MathSE ID Math Question Tag 2488771 What is the general formula for the coefficient of $x^k$ in the expansion of the trinomial $(1+x+x^2)^n$? Unambiguous 746518 How do I do question 8(i) in this picture? I have $x=\delta\tan(\theta)$, but I'm not sure how to turn this into a PDF. Ambiguous 215089 Subset of Power Set of Natural Numbers Ambiguous 219343 How can I Prove that $[(p \to\neg q) \wedge q] \to \neg p$ is a tautology? Ambiguous 1054529 What is the purpose of mathematical induction in proving statements about integers? Unambiguous 1898365 What are the properties of the tensor product of two abelian groups? Unambiguous 2274060 If $x$ and $y$ are positive real numbers, what is the minimum value of the expression $(x+y)\left(\frac{1}{x} + \frac{1}{y}\right)$? Unambiguous 3198849 Evaluate the limit as $x$ approaches $0$ of $(sin(x) - x) / x^3$ using power series without L'Hospital's Rule. Unambiguous 1779361 How to prove $\cos^4x - \sin^4x - \cos^2x + \sin^2x$ is always $0$? Ambiguous 2152735 "What does ""open set"" mean in the concept of a topology?" Ambiguous 1101966 Provide a step-by-step proof that the definite integral $\int_0^1 \frac{x^3\arctan x}{(3-x^2)^2\sqrt{1-x^2}} \, dx$ equals $\frac{\pi\sqrt{2}}{192}(18-\pi-6\sqrt{3})$. Unambiguous 160327 What is the mathematical symbol for the universal quantifier 'for every element' in predicate logic? Unambiguous 2596764 If a differentiable function $f: \mathbb{R} \to \mathbb{R}$ satisfies $\lim_{x \to \pm\infty} f(x) = -\infty$, must the function have a global maximum? Unambiguous 590489 What are some topologies that work on functions? Ambiguous 1761660 How to proceed from the discriminant $1+4y(y+a)\geq 0$ to find the number of integral values of $a$? Ambiguous 279233 Is $m^4+4^n\equiv 1\pmod 8$ an iff condition? Ambiguous 769619 For which Banach spaces $X$ is the closed unit ball of the dual space $X^*$ strictly convex? Unambiguous 3806534 If $\lim_{x\to 0}\left(f(x)+\frac{1}{f(x)}\right)=2,$ show that $\lim_{x\to 0}f(x)=1$. Ambiguous 1890614 What are the mathematical definitions of infinity in set theory and calculus? Unambiguous 217291 How do I find the mathematical expressions for the red and green lines in my image to put into pgfplots? Ambiguous 1726540 How do I find the derivative of the function $f(x) = x^{1/x}$ with respect to $x$? Unambiguous 1409828 How to get the center point of the red area where my ellipses intersect? Ambiguous 1568161 Evaluate the limit as $x$ approaches $\pi$ of $\frac{(\pi - x) \sin(x)}{1 + \cos(x)}$. Unambiguous 884228 For the density f(x) = 1/(b-a) on [a,b], what is R(t)? Ambiguous 2170954 Let $p(n)$ denote the productivity of the most productive $n$-state Turing machine, where productivity is the length of the output string if the machine halts in standard position and $0$ otherwise. Prove that $\forall n\, [p(n+11) \ge 2n]$. Unambiguous 2657826 Given $P(x) = ax^3 + 3x^2 + bx + 3$ with $P(1) = 0$ and remainder $15$ when divided by $(x-2)$, find $a$ and $b$, and solve the inequality $P(x) \le 0$. Unambiguous 2426632 What is the difference between curvature and torsion for a space curve in differential geometry? Unambiguous 571232 Show that a natural number $a$ with base-$10$ representation $(r_k r_{k-1}\cdots r_1 r_0)_{10}$ satisfies $4 \mid a$ if and only if $4 \mid (r_1 r_0)_{10}$. Unambiguous 2007373 Basic Geometric intuition, context is undergraduate mathematics Ambiguous 541081 In the Martingale betting strategy, when is it mathematically optimal to stop doubling down on losses? Unambiguous 2323384 equation of common tangent touching circle and the parabola Ambiguous 1594324 Find all real numbers x such that $|x| 0$ and $N \in \mathbb{N}$, the