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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]]
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• 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
• 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
• Inverse of a function
  ...ule <math>\displaystyle f(x) = x^3 + 6</math>. The function <math>g(x) = \sqrt[3]{x-6}</math> has the property that <math>f(g(x)) = x</math>. In this cas ...on <math>f</math> is denoted by <math>f^{-1}</math>. Note that the <math>-1</math> does ''not'' indicate an [[exponent]].
  1 KB (269 words) - 13:50, 5 March 2007
• 2007 AIME II Problems/Problem 11
  ...rts. The distance <math>x</math> can be expressed in the form <math>a\pi+b\sqrt{c},</math> where <math>a,</math> <math>b,</math> and <math>c</math> are [[i ...48\sqrt{3}</math>. Thus, the total horizontal distance covered is <math>96\sqrt{3}</math>.
  2 KB (407 words) - 16:31, 29 February 2020
• 1976 IMO Problems/Problem 1
  ...n any such case it is clear from the Pythagorean theorem that <math>AC = 8\sqrt 2</math>. Therefore the other diagonal has only one possible length: <math>8\sqrt 2</math>.
  1 KB (255 words) - 21:05, 26 September 2009
• 2007 Cyprus MO/Lyceum/Problem 11
  ...2006}+2006 \sqrt{2007}}</math> and <math>Y=\frac{1}{\sqrt{2006}}-\frac{1}{\sqrt{2007}}</math>, which of the following is correct? ...= \frac{\sqrt{b}}{b} - \frac{\sqrt{a}}{a} = \frac{1}{\sqrt{b}} - \frac{1}{\sqrt{a}} = Y</math>
  808 bytes (139 words) - 16:56, 6 May 2007
• 2007 Cyprus MO/Lyceum/Problem 7
  ...B) \ } \frac{d^2}{2}\qquad \mathrm{(C) \ } 2d^2\qquad \mathrm{(D) \ } d^2 \sqrt{2}\qquad \mathrm{(E) \ } \mathrm{None\;of\;these}</math> ...the sides of the rectangle are <math>\frac{d}{2}</math> and <math>\frac{d\sqrt{3}}{2}</math>.
  736 bytes (112 words) - 20:43, 6 May 2007
• 2007 Cyprus MO/Lyceum/Problem 6
  ...qrt{3}\qquad \mathrm{(D) \ } \sqrt{3} - 1\qquad \mathrm{(E) \ } \sqrt{2} - 1</math> ...triangle]]s. <math>AF = BG = \sqrt{3}</math>, and so <math>DF = CG = 2 - \sqrt{3} \Longrightarrow \mathrm{C}</math>.
  1,013 bytes (152 words) - 10:39, 8 May 2007
• 2007 Cyprus MO/Lyceum/Problem 14
  ...c{1}{\sqrt{2}}\qquad \mathrm{(D) \ } \sqrt2-1\qquad \mathrm{(E) \ } \frac{\sqrt{2}}4</math> ...of <math>S_1:S_2</math> is <math>x\sqrt{2}-x:x</math>, or <math>\sqrt{2}-1:1\Longrightarrow\mathrm{ D}</math>.
