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• Completing the square
  ...asy. In fact, all [[quadratic equation]]s can be solved by completing the square. ...math>(x-3)^2=x^2-6x+9</math>. To do this, add <math>7</math> to each side of the equation to get
  2 KB (422 words) - 16:20, 5 March 2023
• Heron's Formula
  ...formula) is a [[mathematical formula | formula]] for finding the [[area]] of a [[triangle]] given only the three side lengths. The Law of Cosines states that in triangle <math>ABC</math>, <math>c^2 = a^2 + b^2 - 2
  4 KB (675 words) - 00:05, 22 January 2024
• Polynomial
  ...ltiplied by [[coefficient]]s from a predetermined [[set]] (usually the set of integers; [[rational]], [[real]] or [[complex]] numbers; but in [[abstract * <math>x^3 + 3x^2y + 3xy^2 + y^3</math>, in the variables <math>x</math> and <math>y</math>
  6 KB (1,100 words) - 01:44, 17 January 2024
• Quadratic formula
  ...er methods, such as [[completing the square]], [[factoring]], and [[square root property]] do not work or are too tedious. ===General Solution For A Quadratic by Completing the Square===
  2 KB (290 words) - 16:55, 19 February 2024
• Fermat's Little Theorem
  ...more at the introductory level if they have a hard time following the rest of this article). This theorem is credited to [[Pierre de Fermat]]. ...a \pmod {p}</math>. As you can see, it is derived by multipling both sides of the theorem by <math>a</math>. The restated form is nice because we no lon
  16 KB (2,675 words) - 10:57, 7 March 2024
• 1959 IMO Problems/Problem 2
  For what real values of <math>x</math> is ...c) <math>A=2</math>, where only non-negative real numbers are admitted for square roots?
  3 KB (466 words) - 12:04, 12 April 2024
• Function
  ...e 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 its inp ...ubset]] of <math>A\times B</math>, which denotes the [[Cartesian product]] of <math>A</math> and <math>B</math>. (That is, <math>f</math> is a [[relatio
  10 KB (1,761 words) - 03:16, 12 May 2023
• Base numbers
  ...the number 2746. This number can be rewritten as <math>2746_{10}=2\cdot10^3+7\cdot10^2+4\cdot10^1+6\cdot10^0.</math> ...<math>10^2</math>'s, and the fourth digit tells us there are two <math>10^3</math>'s.
  4 KB (547 words) - 17:23, 30 December 2020
• Trigonometric identities
  ...all input values. Trigonometric functions have an abundance of identities, of which only the most widely used are included in this article. ...math>\cos^2 (x)</math> and substitute the respective trigonometry in place of the ratios to obtain the desired result.
  8 KB (1,397 words) - 21:55, 20 January 2024
• Root-Mean Square-Arithmetic Mean-Geometric Mean-Harmonic mean Inequality
  ...], [[arithmetic mean]], [[geometric mean]], and [[harmonic mean]] of a set of [[positive]] [[real number]]s <math>x_1,\ldots,x_n</math> that says: ...math>n_1>1,~~0<n_2<1,~~-1<n_3<0,~~n_4<-1</math>, and <math>n</math> is the root mean power.
  5 KB (912 words) - 20:06, 14 March 2023
• Perfect square
  An [[integer]] <math>n</math> is said to be a '''perfect square''' if there is an integer <math>m</math> so that <math>m^2=n</math>. The fi The sum of the first <math>n</math> square numbers (starting with <math>1</math>) is <math>\frac{n(n+1)(2n+1)}{6}</mat
  954 bytes (155 words) - 01:14, 29 November 2023
• 2022 USAJMO Problems/Problem 1
  ...of integers <math>a_1,a_2,\cdots</math> and an infinite geometric sequence of integers <math>g_1,g_2,\cdots</math> satisfying the following properties? <cmath>m | a+(n+2)d-gr^{n+2} \; (3),</cmath>
  4 KB (792 words) - 00:29, 13 April 2024
• 2005 AIME II Problems/Problem 12
  ...ive]] [[integer]]s and <math> r </math> is not divisible by the [[square]] of any [[prime]], find <math> p+q+r. </math> Let <math>G</math> be the foot of the [[perpendicular]] from <math>O</math> to <math>AB</math>. Denote <math>
  13 KB (2,080 words) - 21:20, 11 December 2022
• 2005 AIME I Problems/Problem 11
  ...e (geometry) | square]] whose sides have length 8. Given the maximum value of <math> d </math> is <math> m - \sqrt{n},</math> find <math> m+n. </math> ...arranged so that the center of the semicircle lies on one diagonal of the square.
  4 KB (707 words) - 11:11, 16 September 2021
• 2005 AIME II Problems/Problem 14
  By the [[Law of Sines]] and since <math>\angle BAE = \angle CAD, \angle BAD = \angle CAE</m Substituting our knowns, we have <math>\frac{CE}{BE} = \frac{3 \cdot 14^2}{2 \cdot 13^2} = \frac{BC - BE}{BE} = \frac{15}{BE} - 1 \Longrig
  13 KB (2,129 words) - 18:56, 1 January 2024
• 2003 AIME II Problems
  ...integers is the sum of the other two. Find the sum of all possible values of <math>N</math>. Let <math>N</math> be the greatest integer multiple of 8, whose digits are all different. What is the remainder when <math>N</math
  7 KB (1,127 words) - 09:02, 11 July 2023
• 1983 AIME Problems/Problem 1
  ...h>, and <math>(xyz)^{12}=w</math>. If we now convert everything to a power of <math>120</math>, it will be easy to isolate <math>z</math> and <math>w</ma <math>x^{120}=w^5</math>, <math>y^{120}=w^3</math>, and <math>(xyz)^{120}=w^{10}</math>.
  4 KB (642 words) - 03:14, 17 August 2022
• 1983 AIME Problems/Problem 3
  What is the product of the [[real]] [[root]]s of the [[equation]] <math>x^2 + 18x + 30 = 2 \sqrt{x^2 + 18x + 45}</math>? ...<math>y</math>, we get <math>y=10</math> or <math>y=-6</math>. The second root is extraneous since <math>2\sqrt{y+15}</math> is always non-negative (and m
  3 KB (532 words) - 05:18, 21 July 2022
• 1983 AIME Problems/Problem 9
  Find the minimum value of <math>\frac{9x^2\sin^2 x + 4}{x\sin x}</math> for <math>0 < x < \pi</math>. ...<math>0 \le x\sin x \le \frac{\pi}{2}</math>, this value of <math>\frac{2}{3}</math> is attainable by the [[Intermediate Value Theorem]]).
  4 KB (722 words) - 20:25, 14 January 2023
• 1984 AIME Problems/Problem 3
  ...4</math>, <math>9</math>, and <math>49</math>, respectively. Find the area of <math>\triangle ABC</math>. D(A--B--C--cycle); D(A+(B-A)*3/4--A+(C-A)*3/4); D(B+(C-B)*5/6--B+(A-B)*5/6);D(C+(B-C)*5/12--C+(A-C)*5/12);
  4 KB (726 words) - 13:39, 13 August 2023

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