sum $\sum_{i=0}^{N} \frac{1}{\sqrt{2\pi\sigma^2}} e^{-(i+0.5)^2/(2\sigma^2)} < 0.5$. Unambiguous 3984395 How to prove my integral $\Gamma(x)$ decays exponentially bounded by $A_d e^{-|x|/L}$? Ambiguous 1524225 Find the least $n$ such that the expression is divisible by $700$. Ambiguous 685435 How can I find an upper bound for the remainder term of the Riemann zeta function series for $\zeta(2)$ after $n$ terms? Unambiguous 4841809 Why is it a problem that the echelon form isn't unique? Ambiguous 612241 Evaluate $\lim_{x\rightarrow 0} \frac{\sin x}{x + \tan x} $ without L'Hopital rule. Unambiguous 2617141 Why is the equation of a circle not a function of $x$, and how do I show this algebraically using the vertical line test or solving for $y$? Unambiguous 44348 Simplify $||x-5|-10|<\epsilon$. Ambiguous 483708 How to prove the identity $\frac{\eta^{14}(q^4)}{\eta^{4}(q^8)}=4\eta^4(q^2)\eta^2(q^4)\eta^4(q^8)+\eta^4(q)\eta^2(q^2)\eta^4(q^4)$ where $\eta(q)$ is the Dedekind eta function? Unambiguous 571232 How do I show that 4|$a$ iff 4|($r_{1}$$r_{0}$)$_{10}$? Ambiguous 1896418 For $n\ge5$ points in the plane with no four collinear, find a closed-form expression for $f(n)$, the maximum number of points guaranteed such that no three of them are collinear, over all such configurations. Unambiguous 3330117 Limit involving prime and composite numbers Ambiguous 3390858 Find the range of values of $k$ for which $kx^2 + 8x + k <6$ for all real values of $k$ Ambiguous 4726367 How do people create trigonometric questions from scratch? Ambiguous 1025196 Let $f:[-1,1]\to\mathbb{R}$ be continuous with $f(-1)=f(1)$. Prove that there exists $y\in[0,1]$ such that $f(y)=f(y-1)$. Unambiguous 4351550 Intersection of perpendicular bisector and circle in a triangle. Ambiguous 3026909 How many presentations are there for $D_8$ using permutations, and can it be done with other groups? Ambiguous 4734106 Are there numerical approximations or algorithms for computing the square root of a number that are computationally faster than the standard library function? Unambiguous 1730258 What is the formula for the arc length of a parabola given by $y = ax^2$ from $x = 0$ to $x = b$? Unambiguous 2092731 Find the radius of convergence of $\sum_{n=1}^\infty \frac{(3n+1)^{10}x^n}{2n^n}$ using the root test. Unambiguous 2587259 How many permutations are there for a 9-player baseball batting order? Unambiguous 215332 Coefficent of an equation Ambiguous 3149675 Open ball in $M=\mathbb{R}\setminus\{-1,1\}$ Ambiguous 2669535 I can show that $\overline{R(A')} \subseteq N(A)^{\perp}$, but I'm not sure how to prove the converse using the fact that $R(A)$ is Banach. Ambiguous 1150363 Is the real number $\pi$ equal to the angle $180^\circ$ in terms of their numerical value? Unambiguous 1810374 How many distinct 7-note musical scales can be formed by choosing subsets of notes from the 12-note chromatic scale? Unambiguous 1745083 What is the optimal path between $2$ fixed points around an invisible obstructing wall? Ambiguous 360449 What does $x \ne y \ne z$ mean? Ambiguous 2660495 Probability of Choosing Exactly 1 pair out of 4 Ambiguous 4083792 Conditional expectation based on two bivariate normal RVs? Ambiguous 3017130 How do you factorize the quadratic expression $m^2 + 2ABm + B^2$? Unambiguous 348715 Evaluate the limit as $x$ approaches infinity: $\lim_{x \to \infty} \left( \sqrt{x^2+x} + 2\sqrt{x^2+2x} - x \right)$ Unambiguous 4520880 Why does the distributive law not apply to the expression $2(2.49 \times 10^8)$ when there is only one term inside the parentheses? Unambiguous 1114879 How to algorithmically detect if two ellipses in a 2D plane intersect? Unambiguous 