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• Algebraic geometry
  ...[prime ideal]] of <math>R</math>. Then <math>V(I)=\{p\in\mathbb{A}^n\mid f(p)=0\mathrm{\ for\ all\ } f\in I\}</math> is called an '''affine algebraic va ...ective varieties are algebraic varieties. A projective space <math>\mathbb{P}^n</math> is a quotient set with an equivalence class satisfying
  2 KB (361 words) - 01:59, 24 January 2020
• Mock AMC
  | [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=102483#p102483 Answers] [http://www.artofproblemsolving.com/community/c5h14 | [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=105741#p105741 Results / Discussion]
  51 KB (6,175 words) - 20:58, 6 December 2023
• Uncountable
  ...example of an uncountable set is the set of [[real number]]s <math>\mathbb{R}</math>. == Proof that <math>\mathbb{R}</math> is uncountable ==
  2 KB (403 words) - 20:53, 13 October 2019
• Subset
  ...} \subset \mathbb{Q} \subset \mathbb{R} \subset \mathbb{C} \subset \mathbb{C}\cup\{\textrm{Groucho, Harpo, Chico}\} \supset \{1, 2, i, \textrm{Groucho}\ ...s called the [[power set]] of <math>S</math> and is denoted <math>\mathcal{P}(S)</math> or <math>2^S</math>. The number of subsets of <math>S</math> is
  1 KB (217 words) - 09:32, 13 August 2011
• 2006 AIME I Problems
  ...sqrt{2}+b\sqrt{3}+c\sqrt{5}, </math> where <math> a, b, </math> and <math> c </math> are positive integers. Find <math> abc. </math> ...ng decimals of the form <math> 0.\overline{abc} </math> where <math> a, b, c </math> are distinct digits. Find the sum of the elements of <math> \mathca
  7 KB (1,173 words) - 03:31, 4 January 2023
• 2006 AIME I Problems/Problem 8
  ...and <math> \mathcal{T,} </math> as shown. Rhombuses <math> \mathcal{P, Q, R,} </math> and <math> \mathcal{S} </math> are [[congruent (geometry) | congr pair A=(0,0), B=(4.2,0), C=(5.85,-1.6), D=(4.2,-3.2), EE=(0,-3.2), F=(-1.65,-1.6), G=(0.45,-1.6), H=(3
  5 KB (730 words) - 15:05, 15 January 2024
• 2006 AIME I Problems/Problem 5
  ...sqrt{2}+b\sqrt{3}+c\sqrt{5}, </math> where <math> a, b, </math> and <math> c </math> are [[positive]] [[integer]]s. Find <math>abc</math>. <cmath> a\sqrt{2}+b\sqrt{3}+c\sqrt{5} = \sqrt{104\sqrt{6}+468\sqrt{10}+144\sqrt{15}+2006}</cmath>
  3 KB (439 words) - 18:24, 10 March 2015
• 2022 AIME I Problems/Problem 1
  ...h the two points <math>(16,54)</math> and <math>(20,53).</math> Find <math>P(0) + Q(0).</math> ...of <math>P(x)</math> and <math>Q(x)</math> cancel, we conclude that <math>R(x)</math> is a linear polynomial.
