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• 2010 AIME II Problems/Problem 8
  <LI> <math>\mathcal{A} \cap \mathcal{B} = \emptyset</math>, </LI>
  4 KB (699 words) - 20:57, 20 July 2023
• Bézout's Identity
  ...a_iu_i: u_1, \dots, u_m \in \mathbb{Z}\}</math>. Obviously, <math>P \neq \emptyset</math>. Thus, because all the elements of <math>P</math> are positive, by t
  4 KB (768 words) - 16:50, 6 September 2023
• 1997 USAMO Problems/Problem 6
  ...th> and <math>k</math> are positive integers, then <math>I_n\cap I_{k}\ne \emptyset</math>. To prove this, we need the following lemma. ...his contradicts the statement of Lemma 1. Therefore, <math>I_k\cap I_n\ne \emptyset</math>.
  6 KB (1,107 words) - 14:12, 12 April 2023
• 2006 Romanian NMO Problems/Grade 10/Problem 1
  ...ve <math>f(\emptyset) = f(\emptyset) + f(\emptyset)</math>, hence <math>f(\emptyset) = 0</math>. Also, if <math>A \subset B \subset M</math>, then <math>f(B) = Let <cmath>A_{0} = \emptyset \subsetneq A_{1} \subsetneq \dotsb \subsetneq A_{n-1} \subsetneq A_{n} = M<
  4 KB (796 words) - 10:53, 8 May 2012
• 2014 AMC 12B Problems/Problem 22
  <cmath>a=0.1\emptyset+0.9b,b=0.2a+0.8c,c=0.3b+0.7d,d=0.4c+0.6e</cmath> <cmath>10a=\emptyset+9b,10b=2a+8c,10c=3b+7d,10d=4c+6e</cmath>
  7 KB (1,082 words) - 22:35, 3 April 2024
• 2014 AIME I Problems/Problem 13
  <cmath>51=\emptyset</cmath> <cmath>68=\emptyset\text{ others}</cmath>
  9 KB (1,404 words) - 21:07, 13 October 2023
• 2006 iTest Problems
  ...by the rule <cmath>P(n):=\begin{cases}\min(s)_{s\in S_n}&\text{if }S_n\neq\emptyset,\\0&\text{otherwise}.\end{cases}</cmath> Let <math>d</math> be the least up
  31 KB (4,811 words) - 00:02, 4 November 2023
• 2016 USAMO Problems
  ...union is not the whole set <math>S</math>, that is, <math>X_i\cap X_{i+1}=\emptyset</math> and <math>X_i\cup X_{i+1}\neq S</math>, for all <math>i\in\{1, \ldot
  4 KB (608 words) - 13:49, 22 November 2023
• 2016 USAJMO Problems
  ...union is not the whole set <math>S</math>, that is, <math>X_i\cap X_{i+1}=\emptyset</math> and <math>X_i\cup X_{i+1}\neq S</math>, for all <math>i\in\{1, \ldot
  3 KB (414 words) - 16:43, 5 August 2023
• 2016 USAMO Problems/Problem 1
  ...union is not the whole set <math>S</math>, that is, <math>X_i\cap X_{i+1}=\emptyset</math> and <math>X_i\cup X_{i+1}\neq S</math>, for all <math>i\in\{1, \ldot \emptyset & i\equiv 0 \text{ (mod } 9\text{)} \\
  7 KB (985 words) - 18:11, 11 June 2017
• 1994 USAMO Problems/Problem 5
  ...can use PIE to show that <math>\left|\bigcup_{k \in U}A_k\right| = \sum_{\emptyset \neq U \subseteq S} (-1)^{|U|+1}\left|\bigcap_{k \in U}A_k\right|</math>. <math>\binom{m}{|S|}-\pi(S) = \sum_{\emptyset \neq U \subseteq S} (-1)^{|U|+1}\binom{m- \sigma(U)}{|S|} \iff \pi(S) = \su
  3 KB (628 words) - 10:42, 4 August 2023
• 2017 AMC 10B Problems/Problem 17
  ...ce it will always be a leading digit and that is not allowed. Also, <math>\emptyset</math> (the empty set) isn't included because it doesn't generate a number. ...s included since we are allowed to end numbers with zeros. However, <math>\emptyset</math> (the empty set) still isn't included because it doesn't generate a n
  7 KB (1,158 words) - 19:34, 27 March 2024
• 2018 AIME I Problems
  ...th>s(T)</math> be the sum of the elements of <math>T</math>, with <math>s(\emptyset)</math> defined to be <math>0</math>. If <math>T</math> is chosen at random
  8 KB (1,284 words) - 14:35, 9 August 2021
• 2018 AIME I Problems/Problem 12
  ...th>s(T)</math> be the sum of the elements of <math>T</math>, with <math>s(\emptyset)</math> defined to be <math>0</math>. If <math>T</math> is chosen at random ...^3=8</math> total subsets, and <math>w(3)=4</math> (the subsets are <math>\emptyset, \{0\}, \{1,2\},</math> and <math>\{1,2,0\}</math>). Now consider the first
  26 KB (4,044 words) - 13:58, 24 January 2024
• 2019 USAMO Problems/Problem 4
  ...h S_{ij}</math>. Begin by considering <math>C_0</math> and <math>S_{00} = \emptyset</math>. Then given <math>S_{i0}</math> we can create <math>S_{(i+1)0}</mat
  7 KB (1,288 words) - 19:17, 26 April 2023
• 2006 iTest Problems/Problem 34
  ...by the rule <cmath>P(n):=\begin{cases}\min(s)_{s\in S_n}&\text{if }S_n\neq\emptyset,\\0&\text{otherwise}.\end{cases}</cmath> Let <math>d</math> be the least up
  2 KB (346 words) - 18:04, 13 September 2020
• 2023 AIME II Problems
  ...<math>X</math> and <math>Y</math> in the collection, <math>X \cap Y \not= \emptyset.</math>
  8 KB (1,370 words) - 21:34, 28 January 2024
• 2022 AIME I Problems/Problem 12
  (A, B) \in \left\{ (\emptyset, \emptyset) , ( \{1\} , \{1\} ), ( \{1\} , \{2\} ) , ( \{2\} , \{1\} ) , ( \{2\} , \{2
  9 KB (1,471 words) - 16:41, 1 February 2024
• 2022 AIME I Problems
  ...sum is taken over the pairs of subsets <cmath>(A,B) \in \left\{(\emptyset,\emptyset),(\{1\},\{1\}),(\{1\},\{2\}),(\{2\},\{1\}),(\{2\},\{2\}),(\{1,2\},\{1,2\})\
  9 KB (1,520 words) - 19:06, 2 January 2023
• 2023 AIME II Problems/Problem 11
  ...<math>X</math> and <math>Y</math> in the collection, <math>X \cap Y \not= \emptyset.</math> This entails <math>\emptyset \in \mathcal C</math>.
  12 KB (2,014 words) - 00:58, 12 February 2024

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