  868 bytes (143 words) - 23:02, 8 May 2007
• 2007 Cyprus MO/Lyceum/Problem 15
  ...\mathrm{(B) \ } 8\qquad \mathrm{(C) \ } 1\qquad \mathrm{(D) \ } \frac{6+2\sqrt{3}}{3}\qquad \mathrm{(E) \ } \mathrm{None\;of\;these}</math> pair a=(0,1), c=(1,0), bb=(a+c)/2, b=bb+dir(225)/sqrt(6);
  2 KB (388 words) - 14:26, 29 January 2009
• 2006 Cyprus MO/Lyceum/Problems
  == Problem 1 == ...ntity of milk with <math>4\%</math> fat adds a quantity of milk with <math>1\%</math> fat and produces <math>1200</math>kg of milk with <math>2\%</math>
  11 KB (1,672 words) - 10:56, 27 April 2008
• 2007 Cyprus MO/Lyceum/Problem 30
  A coin with a shape of a regular hexagon of side 1 is tangent to a square of side 6, as shown in the figure. ...) \ } 24\qquad \mathrm{(C) \ } \frac{28\pi}{3}\qquad \mathrm{(D) \ } 6 \pi\sqrt{2}\qquad \mathrm{(E) \ } \mathrm{None\;of\;these}</math>
  1 KB (168 words) - 00:22, 10 May 2007
• 2004 IMO Shortlist Problems/N1
  === Solution 1 === ...each divisor <math> d > \sqrt{n} </math>, a divisor <math> \frac{n}{d} < \sqrt{n} </math>. Hence we obtain the inequality
  7 KB (1,166 words) - 22:14, 10 May 2007
• 2005 IMO Shortlist Problems/G1
  === Solution 1 === ...h> are congruent. This means that <math> NI = MP = \frac{1}{2} DI = \frac{1}{2} KI </math>, so <math> \displaystyle PN </math> is the perpendicular bis
  7 KB (1,088 words) - 16:57, 30 May 2007
• 2007 Cyprus MO/Lyceum/Problem 22
  ...rt{2}}\qquad \mathrm{(C) \ } \frac{5a}{2}\qquad \mathrm{(D) \ } \frac{3a}{\sqrt{3}}\qquad \mathrm{(E) \ } 2a</math> ...5-45-90</math> triangle, the length of <math>AM</math> is <math>\frac{3a}{\sqrt{2}}</math>, and the answer is <math>\boxed{\mathrm{B}}</math>.
  979 bytes (166 words) - 02:33, 19 January 2024
• 2006 Cyprus MO/Lyceum/Problem 2
  The value of the expression <math>K=\left[\left(1+\sqrt{3}\right)*2\right]*\sqrt{2}</math> is ...B)}\ 0\qquad\mathrm{(C)}\ \sqrt{3}\qquad\mathrm{(D)}\ 9\qquad\mathrm{(E)}\ 1</math>
  552 bytes (78 words) - 10:56, 27 April 2008
• 2006 Cyprus MO/Lyceum/Problem 3
  The [[domain]] of the [[function]] <math>f(x)=\sqrt{4+2x}</math> is
  477 bytes (74 words) - 10:50, 27 April 2008
• 2006 Cyprus MO/Lyceum/Problem 4
  \alpha x^2+9x+\frac{81}{4\alpha}=\left(\sqrt{\alpha}x+\frac{9}{2\sqrt{\alpha}}\right)^2 ...th>x</math> has the unique root of <math>-\frac{\frac{9}{2\sqrt{\alpha}}}{\sqrt{\alpha}} = \frac{-9}{2\alpha}</math>, so it touches the x-axis, <math>\math
  1 KB (216 words) - 10:46, 27 April 2008
• 2006 Cyprus MO/Lyceum/Problem 6
  The value of the expression <math>K=\sqrt{19+8\sqrt{3}}-\sqrt{7+4\sqrt{3}}</math> is <math>\mathrm{(A)}\ 4\qquad\mathrm{(B)}\ 4\sqrt{3}\qquad\mathrm{(C)}\ 12+4\sqrt{3}\qquad\mathrm{(D)}\ -2\qquad\mathrm{(E)}\ 2</math>
  887 bytes (124 words) - 10:43, 27 April 2008
• 2006 Cyprus MO/Lyceum/Problem 7
  ...h>\Delta E\perp A\Gamma</math>, <math>EZ\perp B\Gamma</math>. If <math>EZ=\sqrt{3}</math>, then the length of the side of the triangle <math>AB\Gamma</math <math>\mathrm{(A)}\ \frac{3\sqrt{3}}{2}\qquad\mathrm{(B)}\ 8\qquad\mathrm{(C)}\ 4\qquad\mathrm{(D)}\ 3\qquad
  1 KB (163 words) - 10:42, 27 April 2008
• 2006 Cyprus MO/Lyceum/Problem 9
  If <math>x=\sqrt[3]{4}</math> and <math>y=\sqrt[3]{6}-\sqrt[3]{3}</math>, then which of the following is correct? ...[3]{18}+\sqrt[3]{9})&:(\sqrt[3]{6}-\sqrt[3]{3})(\sqrt[3]{36}+\sqrt[3]{18}+\sqrt[3]{9})\\
  1 KB (182 words) - 10:40, 27 April 2008
• Surface area
  For cones, the surface area is <math>\pi \cdot r \cdot (r+\sqrt{h^2+r^2})</math> ...the surface area is <math>lw+l \cdot \sqrt{(\frac{w}{2})^2+h^2}+w^2 \cdot \sqrt{(\frac{l}{2})^2+h^2}</math>.
