions, process, terminate, iterate, what happens	4 \frac{2}{3}		
A.267	dual of Lagrange dual, primal, duality	L(x,\lambda) = f_0(x) + \sum_{i=1}^m \lambda_i f_i(x)		
A.268	Poisson, expected value	E\left[\frac{1}{1+Z}\right]=\sum_{k=0}^\infty\frac{1}{1+k}\frac{e^{-\lambda}\lambda^k}{k!}		
A.269	conditional probability, both, know one, children, boy, family			
A.270	fair coin, repeatedly until before, turns up five times, expected number of heads			
A.271	Gaussian, mean, zero 0, probability, limit, sup, lim, Borel Cantelli			
A.272	show, not true, when both negative	\sqrt{ab} \neq \sqrt{a}\sqrt{b}		
A.273	decimal, fractional exponents, radicals, inverse, roots exist	\sqrt[\cfrac{b}{a}]{x}=x^{\cfrac{a}{b}}		
A.274	why prove proof show, Heaviside distribution, belong to any Sobolev space	H^{s}(\mathbb{R})		
A.275	why prove proof show	\lim_{A \rightarrow \infty} \int_0^A \frac{\sin(x)}{x} dx = \frac{\pi}{2}		
A.276	proof, prove, show, other approach	\lim_{x \to \infty} \frac{x^k}{e^x}=0		
A.277	AM-GM inequality, sufficient necessary condition, exist sequence positive real numbers			
A.278	find differentiable function, such that  	f(\mathbb Q) \subseteq \mathbb Q	f'(\mathbb Q) \not \subseteq \mathbb Q	
A.279	proof prove show, if, then	\lim_{x\to 0}\left(f(x)+\frac{1}{f(x)}\right)=2.	\lim_{x\to 0}f(x)=1	
A.280	derivatives in discontinuous functions			
A.281	proof prove show	n^{\frac{1}{n}} \rightarrow 1	\lim_{n\rightarrow \infty} n^{\frac{1}{n}} = 1	
A.282	proof prove show, Functional Analysis	\sum_{n=1}^{\infty} \frac{1}{n^2} = \frac{ \pi^2 }{ 6 }		
A.283	sequence converges or diverges, bounded	x_{n+1} = x_n + \frac{1}{x_n}		
A.284	proof rational number, decimal digits don't repeat, periodically			
A.285	determine the sum of a series that is quadratic or cubic, not geometric nor arithmetic	\sum_n a n^2 + bn + c		
A.286	series converges	\lim n a_{n}=0	\sum a_{k}	
A.287	Evaluate	\sum_{i=1}^n\frac{2i}{2^i}		
A.288	alternating sign changing Harmonic Series	\sum\frac{\sin(n)}{n}		
A.289	general	n=x^n+y^n+z^n		
A.290	divisible, induction, 10	7 \times 11^{2n+1}-3^{4n-1}		
A.291	even, odd, diverging, converging	\sum_{n=0}^\infty \frac{\prod_{k=0}^n(2k+1)^2}{(2n+3)!}=\frac{1^2}{3!}+\frac{1^23^2}{5!}+...		
A.292	series, for all	\lim_{n \to \infty} \frac{n^k}{x^n}		
A.293	re-indexing, reindexing	\sum_{j=2}^{n-1}j^2		
A.294	hypercohomology hyper, sheaf, isomorphic, cohomology, resolution, existence, derived category			
A.295	use power series	\sum_{n=1}^{\infty} n(n+1)x^n	\sum_{n=1}^{\infty} \frac{n(n+1)}{3^n}	
A.296	Taylor series, well defined, point	e^x=\frac {x^0}{0!}+\frac {x^1}{1!}+ ...	0^0	
A.297	Find all integers, solutions, satisfies, such that, Euler Phi function value, 320	\phi(n)=320		
A.298	Dirac delta function, Hubble, comoving, redshift, understand, interpret, transformation	\delta(a\chi(z)-b) \rightarrow \delta(z-c)		
A.299	proof prove show, without calculating	\cos \frac{\pi}5-\cos \frac{2 \pi}5= 0.5		
A.300	uniformly continuous, on	\frac{1}{1+\log^2 (x)}		
