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• Quadratic formula
  <cmath>x+\frac{b}{2a}=\pm\sqrt{\frac{b^{2}-4ac}{4a^{2}}}=\frac{\pm\sqrt{b^{2}-4ac}}{2a}</cmath> <cmath>x=-\frac{b}{2a}+\frac{\pm\sqrt{b^{2}-4ac}}{2a}=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}</cmath>
  2 KB (290 words) - 16:55, 19 February 2024
• Sieve of Eratosthenes
  ...umber | composite]], then it has a prime [[divisor]] not exceeding <math>\sqrt n</math>.
  1 KB (212 words) - 21:16, 7 December 2007
• Imaginary unit
  ...ex number itself. It has a [[magnitude]] of 1, and can be written as <math>1 \text{cis } \left(\frac{\pi}{2}\right)</math>. Any [[complex number]] can b #<math>i^1=\sqrt{-1}</math>
  2 KB (321 words) - 15:57, 5 September 2008
• Absolute value
  * (Alternative definition) <math>|x| = \sqrt{x^2}</math> ...ex number]]s <math>z</math>, the absolute value is defined as <math>|z| = \sqrt{x^2+y^2}</math>, where <math>x</math> and <math>y</math> are the real and i
  2 KB (368 words) - 10:37, 5 January 2009
• Rational approximation of famous numbers
  ...<+\infty</math>, the [[inequality]] <math>\left|x-\frac pq\right|\ge \frac 1{q^M}</math> holds for all sufficiently large denominators <math>q</math>. Suppose that there exist <math>0<\beta<\gamma<1</math>, <math>1<Q<+\infty</math>
  8 KB (1,431 words) - 13:48, 26 January 2008
• LaTeX:Symbols
  ...setminus</math>||\setminus||<math>\wr</math>||\wr||<math>\sqrt{x}</math>||\sqrt{x} ...rc}||<math>\triangledown</math>||\triangledown||<math>\sqrt[n]{x}</math>||\sqrt[n]{x}
  16 KB (2,324 words) - 16:50, 19 February 2024
• 2006 AMC 12B Problems/Problem 21
  \mathrm{(A)}\ \frac {16\sqrt {2006}}{\pi} \mathrm{(C)}\ 8\sqrt {1003}
  2 KB (339 words) - 13:15, 12 July 2015
• 3D Geometry
  ...he 30-60-90 triangle, or the Pythagorean Theorem, to find that <math> x = \sqrt{3} </math> units.
  2 KB (263 words) - 12:29, 30 December 2023
• Srinivasa Ramanujan
  <math>\frac{1}{\pi} = \frac{2\sqrt{2}}{9801} \sum^\infty_{k=0} \frac{(4k)!(1103+26390k)}{(k!)^4 396^{4k}}</mat
  496 bytes (62 words) - 23:16, 24 February 2007
• 2004 AIME II Problems/Problem 11
  ...xact opposite side of the cone whose distance from the vertex is <math>375\sqrt{2}.</math> Find the least distance that the fly could have crawled. ...<math>R=\sqrt{r^2+h^2}=\sqrt{600^2+(200\sqrt{7})^2}=\sqrt{360000+280000}=\sqrt{640000}=800</math>. Setting <math>\theta R=C\implies 800\theta=1200\pi\impl
  2 KB (268 words) - 22:20, 23 March 2023
• 1986 AIME Problems/Problem 1
  ...he sum of the solutions to the equation <math>\sqrt[4]{x} = \frac{12}{7 - \sqrt[4]{x}}</math>? Let <math>y = \sqrt[4]{x}</math>. Then we have
  688 bytes (104 words) - 13:34, 22 July 2020
• 1987 AIME Problems/Problem 9
  C = (21*sqrt(3),0);
  1 KB (200 words) - 18:44, 5 February 2024
• 1989 AIME Problems/Problem 12