  1 KB (209 words) - 01:34, 25 January 2016
• Root-mean-square
  ...n of [[real number]]s <math>x_1,\dots , x_n</math> is defined to be <math>\sqrt{\frac{x^2_1+x^2_2+\dots+x^2_n}{n}}</math>. This is the second [[power mean
  543 bytes (86 words) - 13:41, 8 May 2013
• Octagon
  [[Area]]: <math>2s^2(1 + \sqrt{2})</math> [[Apothem]]: <math>\frac{s(1+ \sqrt2 )}{2}</math>
  489 bytes (60 words) - 20:40, 14 October 2007
• 2005 AMC 12A Problems/Problem 22
  ...iameter of the sphere is the space diagonal of the prism, which is <cmath>\sqrt{l^2 + w^2 +h^2}.</cmath> <cmath>\sqrt{l^2 + w^2 +h^2} = \sqrt{400} = 20.</cmath> The radius is half of the diameter, so
  2 KB (334 words) - 10:20, 16 September 2022
• 2005 AMC 12A Problems/Problem 3
  ...ath>. The area of the rectangle is <math>(w)(2w) = 2w^2 = 2\left(\frac{x}{\sqrt{5}}\right)^2 = \frac{2}{5}x^2 \ \mathrm{(B)}</math>.
  739 bytes (113 words) - 21:15, 3 July 2013
• 2006 Cyprus MO/Lyceum/Problem 14
  ...Delta</math> and to the square <math>ZB\Gamma E</math>, so that <math>AE=2\sqrt{5}\ \text{m}</math> and the shaded area of the triangle <math>\Delta BE</ma ...)}\ 16\ \text{m}^2\qquad\mathrm{(D)}\ 32\ \text{m}^2\qquad\mathrm{(E)}\ 10\sqrt{5}\ \text{m}^2</math>
  1 KB (229 words) - 09:11, 12 August 2008
• 2006 Cyprus MO/Lyceum/Problem 15
  ...7} + \frac{1}{\sqrt7+\sqrt{10}}+ \frac{1}{\sqrt{10}+\sqrt{13}} + \frac{1}{\sqrt{13}+4}</math> equals ...{(B)}\ \frac{3}{2}\qquad\mathrm{(C)}\ \frac{2}{5}\qquad\mathrm{(D)}\ \frac{1}{2}\qquad\mathrm{(E)}\ \frac{2}{3}</math>
  985 bytes (124 words) - 10:29, 27 April 2008
• 2006 Cyprus MO/Lyceum/Problem 22
  <math>\mathrm{(A)}\ \frac{5}{9}\qquad\mathrm{(B)}\ \frac{1}{3}\qquad\mathrm{(C)}\ \frac{9}{5}\qquad\mathrm{(D)}\ \frac{3}{5}\qquad\mat ...{\sqrt{9x^4 + 9y^4 + 82x^2y^2}}{18}</math>. Thus the ratio is <math>\frac{\sqrt{9x^4 + 9y^4 + 82x^2y^2}}{18xy}</math>. There is no way we can simplify this
  2 KB (285 words) - 13:19, 26 April 2008
• Magnitude
  ...agnitude <math>|z|</math> of a complex number <math>z</math> equals <math>\sqrt{\mathrm {Re}(z)^2 + \mathrm{Im}(z)^2}</math>. Both types of magnitude are b <cmath>\begin{align*} |z\omega| &= \sqrt{(ac - bd)^2 + (ad + bc)^2} \\
  2 KB (246 words) - 18:27, 2 March 2023
• Torricelli's Law