102524 Finding the formula of an nonlinear function Ambiguous 2334855 What are the bounds of $$\frac{x+1}{xy+x+1}+\frac{y+1}{yz+y+1}+\frac{z+1}{zx+z+1}$$? Ambiguous 1325994 What is the last digit of 347 raised to the power of 61 in base 10? Unambiguous 956087 I found that y' = (1-y)/(x-2). How do I find the tangent line from here? Ambiguous 3542957 What is the difference between $-i$ and $j$? Ambiguous 68897 What are the properties and definitions of a symmetric positive definite matrix in linear algebra? Unambiguous 188945 What are the epistemological and pedagogical disadvantages of relying exclusively on computer algebra systems for mathematical proof and computation compared to manual derivation? Unambiguous 2894360 Provide a step-by-step proof to show that the equation $\frac{x + \frac{1}{2}}{\frac{1}{x} + 2} = \frac{x}{2}$ holds for all $x \neq 0, -\frac{1}{2}$. Unambiguous 1513096 Prove that the stereographic projection of the upper hemisphere from the North Pole is orientation-preserving if and only if the dimension n of the sphere $S^n$ is even. Unambiguous 2140493 Counterintuitive examples in probability Ambiguous 3178233 Prove that there is $c \in (0,1)$ such that $e^{f'(c)}f(0)^{f(c)}=f(1)^{f(c)}$ Ambiguous 1668681 An example that a 2D shape with its centre of mass on its boundary Ambiguous 767709 How to show a function with $x^n$ is uniformly convergent on $[0,1]$? Ambiguous 1040278 Nobody told me that self teaching could be so damaging... Ambiguous 3772635 Find all real solutions to the system of equations $\sqrt{x} + y = 7$ and $x + \sqrt{y} = 11$. Unambiguous 2431076 Prove that if $\sigma^r$ is an $n$-cycle for some permutation $\sigma \in S_m$, then $\sigma$ itself must be an $n$-cycle. Unambiguous 2867689 For the equation $6x^2+8xy+4y^2+17y-6=0$, use implicit differentiation to find $\frac{dy}{dx}$, isolating all terms containing $\frac{dy}{dx}$ on one side. Unambiguous 1901862 For what smooth functions is there a nice closed form for a certain exponential integral, invertible in terms of the upper limit? Ambiguous 2445959 find all natural number such that $n-2\mid n^2-2$ Ambiguous 982093 Proof for $\lim_{n \to \infty}(1+\frac{a_n}{n})^n = e^a$ given that the sequence $a_n$ converges to $a$ as $n \to \infty$. Unambiguous 1554069 On a cubic curve $C$ defined by an elliptic curve group law, prove that the tangent at a point $P$ intersects $C$ at $P$ with multiplicity 2 and at the point $4P$, and that the tangent at $5P$ intersects $C$ at $5P$ with multiplicity 2 and at the point $2P$. Unambiguous 3804598 In a building with elevators following uniform distributions, if the average wait time with two is 1 minute, what is it with one? Ambiguous 128823 What is the limit of $n\int_{0}^{1} x^n f(x) \text{d}x$ as $n\rightarrow \infty$ for a continuous function $f$ on $[0, 1]$? Unambiguous 3570162 $\exists b\in S_2\ \forall a\in S_1: p(a,b)$ Ambiguous 3024780 Given three points forming a triangle in the Cartesian plane, how do you calculate the coordinates of the Fermat point (the point minimizing the sum of distances to the vertices)? Unambiguous 2062239 How to construct the polar decomposition for Wegge-Olsen Exercise 1.A.c? Ambiguous 2599280 How to find $m$ and point $P$ for a tangent line without using differentiation at all? Ambiguous 2538727 Show that the space $X = [0,1]^{[0,1]}$ (the set of all functions from $[0,1]$ to $[0,1]$ equipped with the product topology) is not sequentially compact. Unambiguous 574475 Given the points $P=(2,1)$ and $A=(-1,5)$, find the correct parametric equations of the line through $P$ in the direction of $A$. Unambiguous 2490647 What is the asymptotic behavior of the number of involutions on a set of n elements as n approaches infinity? Unambiguous 2296759 How to prove $B$ is a symmetric matrix from its inner product? Ambiguous 590489 What are the most common topologies used on function spaces, such as the compact-open and product topologies? Unambiguous 3906075 Number of non-identity elements of order $2$ in this set. (Related to abelian group of order $34.