  ...ontsize(10)); label("$7$",A--C,2*dir(210),fontsize(10)); label("$18$",A--D,1.5*dir(30),fontsize(10)); label("$36$",(3,0),up,fontsize(10)); ...h>\triangle CDM</math>. The formula for the length of a median is <math>m=\sqrt{\frac{2a^2+2b^2-c^2}{4}}</math>, where <math>a</math>, <math>b</math>, and
  2 KB (376 words) - 13:49, 1 August 2022
• 2003 AIME I Problems/Problem 8
  <cmath>d=\frac{-3a + \sqrt{9a^2+120a}}{2}.</cmath> <cmath>a=\frac{-120 + \sqrt{14400+36x^2}}{18}</cmath>
  5 KB (921 words) - 23:21, 22 January 2023
• 2003 AIME II Problems/Problem 10
  ...qrt{b}+\sqrt{b+60}=\sqrt{c}</math>. Square both sides to get <math>2b+60+2\sqrt{b^2+60b}=c</math>. Thus, <math>b^2+60b</math> must be a square, so we have
  1 KB (218 words) - 14:14, 25 June 2021
• 2005 AMC 10B Problems/Problem 4
  ...al numbers <math>a</math> and <math>b</math>, define <math>a \diamond b = \sqrt{a^2 + b^2}</math>. What is the value of ...\textbf{(B) } \frac{17}{2} \qquad \textbf{(C) } 13 \qquad \textbf{(D) } 13\sqrt{2} \qquad \textbf{(E) } 26</math>
  833 bytes (110 words) - 13:58, 24 July 2022
• Incircle
  ...formula above can be simplified with Heron's Formula, yielding <math>r = \sqrt{\frac{(s-a)(s-b)(s-c)}{s}}.</math> *The [[area]] of the [[triangle]] by [[Heron's Formula]] is <math>A=\sqrt{s(s-a)(s-b)(s-c)}</math>.
  2 KB (384 words) - 18:38, 9 March 2023
• Hypotenuse
  ...a triangle with legs of length <math>a</math> and <math>b</math> is <math>\sqrt{a^2 + b^2}</math>.
  810 bytes (133 words) - 19:02, 15 October 2018
• 1969 Canadian MO Problems/Problem 9
  ...>1,</math> the length of the shortest side is less than or equal to <math>\sqrt{2}</math>. ...ossible. Thus, at least one of the sides must have length less than <math>\sqrt 2</math>, so certainly the shortest side must.
  795 bytes (138 words) - 22:42, 17 November 2007
• Parallel
  <math>\sqrt{\frac{(y_2-y_1)^2}{m^2+1}}</math>
  2 KB (326 words) - 12:11, 21 May 2009
• Furman University Wylie Mathematics Tournament/1996 Senior Exam/Solutions
  == Problem 1 == Multiplying the denominator by <math> 1-2i </math> gives us that the expression is <math> 2+3i, </math> the coeffic
  9 KB (1,364 words) - 15:59, 21 July 2006
• University of South Carolina High School Math Contest/1993 Exam/Problem 27
  ...mathrm{(C) \ }36 \qquad \mathrm{(D) \ }12\sqrt{2} \qquad \mathrm{(E) \ }12\sqrt{3} </math></center>
  1 KB (149 words) - 16:27, 18 August 2006
• University of South Carolina High School Math Contest/1993 Exam/Problem 26
  ...Then the first nonzero digit in the decimal expansion of <math>\sqrt{n^2 + 1} - n</math> is <center><math> \mathrm{(A) \ }1 \qquad \mathrm{(B) \ }2 \qquad \mathrm{(C) \ }3 \qquad \mathrm{(D) \ }4 \qq
  2 KB (258 words) - 16:34, 18 August 2006
• University of South Carolina High School Math Contest/1993 Exam/Problem 16
  ...r><math> \mathrm{(A) \ }1 \qquad \mathrm{(B) \ }4/3 \qquad \mathrm{(C) \ }\sqrt{2} \qquad \mathrm{(D) \ }3/2 \qquad \mathrm{(E) \ }2 </math></center> ...P</math> is [[parallel]] to <math>AB</math> and thus <math>\frac{CQ}{QN} = 1</math> and <math>CQ = QN = 4</math>. Then <math>CN = CQ + QN = 8</math>.
