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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>? label("$r$",midpoint(O0--P0),NE);
  2 KB (307 words) - 23:58, 17 November 2024
• 2015 IMO Problems
  We say that a finite set <math>\mathcal{S}</math> in the plane is <i> balanced </i> if, for any two different points <math>A</math>, <math>B</math> in <math>\mathcal{S}</math>, there is
  4 KB (709 words) - 14:00, 1 June 2024
• Wilson's Theorem
  Suppose first that <math>p</math> is composite. Then <math>p</math> has a factor <math>d > 1</math> that is less than or equal to <math>p-1</math>. ...ique, and each number is the inverse of its inverse. If one integer <math>a</math> is its own inverse, then
  4 KB (639 words) - 00:53, 2 February 2023
• Trivial Inequality
  ...(x^2+\frac{b}{a}x+\frac{b^2}{4a^2})+c-\frac{b^2}{4a} = a(x+\frac{b}{2a})^2+c-\frac{b^2}{4a}.</cmath> .../math> is positive and <math>ax^2+bx+c\le c-\frac{b^2}{4a}</math> if <math>a</math> is negative.
  3 KB (583 words) - 20:20, 2 August 2024
• Combination
  ...generally concerned with finding the number of combinations of size <math>r</math> from an original set of size <math>n</math> ...ons are, their various types, and how to calculate each type! It serves as a great introductory video to combinations, permutations, and counting proble
  4 KB (615 words) - 10:43, 21 May 2021
• Cauchy-Schwarz Inequality
  ..._2 + \cdots + a_nb_n)^2,</cmath> with equality if and only if there exists a constant <math>t</math> such that <math>a_i = t b_i</math> for all <math>1 ...cdot \overrightarrow{w}|</cmath> with equality if and only if there exists a scalar <math>t</math> such that <math>\overrightarrow{v} = t \overrightarro
  13 KB (2,048 words) - 14:28, 22 February 2024
• Menelaus' theorem
  ...earity]] of points on each of the three sides (extended when necessary) of a [[triangle]]. ...C</math>, <math>Q</math> is on the extension of <math>AC</math>, and <math>R</math> on the intersection of <math>PQ</math> and <math>AB</math>, then
  5 KB (804 words) - 02:01, 12 June 2023
• Fermat's Little Theorem
  ...] (which students should study more at the introductory level if they have a hard time following the rest of this article). This theorem is credited to ...}</math> is not [[divisibility|divisible]] by <math>{p}</math>, then <math>a^{p-1}\equiv 1 \pmod {p}</math>.
  16 KB (2,660 words) - 22:42, 28 August 2024
• Geometric inequality
  A '''geometric inequality''' is an [[inequality]] involving various measures ...level geometry problems. It also provides the basis for the definition of a [[metric space]] in [[analysis]].
  7 KB (1,300 words) - 00:11, 28 October 2024
• Ptolemy's theorem
  ...orem''' gives a relationship between the side lengths and the diagonals of a [[cyclic quadrilateral]]; it is the [[equality condition | equality case]] ...[cyclic quadrilateral]] <math>ABCD</math> with side lengths <math>{a},{b},{c},{d}</math> and [[diagonal]]s <math>{e},{f}</math>:
  7 KB (1,198 words) - 23:28, 19 September 2024
• Number theory/Advanced
  ...in some subfield (like the reals or the rationals). One also needs to add a limit point, called the point at infinity. As <math>x\to \infty</math>, the ...points. This means that given 2 points on the curve, they can be added in a way that satisfies the normal laws of addition, like associativity, commuta
  5 KB (849 words) - 15:14, 18 May 2021
• Real number
  A '''real number''' is a number that falls on the real number line. It can have any value. Some exam ...<math>\mathbb{R}</math>, is a subset of [[complex number]]s(<math>\mathbb{C}</math>). Commonly used subsets of the real numbers are the [[rational numb
  3 KB (496 words) - 22:22, 5 January 2022
• Jensen's Inequality
  ...[convex function]] of one real variable. Let <math>x_1,\dots,x_n\in\mathbb R</math> and let <math>a_1,\dots, a_n\ge 0</math> satisfy <math>a_1+\dots+a_n If <math>{F}</math> is a concave function, we have:
  3 KB (623 words) - 12:10, 20 February 2024
• Function
  A '''function''' is a rule that maps one set of values to another set of values, assigning to eac ...is a ''function from <math>A</math> to <math>B</math>'' (written <math>f: A \to B</math>) if and only if
  10 KB (1,761 words) - 02:16, 12 May 2023
• Circle
  A '''circle''' is a geometric figure commonly used in Euclidean [[geometry]]. {{asy image|<asy>unitsize(2cm);draw(unitcircle,blue);</asy>|right|A basic circle.}}
  9 KB (1,585 words) - 12:46, 2 September 2024
• Muirhead's Inequality
  ...>p</math> [[Majorization|majorizes]] a sequence <math>q</math>, then given a set of positive reals <math>x_1,x_2,\cdots,x_n</math>: A common [[Brute forcing|bruteforce]] technique with inequalities is to clear
  8 KB (1,346 words) - 11:53, 8 October 2023
• Vector
  ...y, but also most abstractly, a vector is any object which is an element of a given vector space. ...(x\,\,y\,\,z\,\,...)</math>. The set of vectors over a [[field]] is called a [[vector space]].
  11 KB (1,876 words) - 18:01, 29 August 2024
• Logarithm
  ...would be a pain to have to calculate any time you wanted to use it (say in a comparison of large numbers). Its natural logarithm though (partly due to ...ly 7 digits before the decimal point. Comparing the logs of the numbers to a given precision can allow easier comparison than computing and comparing th
  4 KB (680 words) - 11:54, 16 October 2023
• Schur's Inequality
  ...ates that for all non-negative <math>a,b,c \in \mathbb{R}</math> and <math>r>0</math>: <cmath>a^r(a-b)(a-c)+b^r(b-a)(b-c)+c^r(c-a)(c-b) \geq 0</cmath>
  2 KB (398 words) - 15:57, 29 December 2021
• Complex number
  ...n has no solutions in the real numbers. However, it is possible to define a number, <math> i </math>, such that <math> i = \sqrt{-1} </math>. If we ad ...s the set <math>\mathbb{R}</math> of the [[real number]]s, since <math>a = a + 0i</math>.
  5 KB (860 words) - 14:36, 10 December 2023

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