$) Ambiguous 1762325 Is the formula $\neg \neg p \rightarrow p$ a tautology in classical logic, and is it provable in intuitionistic logic? Unambiguous 1222768 In the context of mathematical analysis, why are vector-valued functions considered well-defined functions, whereas multivalued functions are not considered functions in the strict set-theoretic sense? Unambiguous 547928 What is the minimum value of $a+b+\frac{1}{ab}$ for positive real numbers $a, b$ such that $a^2 + b^2 = 1$? Unambiguous 1051513 What is the probability that three of the selected items had values less than or equal to five? Ambiguous 1405562 What is the method for calculating the remainder of a large integer when divided by a modulus (modular exponentiation or modular reduction)? Unambiguous 1429993 What is the value of an infinite factorial tower? Ambiguous 541081 When to stop doubling down? Ambiguous 3962287 Find the Maclaurin series expansion of the function $y(t) = (t^2 + t - 2)e^{t^2 + 1}$ centered at $t = 0$. Unambiguous 2332969 Find the radius of convergence of the power series $f(x) = \sum_{n=2}^\infty \log(n) x^n$. Unambiguous 1695625 Does $h_{a,b}(x) = ((ax + b) \bmod 2^w)/2^{w-M}$ satisfy the Uniform Difference Property? Ambiguous 2051197 Can someone explain this formula? Ambiguous 602143 Find the area bounded by the curves $y=e^x$, $y=xe^x$, and $x=0$ Ambiguous 1799824 Determine whether the function $f: \mathbb{R}^2 \to \mathbb{R}^2$ defined by $f(x,y) = (x^2-y^2, 2xy)$ is injective. Unambiguous 1663192 Given a linear map with an associated matrix, with respect to which basis is the matrix representation taken? Ambiguous 291038 Let $A$ and $B$ be a toeplitz and symmetric positive definite $NxN$ matrices. If $\kappa (A) > \kappa (B)$, how to show that: $$\kappa (B^{-1}A) < \kappa (A)$$ where $\kappa $(X) is condition number of matrix $X$? Unambiguous 4095882 I have some people each performing repeated trials. How many will get at least a certain number of successes? Ambiguous 526574 I need to know if every element is either countable or co-countable. Ambiguous 1403539 $\arctan (x) + \arctan(1/x) = \frac{\pi}{2}$ Ambiguous 4628757 Prove $\frac{ab^2+2}{a+c} +\frac{bc^2+2}{b+a} +\frac{ca^2+2}{c+b} \geq \frac{9}{2}$ for $a,b,c\geq1$ Ambiguous 546701 Find the number of positive integers whose digits add up to 42 Ambiguous 1475726 Locating three sets of collinear points Ambiguous 2431076 How do I show that if $\sigma^k$ is an $n$-cycle, then $\sigma$ is also an $n$-cycle? Ambiguous 536735 Find all functions $f: \mathbb{Q} \to \mathbb{R}$ that satisfy the Cauchy functional equation $f(x+y) = f(x) + f(y)$ for all $x, y \in \mathbb{Q}$. Unambiguous 1550929 Determine if a 4-tuple exists Ambiguous 840522 Need explanation for simple differential equation Ambiguous 2865778 Find $a$ such that $$f(x)=\left\lfloor \frac{1}{3}+2a\sin^3x\right\rfloor$$ satisfies a symmetry condition. Ambiguous 1510422 How can I show $f(x)$ is zero if $\int_a^b f(x)g(x) dx=0$ for each continuous $g$? Ambiguous 1601141 How to calculate the character of this series $\displaystyle{\sum _{n=0}^{\infty }\left(-1\right)^n\left(\frac{\ln n-1}{n^3+1}\right)}$? Ambiguous 2885878 Distance between two stations Ambiguous 2512671 I need to find a function in order to satisfy $f(f^{-1}(W)) \subseteq W$. Ambiguous 1542105 