  1 KB (221 words) - 11:45, 17 August 2006
• Distance formula
  ...math>(a_1,a_2,...,a_n)</math> and <math>(b_1,b_2,...,b_n)</math> is <math>\sqrt{(a_1-b_1)^2+(a_2-b_2)^2+\cdots+(a_n-b_n)^2}</math>. <cmath>\dfrac{|ax_1+by_1+c|}{\sqrt{a^2+b^2}}</cmath>
  2 KB (340 words) - 18:34, 8 September 2018
• 2006 Romanian NMO Problems/Grade 8/Problem 1
  ...e and a point <math>Q</math> on the circle, <math>PQ=\sqrt{PO_1^2+O_1Q^2}=\sqrt{PO_1^2+r_1^2}</math>, which does not depend on the point <math>Q</math> cho
  3 KB (509 words) - 23:22, 15 August 2012
• 1969 Canadian MO Problems/Problem 3
  ...have lengths <math>a</math> and <math>b</math>. Prove that <math>a+b\le c\sqrt{2}</math>. When does the equality hold?
  790 bytes (129 words) - 22:39, 17 November 2007
• Equilateral triangle
  ...f an equilateral triangle can be found in terms of a side: <math>\frac{s^2\sqrt{3}}{4}</math>.
  1 KB (186 words) - 19:57, 15 September 2022
• Notation
  ...easier to express using notation: <math>\displaystyle x+x^2=\frac{x}{2}+x\sqrt{x}</math>. The latter is easier to understand, and we can immediately jump
  692 bytes (109 words) - 10:28, 4 August 2006
• Galois theory
  ...ath>. Then the map <math>f:K\to K</math> given by <math>f(a+b\sqrt{2})=a-b\sqrt{2}</math> is a field automorphism; that is, <math>f(\alpha\beta)=f(\alpha)f ...he case. For example, if <math>K=\mathbb{Q}</math> and <math>L=\mathbb{Q}(\sqrt[3]{2})</math>, then <math>Gal(L/K)</math> is the trivial group, so every el
  2 KB (385 words) - 14:09, 5 May 2008
• 1969 Canadian MO Problems
  == Problem 1 == [[1969 Canadian MO Problems/Problem 1 | Solution]]
  3 KB (536 words) - 12:46, 8 October 2007
• Mock USAMO by probability1.01 dropped problems
  == Problem 1 == Let <math>n>1</math> be a fixed positive integer, and let <math>a_1,a_2,\ldots,a_n</math>
  3 KB (572 words) - 02:46, 16 May 2009
• Finite Fourier Analysis
  ...th>f(0) = f(1)</math> and <math>\frac{d^m f}{dx^m}(0) = \frac{d^m f}{dx^m}(1)</math> for all <math>m \in \mathbb{Z}^+</math> . <math>S</math> is a vect ...rac1N</math>, <math>x=\frac2N</math>, <math>\ldots</math>, <math>x=\frac{N-1}N </math>. For each <math>n \in \mathbb{Z}</math>, let <math>f_n</math> be
  4 KB (724 words) - 19:15, 9 September 2006
• Real part
  ...can be written in the form <math>z = a + bi</math> where <math>i = \sqrt{-1}</math> is the [[imaginary unit]] and <math>a</math> and <math>b</math> ar ...athrm{Re}(4(\cos \frac \pi6 + i \sin \frac\pi 6)) = 4 \cos \frac \pi 6 = 2\sqrt 3</math>
  2 KB (281 words) - 15:56, 5 September 2008
• Imaginary part
  ...can be written in the form <math>z = a + bi</math> where <math>i = \sqrt{-1}</math> is the [[imaginary unit]] and <math>a</math> and <math>b</math> ar * <math>\mathrm{Im}((1 + i)\cdot(2 + i)) = \mathrm{Im}(1 + 3i) = 3</math>. Note in particular that <math>\mathrm Im</math> is ''not
  2 KB (269 words) - 15:56, 5 September 2008
• Euclidean metric
  ...{y} = (y_1, y_2, \ldots, y_n)</math> is given by <math>d(\mathbf{x, y}) = \sqrt{(x_1 - y_1)^2 + (x_2 - y_2)^2 + \ldots + (x_n - y_n)^2}</math>. It is stra
  813 bytes (132 words) - 17:49, 28 March 2009
• Mock AIME 5 2005-2006 Problems
  == Problem 1 == ...5</math> and <math>f(101) = 0</math>). Evaluate the remainder when <math>f(1)+f(2)+\cdots+f(99)</math> is divided by <math>1000</math>.
