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• The positive three-digit integer N has a ones digit of 0. What is the probability that N is divisible by 4? Express your answer as a common fraction.
  ...a <math>\boxed{1/2}</math> probability that <math>N</math> is divisible by 4.
  479 bytes (79 words) - 12:48, 30 October 2024

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• Pythagorean Theorem
  B = (4, 3); C = (4, 0);
  5 KB (885 words) - 22:03, 5 October 2024
• Physics
  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) - 06:29, 29 September 2021
• 2005 Alabama ARML TST Problems/Problem 7
  ...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) - 20:13, 18 May 2021
• 2015 IMO Problems
  ...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 (709 words) - 14:00, 1 June 2024
• Wilson's Theorem
  ...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) - 00:53, 2 February 2023
• Trivial Inequality
  *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^2-5</math> for all real <math>x</math> and <math>y</math>.
  3 KB (583 words) - 20:20, 2 August 2024
• Arithmetic mean
  <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) - 11:44, 20 September 2015
• Polynomial
  * <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 (1,100 words) - 14:57, 30 August 2024
• Simon's Favorite Factoring Trick
  *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 (679 words) - 20:05, 16 October 2024
• Factoring
  <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 (532 words) - 21:00, 13 January 2024
• Mathematical problem solving
  ...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) - 05:45, 1 May 2014
• Pigeonhole Principle
  ...e''' states that if <math>n+1</math> or more pigeons are placed into <math>n</math> holes, one hole must contain two or more pigeons. This seemingly tri ...if <math>n</math> balls are to be placed in <math>k</math> boxes and <math>n>k</math>, then at least one box must contain more than one ball.
  11 KB (1,986 words) - 18:13, 19 June 2024
• 1965 AHSME Problems/Problem 40
  ...^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) - 23:42, 21 October 2021
• Geometric mean
  ...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) - 21:04, 11 July 2008
• Principle of Inclusion-Exclusion
  ...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 (1,703 words) - 06:25, 24 March 2024
• Cauchy-Schwarz Inequality
  ...th>t</math> such that <math>a_i = t b_i</math> for all <math>1 \leq i \leq n</math>, or if one list consists of only zeroes. Along with the [[AM-GM Ineq ...ghtarrow{v}</math> and <math>\overrightarrow{w}</math> in <math>\mathbb{R}^n</math>, where <math>\overrightarrow{v} \cdot \overrightarrow{w}</math> is t
  13 KB (2,048 words) - 14:28, 22 February 2024
• Factorial
  ...[[recursion|recursive definition]] for the factorial is <math>n!=n \cdot (n-1)!</math>. * <math>4! = 24</math>
  10 KB (809 words) - 15:40, 17 March 2024
• Discriminant
  ==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 (768 words) - 16:56, 24 June 2024
• 2006 AIME II
  * [[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) - 11:32, 22 March 2011
• Fermat's Little Theorem
  ...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> <math>\square</math>
  16 KB (2,660 words) - 22:42, 28 August 2024

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