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• 2005 AMC 12A Problems/Problem 16
  ...is tangent to both axes and to the second and third circles. What is <math>r/s</math>? draw(O3--P3,linetype("2 2"));
  2 KB (307 words) - 23:58, 17 November 2024
• 2015 IMO Problems
  ==Problem 2== is a power of 2.
  4 KB (709 words) - 14:00, 1 June 2024
• Wilson's Theorem
  ...ath>p-1</math>. Then <math>d</math> divides <math>(p-1)!</math>, so <math>d</math> does not divide <math>(p-1)! + 1</math>. Therefore <math>p</math> d <cmath> 0 \equiv a^2 - 1 \equiv (a-1)(a+1) \pmod{p} , </cmath>
  4 KB (639 words) - 00:53, 2 February 2023
• Cauchy-Schwarz Inequality
  ...)(b_1^2 + b_2^2 + \cdots + b_n^2) \geq (a_1b_1 + a_2b_2 + \cdots + a_nb_n)^2,</cmath> with equality if and only if there exists a constant <math>t</math ...rrightarrow{v}</math> and <math>\overrightarrow{w}</math> in <math>\mathbb{R}^n</math>, where <math>\overrightarrow{v} \cdot \overrightarrow{w}</math> i
  13 KB (2,048 words) - 14:28, 22 February 2024
• Geometric inequality
  ...h sides of length <math>a \leq b \leq c</math> we have <math>a^2 + b^2 = c^2</math>. The Inequality extends this to [[obtuse triangle| obtuse]] and [[a ...n obtuse triangle with sides <math>a \leq b \leq c</math>, <math>a^2+b^2<c^2</math>.
  7 KB (1,300 words) - 00:11, 28 October 2024
• Ptolemy's theorem
  ...lic quadrilateral]] <math>ABCD</math> with side lengths <math>{a},{b},{c},{d}</math> and [[diagonal]]s <math>{e},{f}</math>: == Proof 2 (inversion) ==
  7 KB (1,198 words) - 23:28, 19 September 2024
• Number theory/Advanced
  <cmath>y^2=x^3+g_2x+g^3.</cmath> ...is a natural abelian group structure on its points. This means that given 2 points on the curve, they can be added in a way that satisfies the normal l
  5 KB (849 words) - 15:14, 18 May 2021
• Divisibility rules
  === Divisibility Rule for 2 and Powers of 2 === ...d only if its units digit is divisible by 2, i.e. if the number ends in 0, 2, 4, 6 or 8.
  10 KB (1,572 words) - 21:11, 22 September 2024
• Euclidean algorithm
  ...h>r</math> coincide: <math>GCD(a,b)=GCD(b,r)</math>. But the pair <math>(b,r)</math> consists of smaller numbers than the pair <math>(a,b)</math>! So, w ...to note that <math>a \pmod{b} = r</math><br>is the same as <math>a \equiv {r} \pmod{b}</math>.
  6 KB (923 words) - 16:39, 30 September 2024
• Binomial Theorem
  ...= \binom{5}{0}</math>, <math>5 = \binom{5}{1}</math>, <math>10 = \binom{5}{2}</math>, etc. ...(\binom{n}{0}a^{n}b^{0} + \binom{n}{1}a^{n-1}b^{1} + \binom{n}{2}a^{n-2}b^{2}+\cdots+\binom{n}{n}a^{0}b^{n})(a+b)</cmath>
  5 KB (935 words) - 12:11, 20 February 2024
• Function
  ...actly one value in the second. For instance, one function may map 1 to 1, 2 to 4, 3 to 9, 4 to 16, and so on. This function has the rule that it takes ...hat function. For an example, consider the function: <math>f(x) = \sqrt{x^2-9}</math>. The domain of the function is the set <math>\{x:|x| \geq 3\}</m
  10 KB (1,761 words) - 02:16, 12 May 2023
• Circle
  ...the [[distance formula]], this gives <math>\sqrt{(x - h)^2 + (y - k)^2} = r</math> which is more commonly written as <cmath>(x - h)^2 + (y - k)^2 = r^2.</cmath>
  9 KB (1,585 words) - 12:46, 2 September 2024
• Pi
  ...places, such as in the sum <math>\sum_{n=1}^\infty \frac{1}{n^2}=\frac{\pi^2}{6}</math>. Some common [[fraction]]al approximations for pi are <math>\fr ...e <math>\pi</math> is to inscribe a unit circle in a square of side length 2. Using a computer, random points are placed inside the square. Because th
  8 KB (1,469 words) - 20:11, 16 September 2022
• Muirhead's Inequality
  ...uence <math>(5,1)</math> majorizes <math>(4,2)</math> (as <math>5>4, 5+1=4+2</math>), Muirhead's inequality states that for any positive <math>x,y</math <cmath>x^5y^1+y^5x^1\geq x^4y^2+y^4x^2</cmath>
  8 KB (1,346 words) - 11:53, 8 October 2023
• Vector
  ...ctors can be uniquely described in many ways. The two most common is (for 2-dimensional vectors) by describing it with its length (or magnitude) and th ...h> formed by them, <math>\|\vec{v}+\vec{w}\|^2=\|\vec{v}\|^2+\|\vec{w}\|^2+2\|\vec{v}\|\|\vec{w}\|\cos\theta</math>.
  11 KB (1,876 words) - 18:01, 29 August 2024
• Complex number
  ...mplex numbers''' arise when we try to solve [[equation]]s such as <math> x^2 = -1 </math>. ...from this addition, we are not only able to find the solutions of <math> x^2 = -1 </math> but we can now find ''all'' solutions to ''every'' polynomial.
  5 KB (860 words) - 14:36, 10 December 2023
• Law of Sines
  ...<math>B</math>, <math>c</math> opposite to <math>C</math>, and where <math>R</math> is the circumradius: real r = 1;
  4 KB (658 words) - 15:19, 28 April 2024
• Area
  ...^{\circ}</math> and the area of the entire polygon is <math>\frac{n\cdot s^2}{4} \cot\frac{180}{n}^{\circ}</math>. * <math>K=\frac{bh}{2}</math> where <math>b</math> is a base and <math>h</math> is the altitude o
  6 KB (1,181 words) - 13:01, 23 October 2024
• Exponential form
  ...s\theta+i\sin\theta)=r cis\theta</math>, where <math>r=|z| = \sqrt{a^2 + b^2}</math>. By [[Euler's identity|Euler's formula]], which states that <math> pair A,B,C,D,E;
  1 KB (238 words) - 21:51, 20 February 2022
• Goldbach Conjecture
  <math>4 = 2 + 2</math> Every [[integer]] greater than 2 can be written as the sum of three primes.
  7 KB (1,201 words) - 15:59, 19 February 2024

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