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• 2005 AMC 12B Problems/Problem 15
  <math>221</math> can be written as the sum of four two-digit numbers, let's say <math>\overline{ae}</math>, <math>\overline{bf}</math>, <math>\overline
  2 KB (411 words) - 21:02, 21 December 2020
• 2005 AMC 12B Problems/Problem 18
  ...ath> as the hypotenuse. Note <math>C</math> cannot be inside this triangle's circumscribed circle or else <math>\angle C > 90^\circ</math>. Hence, the a
  2 KB (262 words) - 21:20, 21 December 2020
• 2005 AMC 12B Problems/Problem 23
  Let <math>S</math> be the set of ordered triples <math>(x,y,z)</math> of real numbers f ...h>b</math> such that for all ordered triples <math>(x,y,z)</math> in <math>S</math> we have <math>x^{3}+y^{3}=a \cdot 10^{3z} + b \cdot 10^{2z}.</math>
  5 KB (786 words) - 11:36, 19 May 2024
• 2005 AMC 12B Problems/Problem 24
  dot("$A(a,a^2)$", A, S);
  4 KB (761 words) - 09:10, 1 August 2023
• Power of a Point Theorem/Introductory Problem 4
  label("3", E--A, S); ...em]], <math> AE = 3 </math> (or by just knowing your [[Pythagorean Triple]]s). Applying the Power of a Point Theorem gives <math> AE\cdot BE = CE\cdot
  1 KB (177 words) - 02:14, 26 November 2020
• Root (polynomial)
  ...x)</math> whose range is the [[real number | real]], the [[complex number]]s or any abstract [[field]] is a value <math>a</math> in the [[domain]] of th ...rivative <math>P'(x)</math> has at least one root in the interval <math>(r,s)</math>. In particular, if <math>P'(x)</math> has no roots in an interval <
  8 KB (1,427 words) - 21:37, 13 March 2022
• 2006 AMC 10A Problems
  Sandwiches at Joe's Fast Food cost <math> \textdollar 3 </math> each and sodas cost <math> \tex The ratio of Mary's age to Alice's age is <math>3:5</math>. Alice is <math>30</math> years old. How many years
  13 KB (2,028 words) - 16:32, 22 March 2022
• 1995 AHSME Problems/Problem 6
  ...A</math> upwards, and thus <math>x</math> is touching <math>B</math> on it's left side. We now fold <math>B</math> up, and we realize that <math>x</math
  1 KB (168 words) - 00:49, 14 October 2013
• 2006 AMC 10A Problems/Problem 12
  label('Rope', (20,-8),S); label("Rope", (20,-24), S);
  3 KB (424 words) - 10:14, 17 December 2021
• 2006 AMC 10A Problems/Problem 15
  We first find the amount of minutes, <math>k</math>, until Odell and Kershaw's next meeting. Let <math>a</math> be the angle in [[radians]] between their Since Odell's rate is <math>5/6</math> that of Kershaw, but Kershaw's lap distance is <math>6/5</math> that of Odell, they each run a lap in the
  3 KB (532 words) - 17:49, 13 August 2023
• 2006 AMC 10A Problems/Problem 17
  D('W',W,1.6*N); D('X',X,1.6*plain.E); D('Y',Y,1.6*S); D('Z',Z,1.6*plain.W); D('W',W,1.6*N); D('X',X,1.6*plain.E); D('Y',Y,1.6*S); D('Z',Z,1.6*plain.W);
  6 KB (1,066 words) - 00:21, 2 February 2023
• 2006 AMC 10A Problems/Problem 21
  ...btracting off those which do not have any <math>2</math>s or <math>3</math>s as digits. Case <math>2</math> : There are two <math>2</math>s OR two <math>3</math>s. If the <math>2</math> or <math>3</math> is occupying the first digit, we h
  3 KB (525 words) - 20:25, 30 April 2024
• Set
  ...all it an [[infinite]] set. The objects in a set are called the [[element]]s of the set. A common misconception is that a set can have multiple indistin ...ect <math>x</math>, we have <math>x\in S'</math> if and only if <math>x\in S''</math>.
  11 KB (2,021 words) - 00:00, 17 July 2011
• Newman's Tauberian Theorem
  '''Newman's Tauberian Theorem''' is a [[tauberian theorem]] its [[Laplace transform]] <math>F(s) = \int_0^\infty f(t)e^{-st}dt</math>
  6 KB (1,034 words) - 07:55, 12 August 2019
• 2006 USAMO Problems
  .../math> be a prime number and let <math>s</math> be an integer with <math>0<s<p</math>. Prove that there exist integers <math>m</math> and <math>n</math> ...brace\frac{sm}{p}\right\rbrace<\left\lbrace\frac{sn}{p}\right\rbrace<\frac{s}{p}</cmath>
  3 KB (520 words) - 09:24, 14 May 2021
• 1991 AJHSME Problems
  ...$1'$",(.5,0),S); label("$3'$",(1,1.5),E); label("$9'$",(1+9*sqrt(3)/4,9/4),S); ...,S); label("$1'$",(1+5*sqrt(3)/2,5),E);label("$1'$",(1/2+5*sqrt(3)/2,11/2),S);
  17 KB (2,246 words) - 13:37, 19 February 2020
• 3D Geometry
  '''3D Geometry''' deals with objects in 3 [[dimension]]s. For example, a drawing on a piece of paper is 2-dimensional since it has ...lems is to make it 2D. We can do this by looking at certain cross-section(s) of the diagram one at a time.
  2 KB (264 words) - 06:43, 24 May 2024
• 2005 AIME II Problems/Problem 2
  ...<math> m </math> and <math> n </math> are [[relatively prime]] [[integer]]s, find <math> m+n. </math> We then label three "zones" corresponding to the three people's meals as the first, second, and third objects in the line corresponding to
  4 KB (628 words) - 11:28, 14 April 2024
• 2005 AIME II Problems/Problem 3
  ...<math> m </math> and <math> n </math> are [[relatively prime]] [[integer]]s. Find <math> m+n. </math> Let's call the first term of the original [[geometric series]] <math>a</math> and
  3 KB (581 words) - 07:54, 4 November 2022
• 2005 AIME II Problems/Problem 13
  ...8a-10=c-3</math>. Thus <math>P(n)=an^2-(41a)n+(408a-10)=0</math>. By Vieta's formulas, we know that the sum of the roots(<math>n</math>) is equal to 41
  4 KB (642 words) - 02:14, 1 June 2024

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