For a four digit number using {2,3,5,7,9}, is the last digit choice 3 if the remainder is 2 when divided by 3 or 5? Should the answer be (4)(3)(2)(3)? Ambiguous 970872 Provide a mathematical proof that the standard solution to the 'Burning Ropes' brain teaser is correct or incorrect, specifically regarding the method of measuring 45 minutes using two non-uniform ropes. Unambiguous 1900241 How to find the expected value of a discrete random variable using its probability generating function? Unambiguous 254347 Prove that the group isomorphism $\mathbb{Z}^m \cong \mathbb{Z}^n$ implies that $m = n$ Ambiguous 2775319 For a polynomial $f(x)$, is the root of $f''$ always equal to $\frac13$ of the sum of the roots of $f$? Ambiguous 2865778 Find the range of real values for $a$ such that the function $f(x) = \lfloor \frac{1}{3}+2a \sin ^3 x \rfloor$ is an even function. Unambiguous 351359 How do I prove my metric space problem using sequential criteria? Ambiguous 2093269 Formulas to find bisector coordinates from 3 points without using matrices or vectors? Ambiguous 646238 How can I prove to my students that the value of a definite integral is independent of the dummy variable of integration? Unambiguous 2634909 How to prove that $e^x > x$ for all real numbers $x$? Unambiguous 2619473 Suppose a power series converges at its radius of convergence $R=1$. Does the derivative exist at 1? If not, are there counter-examples? Ambiguous 264371 Fun but serious mathematics books to gift advanced undergraduates. Ambiguous 2387841 Are the remainders $\beta \mathbb{N} \setminus \mathbb{N}$ and $\beta K_0 \setminus K_0$ homeomorphic? Ambiguous 672529 Verify that $-3x+5y-6z=-8$ is the correct equation of the plane consisting of all points equidistant from $(4,3,-2)$ and $(7,-2,4)$. Unambiguous 5877 What is an efficient algorithm to find two integers $x$ and $y$ such that $x^2 + y^2 = p$ for a given prime $p \equiv 1 \pmod 4$? Unambiguous 772475 Is it true that for a differentiable function $f(x)$, $\lim_{x\to\infty} f(x)=0$ implies $\lim_{x\to\infty} f'(x)=0$? If not, provide a counterexample. Unambiguous 4401632 This is a surface on a sphere. What are the equations of the surface in yellow? Ambiguous 3348703 How to perform bivariate interpolation to find the function $y(x,z)$ from a given set of scattered data points? Unambiguous 4008509 Find the critical points of the multivariable function $f(x,y) = (x-2)(y-2)(x+y-2)$. Unambiguous 1822028 Is the closed convex hull of a compact set in a dual space compact? Ambiguous 2361362 How is this a bijection? Ambiguous 2396015 What is the probability that the sum of three fair six-sided dice is eleven? Unambiguous 4253329 For what values of p is the series $\sum_{n=2}^\infty \frac{1}{n(\log n)^p}$ convergent or divergent, and how do I prove it? Unambiguous 1779361 How to prove that the trigonometric identity $\cos^4x - \sin^4x - \cos^2x + \sin^2x = 0$ holds for all real numbers $x$? Unambiguous 614812 How do you derive the formula $2\cdot(2\cdot(n/4) + \min(n\%4, 2))$? Ambiguous 2768349 Evaluate the limit as $n$ approaches infinity of $\left(1 + \frac{a}{n^2}\right)^{n/b}$, assuming $a$ and $b$ are constants with $b \neq 0$. Unambiguous 2591017 What is the definition and formula for the nth roots of unity in the complex plane? Unambiguous 1987713 If a continuous function $f: \mathbb{R} \to \mathbb{R}$ has a square $f^2$ that is uniformly continuous, must $f$ itself be uniformly continuous? Unambiguous 119307 Cauchy Sequence that Does Not Converge Ambiguous 1295010 What are statements about the natural numbers where induction is impossible or unnecessary to prove? Ambiguous 3178788 How do I determine the type of