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• Pigeonhole Principle
  ...most one ball implies that <math>b_r \leq 1</math> for all <math>1 \leq r \leq n</math>, so <cmath>b_1 + b_1 + \cdots + b_n \geq n.</cmath> However, we kn ...ere exists a rational number <math>\frac{p}{q}</math> with <math>1 \leq q \leq n</math> such that <math>|x - \frac{p}{q}| < \frac{1}{nq}</math>.''
  11 KB (1,986 words) - 18:13, 19 June 2024
• 1965 AHSME Problems/Problem 40
  ...\neq 0</math> and <math>x^2+3x+1-|a| \geq 1</math> or <math>x^2+3x+1-|a| \leq -1</math>.
  3 KB (571 words) - 23:42, 21 October 2021
• Principle of Inclusion-Exclusion
  If <math>(A_i)_{1\leq i\leq n}</math> are finite sets, then:
  9 KB (1,703 words) - 00:20, 7 December 2024
• 2020 CIME II Problems/Problem 10
  ...n\{c,2c,3c,4c,5c\}</math>, for if <math>a+b\geq 6c</math> we have <math>6c\leq 2c+6\Rightarrow c\leq3/2\Rightarrow c=1</math>; a contradiction since <math
  2 KB (332 words) - 08:37, 30 December 2021
• Cauchy-Schwarz Inequality
  ...<math>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 In <cmath>|\langle \mathbf{a}, \mathbf{b}\rangle| \leq ||\mathbf{a}||||\mathbf{b}||</cmath> with equality holding only when <math>
  13 KB (2,048 words) - 14:28, 22 February 2024
• Fermat's Little Theorem
  ...iv j \pmod{p},</math> which means <math>i = j</math> as <math>1 \leq i, j \leq p-1.</math>
  16 KB (2,660 words) - 22:42, 28 August 2024
• Chebyshev's Inequality
  <math>n \left(\sum_{i=1}^{n}a_ib_i\right)\leq\left(\sum_{i=1}^{n}a_i\right)\left(\sum_{i=1}^{n}b_i\right)</math>.
  1 KB (214 words) - 19:32, 13 March 2022
• Geometric inequality
  ...m states that in a [[right triangle]] with sides of length <math>a \leq b \leq c</math> we have <math>a^2 + b^2 = c^2</math>. The Inequality extends this ...math>a^2+b^2>c^2</math>. For an obtuse triangle with sides <math>a \leq b \leq c</math>, <math>a^2+b^2<c^2</math>.
  7 KB (1,300 words) - 00:11, 28 October 2024
• Elementary symmetric sum
  ...he sum of all products of <math>k</math> of those numbers (<math>1 \leq k \leq n</math>). For example, if <math>n = 4</math>, and our set of numbers is <
  2 KB (275 words) - 11:51, 26 July 2023
• Euler's totient function
  <cmath>A_d=\{x:1\leq x\leq n\quad\text{and}\quad \operatorname{syt}(x,n)=d \}.</cmath> ...th>. Furthermore, <math>1\leq dx\leq n</math> if and only if <math>1\leq x\leq n/d</math>. Now one can see that the number of elements of <math>A_d</math>
  5 KB (903 words) - 14:49, 27 July 2024
• Rearrangement Inequality
  ...1 \leq a_2 \leq \cdots \leq a_n</math> and <math>b_1 \leq b_2 \leq \cdots \leq b_n</math>; and suppose the grouping that maximizes the desired sum of prod
  5 KB (804 words) - 12:54, 26 January 2023
• Divisibility rules
  Rule 3: Works for <math>1 \leq N \leq 1000</math>. Let <math>K = 3N</math>. If <math>K</math> is odd add 39 to
  10 KB (1,572 words) - 21:11, 22 September 2024
• Jensen's Inequality
  ...t states that <math>x_1^{\lambda_1}x_2^{\lambda_2} \cdots x_n^{\lambda_n} \leq \lambda_1 x_1 + \lambda_2 x_2 +\cdots+ \lambda_n x_n</math> when <math>\lam ...for any <math>\triangle ABC</math>, we have <math>\sin{A}+\sin{B}+\sin{C}\leq \frac{3\sqrt{3}}{2}</math>.
  3 KB (628 words) - 05:29, 28 November 2024
• Binomial Theorem
  ...<math>\binom{n}{m} + \binom{n}{m+1} = \binom{n+1}{m+1} </math> for <math>m\leq n</math>)
  5 KB (935 words) - 12:11, 20 February 2024
• Function
  ...A</math> and <math>B</math> are finite sets, injectivity implies <math>|A|\leq |B|</math>.
  10 KB (1,761 words) - 02:16, 12 May 2023
• Combinatorial identity
  ...\binom{n - 1}{k}</math> and so the result is trivial. So assume <math>k \leq n</math>. Then
  12 KB (1,993 words) - 22:47, 27 October 2024
• Henstock-Kurzweil integral
  <math>S(f,\mathcal{\dot{P}})\leq\sum_{k=1}^{\infty}\frac{\varepsilon}{2^k}<\varepsilon</math>
  2 KB (401 words) - 08:46, 31 January 2018
• Injection
  ...and <math>|X|\leq|Y|</math> and <math>|Y|\leq|Z|</math> implies <math>|X|\leq|Z|</math> because the composition of injections is again an injection.
  1 KB (228 words) - 00:01, 17 November 2019
• Isoperimetric Inequalities
  ...<math>A</math> and perimeter <math>P</math> then <math>\frac{4\pi A}{P^2} \leq 1</math>. This means that given a perimeter <math>P</math> for a plane figu
  789 bytes (115 words) - 16:08, 29 December 2021
• Sequence
  * '''increasing''' if <math>p_n\leq p_{n+1}</math> for all <math>n\in\mathbb{N}</math> and '''strictly increasi
  2 KB (413 words) - 20:18, 13 November 2022

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