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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.
  ...as odd digits, there is 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
  ...n [[geometry]], and is one of the many tools in a good geometer's arsenal. A very large number of geometry problems can be solved by building right tria In these proofs, we will let <math>ABC </math> be any right triangle with a right angle at <math>\angle ACB</math>.
  5 KB (885 words) - 22:03, 5 October 2024
• 2016 AMC 10A Problems/Problem 9
  ...coins in the <math>N</math>th row. What is the sum of the digits of <math>N</math>? <math>\textbf{(A)}\ 6\qquad\textbf{(B)}\ 7\qquad\textbf{(C)}\ 8\qquad\textbf{(D)}\ 9\qquad\t
  2 KB (315 words) - 14:34, 18 June 2022
• Physics
  <strong>Physics</strong> is a branch of [[science]] that studies the properties of matter, energy, and ma Modern Physics is also a group of different subjects in physics:
  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
• Newton
  A '''Newton''' (abbreviated N) is the [[Système international|metric]] measure of [[force]], named for [ ...[[kilogram]] times one [[meter]] per [[second]] squared, or <math>\mathrm{N}=\mathrm{kg}\times \frac{\mathrm{m}}{\mathrm{s}^2}</math>. This is because
  665 bytes (96 words) - 22:17, 2 February 2021
• 1954 AHSME Problems/Problem 46
  In the diagram, if points <math>A, B</math> and <math>C</math> are points of tangency, then <math>x</math> eq pair A=(-3*sqrt(3)/32,9/32), B=(3*sqrt(3)/32, 9/32), C=(0,9/16);
  3 KB (415 words) - 17:01, 24 May 2020
• 2015 IMO Problems
  We say that a finite set <math>\mathcal{S}</math> in the plane is <i> balanced </i> if, for any two different points <math>A</math>, <math>B</math> in <math>\mathcal{S}</math>, there is
  4 KB (709 words) - 14:00, 1 June 2024
• Power Mean Inequality
  The '''Power Mean Inequality''' is a generalized form of the multi-variable [[Arithmetic Mean-Geometric Mean]] I ...>n</math> positive real weights <math>w_i</math> with sum <math>\sum_{i=1}^n w_i=1</math>, the power mean with exponent <math>t</math>, where <math>t\in
  3 KB (619 words) - 12:57, 23 August 2024
• Wilson's Theorem
  Suppose first that <math>p</math> is composite. Then <math>p</math> has a factor <math>d > 1</math> that is less than or equal to <math>p-1</math>. ...ique, and each number is the inverse of its inverse. If one integer <math>a</math> is its own inverse, then
  4 KB (639 words) - 00:53, 2 February 2023
• Trivial Inequality
  <cmath>ax^2+bx+c = a(x^2+\frac{b}{a}x+\frac{b^2}{4a^2})+c-\frac{b^2}{4a} = a(x+\frac{b}{2a})^2+c-\frac{b^2}{4a}.</cmath> .../math> is positive and <math>ax^2+bx+c\le c-\frac{b^2}{4a}</math> if <math>a</math> is negative.
  3 KB (583 words) - 20:20, 2 August 2024
• Arithmetic mean
  The '''arithmetic mean''' of a [[set]] of numbers (or variables) is the sum of all the numbers, divided by <center><math>AM=\frac{x_1+x_2+\cdots+x_n}{n}</math></center>
  699 bytes (110 words) - 11:44, 20 September 2015
• Polynomial
  ...algebra]] often an arbitrary [[field]]). Note that a [[constant]] is also a polynomial. ===A More Precise Definition===
  6 KB (1,100 words) - 14:57, 30 August 2024
• Simon's Favorite Factoring Trick
  <math> \mathrm{(A) \ 22 } \qquad \mathrm{(B) \ 60 } \qquad \mathrm{(C) \ 119 } \qquad \mathrm ...when divided by <math>5</math>, find the value of the remainder when <math>n</math> is divided by <math>5</math>.
  4 KB (678 words) - 12:12, 26 November 2024
• Factoring
  ...rs, is to write the expression (often an [[integer]] or [[polynomial]]) as a product of different terms. This often allows one to find information abou Using the formula for the sum of a [[geometric sequence]], it's easy to derive the general formula for differe
  3 KB (532 words) - 21:00, 13 January 2024
• Proof by contradiction
  ...fs look better --> This technique usually works well on problems where not a lot of information is known, and thus we can create some using proof by con ...d as a rational fraction of the form <math>\frac{b}{a}</math>, where <math>a</math> and <math>b</math> are two [[relatively prime]] integers. Now, sinc
  2 KB (374 words) - 13:01, 21 August 2022
• Mathematical problem solving
  ...ng]] involves using all the tools at one's disposal to attack a problem in a new way. == A Historical Example ==
  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
• Relatively prime
  ...ath> and <math>n</math> are relatively prime if and only if <math>\frac{m}{n}</math> is in lowest terms. ...th> and <math>n+1</math>, then it must divide their difference <math>(n+1)-n = 1</math>, which is impossible since <math>p > 1</math>.
  2 KB (245 words) - 17:45, 14 October 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
  .../math> and <math>c</math> is the number <math>a</math> such that <math>a + a = b + c</math>, while the geometric mean of the numbers <math>b</math> and ...length of [[line segment]] <math>AM</math> is the geometric mean of <math>a</math> and <math>b</math>.
  2 KB (282 words) - 21:04, 11 July 2008

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