  <cmath>v_2=\sqrt{2gh+v_1^2}</cmath> ...2 v_1^2)=\frac 12 mv_2^2</math>; solving for <math>v_2</math>, <cmath>v_2=\sqrt{2gh+v_1^2}</cmath>
  1 KB (184 words) - 13:58, 22 December 2007
• 1998 IMO Shortlist Problems/A2
  ...math>r_1, r_2, \dotsc, r_n</math> be real numbers greater than or equal to 1. Prove that ...} + \dotsb + \frac{1}{r_n + 1} \ge \frac{n}{\sqrt[n]{r_1 r_2 \dotsm r_n} + 1} . </cmath>
  2 KB (272 words) - 22:03, 30 December 2007
• 1998 BMO Problems/Problem 2
  If <math>n \ge 2</math> is an integer and <math>0 < a_1 < a_2 \dotsb < a_{2n+1}</math> are real numbers, prove the inequality: ...n}} + \sqrt[n]{a_{2n+1}} < \sqrt[n]{a_1 -a_2 +a_3 - \dotsb -a_{2n} + a_{2n+1}} . </cmath>
  2 KB (346 words) - 00:08, 31 December 2007
• 2007 AMC 10A Problems/Problem 17
  ...of <math>m</math> is <math>3^2 \cdot 5 = 45</math>, which makes <math>n = \sqrt[3]{3^3 \cdot 5^3} = 15</math>. The minimum possible value for the sum of <m
  1 KB (201 words) - 08:04, 11 February 2023
• 1995 AHSME Problems/Problem 23
  <cmath>15^2 >11^2 + k^2 \Longrightarrow k < \sqrt{104}</cmath> <cmath>k^2 >11^2 + 15^2 \Longrightarrow k > \sqrt{346}</cmath>
  1 KB (183 words) - 14:05, 5 July 2013
• 1970 AMC 12 Problems/Problem 2
  <math>\mathrm{(A)}\ \frac{2}{\pi}\qquad \mathrm{(B)}\ \frac{\pi}{\sqrt{2}}\qquad \mathrm{(C)}\ \frac{4}{5}\qquad \mathrm{(D)}\ \frac{5}{6}\qquad \
  613 bytes (104 words) - 15:49, 9 January 2008
• 1951 AHSME Problems/Problem 16
  ...nly one root (applying the quadratic formula, we get <math>x = \frac{-b + \sqrt{0}}{2a} = -b/2a</math>). Thus it follows that <math>f(x)</math> touches the
  1 KB (166 words) - 12:20, 5 July 2013
• Successor set
  :(i) <math>1\in S</math> :(ii) <math>\forall n\in S</math>; <math>n+1\in S</math>
  546 bytes (92 words) - 11:36, 26 January 2008
• Bounded
  ...lane]], <math>\mathbb{R}^2</math> with distance <math>d((x, y), (w, z)) = \sqrt{(x - w)^2 + (y - z)^2}</math>), is bounded if for some <math>x \in X</math>
  847 bytes (159 words) - 17:37, 15 February 2008
• Momentum
  ...math>\vec{p} = \gamma m\vec{v}</math>, where <math>\gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}</math> is the [[Lorentz factor]] dependent on the magnitu
  2 KB (271 words) - 09:02, 11 March 2008
• 2005 Alabama ARML TST Problems/Problem 9
  ...=6+3\sqrt{2}</math>. We multiply and divide by 2 to get <math>\boxed{18+18\sqrt{2}}</math>.
