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• Physics
  Classical Physics can be split even further into its own branches: ...n's most famous achievement is the formal statement of three basic, almost trivial laws of motion:
  9 KB (1,355 words) - 06:29, 29 September 2021
• Proof writing
  ...or mathematical culture; for instance, when a mathematician uses the word "trivial" in a proof, they intend a different meaning to how the word is understood ...[induction|proof by induction]]), and [[proof by contradiction]] (which in its simplest form requires only the demonstration of a counterexample).
  3 KB (501 words) - 23:46, 6 September 2024
• Trivial Inequality
  ...tes that the square of any real number is nonnegative. Its name comes from its simplicity and straightforwardness. The trivial inequality is one of the most commonly used theorems in mathematics. It is
  3 KB (583 words) - 20:20, 2 August 2024
• Pigeonhole Principle
  ...>n</math> holes, one hole must contain two or more pigeons. This seemingly trivial statement may be used with remarkable creativity to generate striking count ...or blue. Prove that there exists a rectangle in the plane such that all of its vertices are the same color. ([[Pigeonhole Principle/Solutions#O5|Solution]
  11 KB (1,986 words) - 18:13, 19 June 2024
• Cauchy-Schwarz Inequality
  ...t algebra, calculus, and contest mathematics. In high-school competitions, its applications are limited to elementary and linear algebra. Its elementary algebraic formulation is often referred to as '''Cauchy's Inequa
  13 KB (2,048 words) - 14:28, 22 February 2024
• Prime number
  ...ed as being neither prime nor [[composite number|composite]] because it is its only factor among the [[natural number|natural numbers]]. ...)\cdot (2b+1)=2(2ab+a+b)+1</cmath> and if <math>a,b>0</math> this is a non-trivial factorization.
  6 KB (1,036 words) - 17:26, 2 September 2024
• Ellipse
  ...|foci]] is a constant. (The equivalence of these two definitions is a non-trivial fact.) ...tangent line | tangent]] to the <math>x</math>-axis. What is the length of its [[major axis]]? ([[1985 AIME Problems/Problem 11|Source]])
  5 KB (892 words) - 20:52, 1 May 2021
• Rational approximation
  ...real line, we, surely, can make the difference between <math>x</math> and its rational approximation <math>\frac pq</math> as small as we wish. The probl ==Trivial theorem==
  7 KB (1,290 words) - 11:18, 30 May 2019
• Goldbach Conjecture
  ...o odd, and if <math>m</math> is even, then <math>n-m</math> is even, a non-trivial relation because (besides 2) only odd numbers can be prime. Similarly, if < ...p,q,2p+1,2q+1\in\mathbb{P}</math> then <math>4n+2</math> can be skipped as its the sum of the latter two primes. </li>
  7 KB (1,201 words) - 15:59, 19 February 2024
• Prime Number Theorem
  is easier, though still non-trivial!) It gives an (In fact, <math>A(x)</math> does not seem to tend to this value, but its
  10 KB (1,729 words) - 18:52, 21 October 2023
• Riemann zeta function
  by [[l'Hôpital's Rule]], so the pole at <math>s=1</math> is simple, and its <math>n</math> a postive integer. These are called the trivial zeros.
  9 KB (1,547 words) - 02:04, 13 January 2021
• 2000 AIME II Problems/Problem 12
  ...math>OA = OB = OC = 20</math>. First we can break up <math>ABC</math> into its two component right triangles <math>5-12-13</math> and <math>9-12-15</math> this is highly trivial for an AIME #12
  4 KB (614 words) - 17:12, 17 November 2024
• Centroid
  ...gure is cut out from uniform cardboard, say, and you connected a string to its centroid and held the other end of the string, the figure would be perfectl ...Theorem]]. However, there are many interesting and elegant ways to prove its existence, such as those shown below.''
  3 KB (612 words) - 16:21, 11 September 2024
• Yang-Mills Theory
  ...satisfy the axioms of axiomatic quantum field theory, it would have to be trivial (i.e. a free field theory). ...pin our hopes on for a nontrivial constructive QFT model in 4D. (QCD, with its fermionic quarks is obviously more complicated).
  2 KB (363 words) - 14:31, 1 December 2015
• Orthocenter
  The '''orthocenter''' of a [[triangle]] is the point of intersection of its [[altitude|altitudes]]. It is [[mathematical convention | conventionally]] ''Note: The orthocenter's existence is a trivial consequence of the trigonometric version of [[Ceva's Theorem]]; however, th
  5 KB (828 words) - 13:09, 18 October 2024
• Euler line
  ...h>L</math>. It is named after [[Leonhard Euler]]. Its existence is a non-trivial fact of Euclidean [[geometry]]. Certain fixed orders and distance [[ratio]]
  59 KB (10,203 words) - 03:47, 30 August 2023
• Nine-point circle
  ...oints of the segments joining the [[vertex | vertices]] of the triangle to its [[orthocenter]]. (These points are sometimes known as the [[Euler point]]s ...tangent to the incircle, has a radius equal to half the circumradius, and its center is the midpoint of the segment connecting the orthocenter and the ci
  6 KB (994 words) - 15:02, 12 March 2024
• 2003 AMC 12A Problems/Problem 25
  ...{2a}\right)^2 - \frac{b^2}{4a}} \le \sqrt{\frac{-b^2}{4a}}</math> by the [[Trivial Inequality]] (remember that <math>a \le 0</math>). Since <math>f</math> is ...}{a}</math>. Now, we can use the function we deduced to figure out some of its properties when:
  9 KB (1,606 words) - 10:34, 10 July 2020
• Ordered pair
  ...ssibilities, we should order <math>b</math> after <math>a</math> (to input its value into calculating the minimum b) in any programming or math. We don't In the language of set theory, it is not trivial to define an ordered pair since the set <math>\{a,b\}</math> and <math>\{b,
  2 KB (303 words) - 17:15, 29 August 2024
• 2002 USAMO Problems/Problem 1
  The base case, <math>n = 0 </math>, is trivial. ...of the union of two red (or blue) sets will have a red (or blue) number as its largest element. In addition, for each integer <math>n \in S</math>, there
  3 KB (498 words) - 13:57, 16 June 2020

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