  8 KB (1,370 words) - 21:52, 27 February 2007
• Hexagon
  [[Area]]: <math>\frac{3s^2\sqrt{3}}{2}</math> Where <math>s</math> is the side length of the hexagon. [[Apothem]], or [[inradius]]: <math>\dfrac{s\sqrt{3}}{2}</math>
  688 bytes (94 words) - 21:00, 14 December 2018
• Coefficient
  ...</math>, and <math>c</math> in the [[Quadratic Formula]], <math>\frac{b\pm\sqrt{b^2-4ac}}{2a}</math> are all coefficients of the polynomial <math>ax^2+bx+c
  1 KB (227 words) - 22:38, 6 October 2020
• Rectangular prism
  ...e interior [[diagonal]]s can be determined by using the formula <math>d = \sqrt{l^2 + w^2 + h^2}</math>. Proof: To get a base diagonal, we use the [[pythagorean theorem]]: <math> \sqrt{l^2+w^2}</math>. We call that v. Then we use the pythagorean theorem again
  883 bytes (132 words) - 09:04, 12 September 2007
• 2007 BMO Problems
  == Problem 1 == [[2007 BMO Problems/Problem 1 | Solution]]
  2 KB (297 words) - 23:44, 4 May 2007
• Specimen Cyprus Seniors Provincial/2nd grade/Problem 3
  ...lambda^2} = \sqrt{(\kappa\lambda)^2(\nu^2 + 4\nu + 4 - 4)} = \kappa\lambda\sqrt{\nu^2+4\nu}</math> <math>(\nu+1)^2 < \nu^2+4\nu < (\nu+2)^2</math>.
  1 KB (184 words) - 22:54, 22 May 2009
• Specimen Cyprus Seniors Provincial/2nd grade/Problem 4
  If <math>\rho_{1}, \rho_{2}</math> are the roots of equation <math>x^2-x+1=0</math> then: a) Prove that <math>\rho_{1}^3=\rho_{2}^3 = -1</math> and
  2 KB (217 words) - 23:28, 22 May 2009
• 2005 Alabama ARML TST Problems/Problem 3
  ...ath>r = 10</math> and the side length of the triangle is equal to <math>20\sqrt 3</math>.
  1 KB (221 words) - 19:38, 6 February 2010
• Mock AIME 6 Pre 2005/Problems
  <math>1</math>. Find the remainder when <math>\displaystyle 11^{2005}</math> is div <math>5</math>. Let <math>\displaystyle f(x)=\sqrt{x+2005\sqrt{x+2005\sqrt{\cdots}}}</math> where <math>\displaystyle f(x)>0</math>. Find the remainde
  7 KB (1,110 words) - 05:15, 31 December 2006
• Mock AIME 4 2006-2007 Problems
  ==Problem 1== ...first digit of 1. He writes <math>1, 10, 11, 12, \ldots</math> but by the 1,000th digit he (finally) realizes that the list would contain an infinite n
  6 KB (923 words) - 14:17, 16 January 2007
• Mock AIME 4 2006-2007 Problems/Problem 11
  ...e area of <math>\triangle ACF</math> is 3. If <math>\frac{CE}{EA}=\frac{p+\sqrt{q}}{r}</math>, where <math>p</math>, <math>q</math>, and <math>r</math> are ...}-3}{6}</math>, so <math>\frac{a}{b} = \frac{6}{\sqrt{129}-3} = \frac{3 + \sqrt{129}}{20}</math>.
  2 KB (325 words) - 19:33, 9 February 2017
• 2005 IMO Shortlist Problems/A1
  ...ystyle p(x)q(x) </math> is a polynomial having all coefficients <math> \pm 1 </math>. ...the constant term of <math> \displaystyle p(x) </math> must be <math> \pm 1 </math>.
  2 KB (363 words) - 22:57, 31 January 2007
• Mock AIME 3 Pre 2005 Problems/Problem 1
  ...>, <math>3+r</math>, and <math>7+r</math>. From Heron's Formula, <math>84=\sqrt{(10+r)(r)(7)(3)}</math>, or <math>84*84=r(10+r)*21</math>, or <math>84*4=r(
  795 bytes (129 words) - 10:22, 4 April 2012
• Mock AIME 4 2005-2006/Problems
  == Problem 1 == 1. A 5-digit number is leet if and only if the sum of the first 2 digits, the
  7 KB (1,176 words) - 04:44, 26 February 2007

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