  776 bytes (136 words) - 00:19, 4 March 2008
• Mock AIME 1 Pre 2005 Problems/Problem 5
  <cmath>\sqrt[3]{x} + \sqrt[3]{20 - x} = 2</cmath> The smaller of the two values can be expressed as <math>p - \sqrt{q}</math>, where <math>p</math> and <math>q</math> are integers. Compute <m
  979 bytes (159 words) - 17:46, 21 March 2008
• 2008 AIME I Problems/Problem 5
  ...tations. The value of <math>h/r</math> can be written in the form <math>m\sqrt {n}</math>, where <math>m</math> and <math>n</math> are positive integers a ...h>\sqrt {r^{2} + h^{2}}</math>. Thus, the length of the path is <math>2\pi\sqrt {r^{2} + h^{2}}</math>.
  1 KB (230 words) - 20:18, 4 July 2013
• 2006 IMO Shortlist Problems/A5
  ...\sqrt{c}+\sqrt{a} - \sqrt{b}} + \frac{\sqrt{a+b-c}}{\sqrt{a} + \sqrt{b} - \sqrt{c}} \le 3. </cmath> <cmath> \sum_{\rm cyc} \left( \frac{y}{x} - 1 \right) \left( \frac{z}{x} -1 \right) \ge 0 . </cmath>
  3 KB (529 words) - 08:03, 29 March 2008
• 1966 AHSME Problems/Problem 3
  <cmath>\sqrt{\eta\zeta}=10\Rightarrow \eta\zeta=100</cmath>.
  942 bytes (143 words) - 01:27, 26 June 2016
• Mock AIME 1 2007-2008 Problems/Problem 5
  Let <math>S = (1+i)^{17} - (1-i)^{17}</math>, where <math>i=\sqrt{-1}</math>. Find <math>|S|</math>. ...m, <math>1+i = \sqrt{2}\,\text{cis}\,\frac{\pi}{4}</math> and <math>1-i = \sqrt{2}\,\text{cis}\,-\frac{\pi}{4}</math>, where <math>\text{cis}\,\theta = \co
  1,008 bytes (141 words) - 21:13, 2 April 2008
• 2007 Alabama ARML TST Problems
  ==Problem 1== Compute: <math>\sqrt{2000\cdot 2007\cdot 2008\cdot 2015+784}.</math>
  5 KB (841 words) - 11:33, 28 January 2009
• 2008 Mock ARML 1 Problems
  = Set 1 = == Problem 1 ==
  3 KB (409 words) - 16:41, 29 May 2008
• 2007 Alabama ARML TST Problems/Problem 1
  Compute: <math>\sqrt{2000\cdot 2007\cdot 2008\cdot 2015+784}.</math> <center><math>\sqrt{2000(2000+7)(2000+8)(2000+15)+784}</math>
  454 bytes (40 words) - 09:26, 18 June 2008
• 2006 Alabama ARML TST Problems/Problem 7
  ...s <math>x^2 = 10^2 + 10^2 - 2\cdot10\cdot10\cdot \cos{150^\circ} = 200+100\sqrt{3}</math>.
  1 KB (185 words) - 01:57, 17 March 2021
• KGS math club/solution 3 1
  ...of a little sphere is sqrt(1²+1²+...1²) = sqrt(n) The radius of the big sphere is sqrt(n)-1
  221 bytes (41 words) - 13:33, 1 July 2008
• 1966 AHSME Problems/Problem 8
  <math>\text{(A)} \ 27 \qquad \text{(B)} \ 21 \qquad \text{(C)} \ \sqrt {389} \qquad \text{(D)} \ 15 \qquad \text{(E)} \ \text{undetermined}</math> ...\sqrt {10^2 - (\frac{16}{2})^2} = \sqrt {100 - 8^2} = \sqrt {100 - 64} = \sqrt {36} = 6</math>
  2 KB (256 words) - 01:45, 26 June 2016
• 1985 IMO Problems/Problem 4
  ...square. Let the elements be <math>x,y</math> and let <math>x=\sqrt{ab},y=\sqrt{cd}</math> where <math>a,b,c,d\in M</math>. Then, we have <math>xy=z^2</ma
  1 KB (261 words) - 23:56, 29 January 2021

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