stationary point without calculating second derivative? Ambiguous 2346308 What are the parts of a number in a given base called? Ambiguous 2298323 How to merge $C_1, C_2, \dots, C_k$ to form $C_m = C(D)$ for my vertically divided dataset? Ambiguous 916805 Prove that for a subset $X$ of a metric space, the distance function $d(a, X) = \inf\{d(x, a) : x \in X\}$ equals $0$ if and only if $a$ is in the closure of $X$. Unambiguous 2138866 $\int_{0}^{\frac{\pi}{4}}\frac{\tan^2 x}{1+x^2}\text{d}x$ on 2015 MIT Integration Bee Ambiguous 2492396 How do I transform the first-order differential equation $(\sin^2y + x \arctan y)y' = 1$ into the form $M(x,y)dx + N(x,y)dy = 0$ to apply Lagrange's method for finding an integrating factor? Unambiguous 218977 For $\Omega = (-1,1)\times(-1,1)$ and the Dirac delta $\delta_0$ defined by $\langle \delta_0, \phi\rangle = \phi(0)$ for all $\phi \in C_0^\infty(\Omega)$, determine the smallest distribution space containing $\delta_0$. Unambiguous 2769775 Let $Z=X/(X+Y),$ find the pdf of $Z$. Ambiguous 2121133 Given $N$ numbers, how many arrangements are strictly increasing or decreasing? Ambiguous 2278214 Let $Y$ be a discrete random variable. What is $H(Y)$? Ambiguous 2309119 Prove that the polynomial equation $r^3-7r+1=0$ has no rational roots. Unambiguous 2934174 Find all pairs of prime numbers $(p, q)$ that satisfy the equation $pq = 3p + 7q$. Unambiguous 1694294 Show that $\pi(p_x^2) \geq 2p_x -2$ Ambiguous 2093269 What is the formula to find the length of the internal angle bisector segment between two sides of a triangle given the lengths of those two sides and the included angle? Unambiguous 249834 If a continuous function goes from $5$ to $7$, why does it have to hit $2x+3$ somewhere? Ambiguous 2775319 What is the general formula for the second derivative of a third degree polynomial function? Unambiguous 681124 Find all $x$ satisfying $\frac{\tan x}{x}=2$, arising from setting $f'(x)=0$ for $f(x)=\frac{\sin^2x}{x}$. Unambiguous 599095 How do I calculate the required sample size to achieve a specific margin of error for a confidence interval of a population mean? Unambiguous 963503 If the sum of n vectors in a Euclidean space is zero and all their norms are equal, what can be said about their geometric arrangement? Unambiguous 1310890 Compute $\lim_{x\to\infty}\left(\sin{\frac 1x}+\cos{\frac 1x}\right)^x$ Unambiguous 889570 Prove the trigonometric identity $\cot(\theta) \sec(\theta) = \frac{1}{\sin(\theta)}$. Unambiguous 291038 Let $A$, $B$ be matrices with $\kappa(A) > \kappa(B)$. Show $\kappa(B^{-1}A) < \kappa(A)$. Ambiguous 1592652 Provide an example of a binary relation on a non-empty set that is symmetric and transitive, but not reflexive. Unambiguous 385679 What are the properties of matrices $A$ and $B$ that satisfy the commutation relation $AB-BA=B$? Unambiguous 1982237 What is the general solution to the Bessel differential equation of order n? Unambiguous 269798 How do you calculate the expected value of a continuous random variable? Unambiguous 2175735 For non-negative integers $n, p, k$, find a closed-form expression for $A(p,n,k) = \frac{(pn)!}{(p!)^k(p(n-k))!}$ and for $A(p,n) = \sum_{k=1}^n A(p,n,k)$. Unambiguous 2237243 What are effective study habits and problem-solving strategies for improving one's mathematical practice and proficiency? Unambiguous 2146495 What are the basic steps for finding the convolution of $(f*f)(t)$ for a fraction? Ambiguous 754268 What is the formal statement of the Continuum Hypothesis and its status in ZFC set theory? Unambiguous 639890 Prove that the minimum value of $\cos(A-B)+\cos(B-C) +\cos(C-A)$ for any real numbers $A, B, C$ is $-3/2$. Unambiguous