  4 KB (670 words) - 13:03, 13 November 2023
• 2021 JMC 10
  <math>\textbf{(A) } 3 \qquad\textbf{(B) } 7 \qquad\textbf{(C) } 8 \qquad\textbf{(D) } 9 \qquad\textbf{(E) } 13</math> <math>\textbf{(A) }\pi-e \qquad\textbf{(B) }2\pi-2e\qquad\textbf{(C) }2e\qquad\textbf{(D) }2\pi \qquad\textbf{(E) }2\pi +e</math>
  12 KB (1,784 words) - 16:49, 1 April 2021
• 2005 AMC 12A Problems
  (\mathrm {A}) \ 1 \qquad (\mathrm {B}) \ 2 \qquad (\mathrm {C})\ 5 \qquad (\mathrm {D}) \ 10 \qquad (\mathrm {E})\ 20 (\mathrm {A}) \ -8 \qquad (\mathrm {B}) \ -4 \qquad (\mathrm {C})\ 2 \qquad (\mathrm {D}) \ 4 \qquad (\mathrm {E})\ 8
  13 KB (1,971 words) - 13:03, 19 February 2020
• 2003 AMC 12A Problems
  <math> \mathrm{(A) \ } 0\qquad \mathrm{(B) \ } 1\qquad \mathrm{(C) \ } 2\qquad \mathrm{(D) \ } 2003\qquad \mathrm{(E) \ } 4006 </math> <math> \mathrm{(A) \ } 77\qquad \mathrm{(B) \ } 91\qquad \mathrm{(C) \ } 143\qquad \mathrm{(D) \ } 182\qquad \mathrm{(E) \ } 286 </math>
  13 KB (1,955 words) - 21:06, 19 August 2023
• 2002 AMC 12A Problems
  <math> \mathrm{(A) \ } \frac{7}{2}\qquad \mathrm{(B) \ } 4\qquad \mathrm{(C) \ } 5\qquad \mathrm{(D) \ } 7\qquad \mathrm{(E) \ } 13 </math> <math> \mathrm{(A) \ } 15\qquad \mathrm{(B) \ } 34\qquad \mathrm{(C) \ } 43\qquad \mathrm{(D) \ } 51\qquad \mathrm{(E) \ } 138 </math>
  12 KB (1,792 words) - 13:06, 19 February 2020
• 2000 AMC 12 Problems
  <math>\textbf{(A)}\ 23 \qquad \textbf{(B)}\ 55 \qquad \textbf{(C)}\ 99 \qquad \textbf{(D)}\ 111 \qquad \textbf{(E)}\ 671</math> ...\textbf{(A)}\ 2000^{2001} \qquad \textbf{(B)}\ 4000^{2000} \qquad \textbf{(C)}\ 2000^{4000} \qquad \textbf{(D)}\ 4,000,000^{2000} \qquad \textbf{(E)}\ 2
  13 KB (1,948 words) - 12:26, 1 April 2022
• 2002 AMC 12B Problems
  \qquad\mathrm{(C)}\ 4 \qquad\mathrm{(C)}\ 10
  10 KB (1,547 words) - 04:20, 9 October 2022
• 2003 AMC 12B Problems
  \text {(A) } -1 \qquad \text {(B) } -\frac{2}{3} \qquad \text {(C) } \frac{2}{3} \qquad \text {(D) } 1 \qquad \text {(E) } \frac{14}{3} \text {(A) } 7 \qquad \text {(B) } 14 \qquad \text {(C) } 19 \qquad \text {(D) } 20 \qquad \text {(E) } 39
  13 KB (1,987 words) - 18:53, 10 December 2022
• 2005 AMC 12B Problems
  \mathrm{(C)}\ 300 \qquad \mathrm{(C)}\ 10 \qquad
  12 KB (1,781 words) - 12:38, 14 July 2022
• 2006 AMC 12A Problems/Problem 22
  ...m a randomly chosen point on the circle is <math>1/2</math>. What is <math>r</math>? ...\sqrt{2}+2\sqrt{3}\qquad \mathrm{(B) \ } 3\sqrt{3}+\sqrt{2}\qquad \mathrm{(C) \ } 2\sqrt{6}+\sqrt{3}\qquad \rm{(D) \ } 3\sqrt{2}+\sqrt{6}\qquad \mathrm{
  2 KB (343 words) - 15:39, 14 June 2023
• Root (polynomial)
  ...and so on, there are in fact exactly <math>n</math> complex roots of <math>P(x)</math>. ...math>Q(x)</math> with degree <math>n</math> having the same roots as <math>P(x)</math>, given by <cmath>Q(x) = x^n + \frac{c_{n-1}x^{n-1}}{c_n} + \dots
  8 KB (1,427 words) - 21:37, 13 March 2022
• Set
  ...ion <math>S</math> of elements of <math>T</math> which have property <math>P</math> will always be a set. (In particular, for our previous example to l <math>\{ x \in \mathbb {R} \mid x^2>0 \}</math>
  11 KB (2,021 words) - 00:00, 17 July 2011
• 2005 AIME II Problems
  ...th> n </math> is not divisible by the square of any prime, find <math> m+n+p. </math> ...th> and that the ratio of the volume of <math> O </math> to that of <math> C </math> is <math> \frac mn, </math> where <math> m </math> and <math> n </m
  7 KB (1,119 words) - 21:12, 28 February 2020

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