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• 2009 AIME II Problems
  ...ng a blue stripe, a red stripe, a white stripe, and a pink stripe. Pink is a mixture of red and white, not necessarily in equal amounts. When Bill finis ...ath>b</math>, and <math>c</math> are positive real numbers such that <math>a^{\log_3 7} = 27</math>, <math>b^{\log_7 11} = 49</math>, and <math>c^{\log_
  8 KB (1,366 words) - 21:33, 3 January 2021
• 2000 AMC 8 Problems
  <math>\mathrm{(A)}\ 15\qquad\mathrm{(B)}\ 16\qquad\mathrm{(C)}\ 17\qquad\mathrm{(D)}\ 21\qqu <math>\text{(A)}\ -2 \qquad \text{(B)}\ -1 \qquad \text{(C)}\ 0 \qquad \text{(D)}\ 1 \qqua
  15 KB (2,165 words) - 18:29, 5 June 2024
• 2009 AIME II Problems/Problem 13
  Let <math>A</math> and <math>B</math> be the endpoints of a semicircular arc of radius <math>2</math>. The arc is divided into seven co ...th>A</math> and <math>B</math>. Assume <math>C_1</math> is closer to <math>A</math> instead of <math>B</math>. <math>\angle AOC_1</math> = <math>\frac
  8 KB (1,279 words) - 20:27, 17 May 2024
• 2009 USAMO Problems/Problem 1
  ...th>S</math>. Prove that if <math>P, Q, R</math> and <math>S</math> lie on a circle then the center of this circle lies on line <math>XY</math>. ...and <math>O_2</math> , <math>O_3</math> respectively. Hence, by [[power of a point]](the power of <math>O_1</math> can be expressed using circle <math>\
  2 KB (290 words) - 13:16, 17 April 2021
• Radical axis
  The theory of radical axis is a priceless geometric tool that can solve formidable geometric problems fairl ...rtains to that situation. I hope after you read this text, you will become a better math student, armed with another tool to solve difficult problems. B
  12 KB (2,125 words) - 08:38, 23 May 2024
• KGS math club/solution 5 1
  ...nce P has nonzero degree, n>0. The Taylor polynomial of P about the point a is <math>F(x)=P(a)+P'(a)(x-a)+\dotsb +\dfrac{P^{(n)}(a)(x-a)^n}{n!}=P(x)</math>.
  2 KB (305 words) - 19:24, 21 December 2015
• Base introduction
  Bases. These confuse a lot of us. Hopefully this will make it better. ...as a subscript. Every number in base 10 (our normal numbers) does not have a subscript.
  3 KB (552 words) - 20:57, 9 July 2009
• 1965 IMO Problems/Problem 4
  Then, for <math>i = 1,2,3,4</math> we have: <math>x_i + \prod_{j \neq i}x_j = 2</math>. ...rightarrow x_i = 1 \pm t</math> where <math>t = \pm \sqrt{1-P} \in \mathbb{R}</math>.
  2 KB (382 words) - 12:51, 29 January 2021
• 2009 USAMO Problems
  ...th>S</math>. Prove that if <math>P, Q, R</math> and <math>S</math> lie on a circle then the center of this circle lies on line <math>XY</math>. ...elements <math>a, b, c</math> (not necessarily distinct) satisfying <math>a + b + c = 0</math>.
  4 KB (718 words) - 18:16, 17 September 2012
• 2009 USAMO Problems/Problem 6
  ...i - s_j)r</math> and <math>(t_i - t_j)/r</math> are integers for all <math>i</math> and <math>j</math>. ...ac{c_2}{d_2} - \frac{c_1}{d_1})</math> is an integer. Now we can set <math>r = \frac{b_1 b_2}{d_1 d_2}</math>, because the least common denominator of <
  2 KB (463 words) - 12:19, 21 August 2020
• Schroeder-Bernstein Theorem
  ...orem''' (sometimes called the '''Cantor-Schroeder-Bernstein Theorem''') is a result from [[set theory]], named for Ernst Schr&ouml;der and Felix Bernste ...ath> and <math>g : B \to A</math>, then there is a [[bijection]] <math>h : A \to B</math>.
  4 KB (805 words) - 13:09, 20 February 2024
• 1992 AJHSME Problems
  <math>\text{(A)}\ -1 \qquad \text{(B)}\ 1 \qquad \text{(C)}\ 5 \qquad \text{(D)}\ 9 \qquad <math>\text{(A)}\ \dfrac{10}{8} \qquad \text{(B)}\ 1\dfrac{1}{4} \qquad \text{(C)}\ 1\dfra
  17 KB (2,346 words) - 13:36, 19 February 2020
• Semisimple module
  A '''semisimple module''' is, informally, a [[module]] that is not Specifically, it is a module <math>M</math> with the following property:
  6 KB (1,183 words) - 15:02, 18 August 2009
• Constructible number
  ...can be constructed with a [[straight edge]] and [[compass]] starting with a segment of length <math>1</math>. ...e as a real number if and only if it is constructible as a complex number, i.e., our two definitions coincide in this case.
  8 KB (1,305 words) - 08:39, 21 August 2009
• Unique factorization domain
  ...ization domain if for any nonzero element <math>r\in R</math> which is not a [[unit (ring theory)|unit]]: ..._n</math> are (not necessarily distinct) [[irreducible element]]s in <math>R</math>.
  6 KB (1,217 words) - 23:05, 23 August 2009
• 1994 IMO Problems/Problem 3
  * (a) Prove that, for each positive integer <math>m</math>, there exists at leas '''a) Surjectivity of f'''
  7 KB (1,298 words) - 19:59, 9 February 2024
• 2010 AMC 12A Problems
  {{AMC12 Problems|year=2010|ab=A}} <math>\textbf{(A)}\ -4020 \qquad \textbf{(B)}\ 0 \qquad \textbf{(C)}\ 40 \qquad \textbf{(D)}
  12 KB (1,817 words) - 15:00, 12 August 2020
• 2010 AMC 10A Problems
  {{AMC10 Problems|year=2010|ab=A}} \mathrm{(A)}\ 1
  13 KB (1,902 words) - 11:20, 5 March 2023
• Uniform convergence
  ...f(x)| < \varepsilon</math>. (More generally, we can replace <math>\mathbb{R}</math> with any [[metric space]] <math>Y</math>.) ...r condition, because (speaking informally) the function has to converge at a similar rate everywhere on its [[domain]].
  4 KB (622 words) - 19:20, 3 March 2010
• Characteristic polynomial
  The '''characteristic polynomial''' of a linear [[operator]] refers to the [[polynomial]] whose roots are the [[eige ...ich is a <math>n</math>th degree polynomial in <math>t</math>. Here, <math>I</math> refers to the <math>n\times n</math> [[identity matrix]].
  19 KB (3,412 words) - 14:57, 21 September 2022

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