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  • B = (4, 3); C = (4, 0);
    5 KB (842 words) - 19:38, 26 January 2020
  • Isaac Newton was born on January 4, 1643, in Lincolnshire, England. Newton was born very shortly after the dea ...places a force on the matter with the same mass <math>n</math>, then <math>n</math> will put an equivalent force in the opposite direction.
    9 KB (1,355 words) - 23:53, 2 August 2020
  • ...e series: <center><math>3+\frac{11}4+\frac 94 + \cdots + \frac{n^2+2n+3}{2^n}+\cdots</math>.</center> ...^{\infty} \left(\frac{2n}{2^n}\right)+\sum_{n=1}^{\infty} \left(\frac{3}{2^n}\right)</math>
    1 KB (193 words) - 21:13, 18 May 2021
  • ...gers <math>n\geq 3</math>, there exists a balanced set consisting of <math>n</math> points. </li> ...ath> for which there exists a balanced centre-free set consisting of <math>n</math> points. </li>
    4 KB (692 words) - 22:33, 15 February 2021
  • ...emainder 1. Show that there is an integer <math>{n}</math> such that <math>n^2 + 1</math> is divisible by <math>{p}</math>.
    4 KB (639 words) - 22:48, 9 January 2020
  • *Show that <math>\sum_{k=1}^{n}a_k^2 \geq a_1a_2+a_2a_3+\cdots+a_{n-1}a_n+a_na_1</math>. [[Inequality_Introductory_Problem_2|Solution]] *Show that <math>x^2+y^4\geq 2x+4y^2-4</math> for all real <math> x </math>.
    2 KB (399 words) - 22:10, 29 May 2021
  • <center><math>AM=\frac{x_1+x_2+\cdots+x_n}{n}</math></center> is the arithmetic mean of the <math>{n}</math> numbers <math>x_1,x_2,\ldots,x_n</math>.
    699 bytes (110 words) - 12:44, 20 September 2015
  • * <math>5x^4 - 2x^2 + 9</math>, in the variable <math>x</math> A polynomial in one variable is a function <math>P(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_2x^2 + a_1x + a_0</math>. Here, <math>a_i</math> is the <m
    6 KB (1,046 words) - 13:07, 14 July 2021
  • *Two different [[prime number]]s between <math>4</math> and <math>18</math> are chosen. When their sum is subtracted from th ...n divided by <math>5</math>, find the value of the remainder of when <math>n</math> is divided by <math>5</math>.
    4 KB (631 words) - 13:42, 14 July 2021
  • <cmath>a^n-b^n=(a-b)(a^{n-1}+a^{n-2}b + \cdots + ab^{n-2} + b^{n-1})</cmath> If <math>n=2</math>, this creates the difference of squares factorization, <cmath>a^2-
    3 KB (530 words) - 13:34, 14 July 2021
  • ...um of the [[series]] <math>\frac11 + \frac14 + \frac19 + \cdots + \frac{1}{n^2} + \cdots</math><br> ...}+\frac{x^4}{5!}-\cdots=\left(1-\frac{x^2}{\pi^2}\right)\left(1-\frac{x^2}{4\pi^2}\right)\left(1-\frac{x^2}{9\pi^2}\right)\cdots</math><br>
    2 KB (314 words) - 06:45, 1 May 2014
  • ...mong any <math>n</math> integers, there are two with the same modulo-<math>n-1</math> residue. ...boxes then at least one box must hold at least <math>\left\lceil \frac{k}{n} \right\rceil</math> objects. Here <math>\lceil \cdot \rceil</math> denote
    4 KB (691 words) - 00:54, 2 April 2021
  • ...^4 + 6x^3 + 11x^2 + 3x + 31</math> is the square of an integer. Then <math>n</math> is: <math>\textbf{(A)}\ 4 \qquad
    3 KB (571 words) - 17:00, 9 July 2018
  • ...numbers. Note that if <math>n</math> is even, we take the positive <math>n</math>th root. It is analogous to the [[arithmetic mean]] (with addition r ...1 and 2 is <math>\sqrt[4]{6\cdot 4\cdot 1 \cdot 2} = \sqrt[4]{48} = 2\sqrt[4]{3}</math>.
    2 KB (282 words) - 22:04, 11 July 2008
  • ...four guys in order. By the same logic as above, this is <math>2!\binom{6}{4}=30</math>. Again, <math>|A\cap C|</math> would be putting five guys in ord If <math>(A_i)_{1\leq i\leq n}</math> are finite sets, then:
    9 KB (1,733 words) - 20:46, 26 October 2020
  • <cmath> \frac{a_1+a_2+\ldots+a_n}{n}\geq\sqrt[n]{a_1a_2\cdots a_n} </cmath> <cmath> \sum_{i=1}^{n}\frac{a_i}{n} \geq \prod\limits_{i=1}^{n}a_i^{\frac{1}{n}} . </cmath>
    4 KB (562 words) - 01:30, 6 March 2021
  • ...[[recursion|recursive definition]] for the factorial is <math>n!=n \cdot (n-1)!</math>. * <math>4! = 24</math>
    10 KB (799 words) - 10:45, 12 May 2021
  • ==Discriminant of polynomials of degree n== .../math> with all the coefficients being real. But for polynomials of degree 4 or higher it can be difficult to use it.
    4 KB (734 words) - 12:06, 15 July 2021
  • * [[2006 AIME II Problems/Problem 4]] {{AIME box|year=2006|n=II|before=[[2006 AIME I]]|after=[[2007 AIME I]], [[2007 AIME II|II]]}}
    1 KB (133 words) - 12:32, 22 March 2011
  • ...s, then <math>a^{\varphi(n)} \equiv 1 \pmod{n}</math>, where <math>\varphi(n)</math> denotes [[Euler's totient function]]. In particular, <math>\varphi( ...let <math>S = \{\text{natural numbers relatively prime to and less than}\ n\}</math>.
    11 KB (1,827 words) - 02:02, 15 October 2020

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