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• 2006 AIME I Problems/Problem 15
  Suppose <math>b_{i} = \frac {x_{i}}3</math>. ...= \sum_{i = 0}^{2005}(b_{i} + 1)^{2} = \sum_{i = 0}^{2005}(b_{i}^{2} + 2b_{i} + 1)
  6 KB (910 words) - 19:31, 24 October 2023
• 2006 AIME I Problems/Problem 13
  ...s]] <math> x. </math> For example, <math> g(20)=4 </math> and <math> g(16)=16. </math> For each positive integer <math> n, </math> let <math> S_n=\sum_{k ...rac{n+1}{16} \le 62</math>, so <math>n+1 = 16, 16 \cdot 3^2, 16 \cdot 5^2, 16 \cdot 7^2</math>.
  10 KB (1,702 words) - 00:45, 16 November 2023
• 2006 AIME I Problems/Problem 5
  <cmath>2006=13^2x^2+4^2y^2+18^2z^2=169\cdot2+16\cdot3+324\cdot5</cmath> {{AIME box|year=2006|n=I|num-b=4|num-a=6}}
  3 KB (439 words) - 18:24, 10 March 2015
• 2022 AIME I Problems/Problem 1
  ...ctively. The graphs of both polynomials pass through the two points <math>(16,54)</math> and <math>(20,53).</math> Find <math>P(0) + Q(0).</math> R(16) &= P(16)+Q(16) &&= 54+54 &&= 108, \\
  4 KB (670 words) - 13:03, 13 November 2023
• 2021 JMC 10
  ...ts, P_8</math> stand in a line, and person <math>P_i</math> calls <math>P_{i+1}</math> a liar where <math>P_1 = P_9.</math> Out of these eight people, h ...{(B) } 9 \qquad\textbf{(C) } 12 \qquad\textbf{(D) } 15 \qquad\textbf{(E) } 16</math>
  12 KB (1,784 words) - 16:49, 1 April 2021
• 2004 AMC 12A Problems
  ...xt{(A) } 11 \qquad \text{(B) } 13 \qquad \text{(C) } 14 \qquad \text{(D) } 16 \qquad \text{(E) } 17</math> == Problem 16 ==
  13 KB (1,953 words) - 00:31, 26 January 2023
• 2000 AMC 12 Problems
  ...cdot O = 2001 </math>. What is the largest possible value of the sum <math>I + M + O</math>? == Problem 16 ==
  13 KB (1,948 words) - 10:35, 16 June 2024
• 2001 AMC 12 Problems
  ...that is, <math>A > B > C</math>, <math>D > E > F</math>, and <math>G > H > I > J</math>. Furthermore, ...</math> are consecutive even digits; <math>G</math>, <math>H</math>, <math>I</math>, and <math>J</math> are consecutive odd
  13 KB (1,957 words) - 12:53, 24 January 2024
• 2004 AMC 12B Problems
  ...A}) 13\qquad (\mathrm {B}) 14 \qquad (\mathrm {C}) 15 \qquad (\mathrm {D}) 16 \qquad (\mathrm {E}) 17</math> == Problem 16 ==
  13 KB (2,049 words) - 13:03, 19 February 2020
• 2005 AMC 12B Problems
  \mathrm{(C)}\ 16 \qquad ...\mathrm{(C)}\ {{{4}}} \qquad \mathrm{(D)}\ {{{8}}} \qquad \mathrm{(E)}\ {{{16}}}</math>
  12 KB (1,781 words) - 12:38, 14 July 2022
• 2006 AMC 12B Problems/Problem 17
  \mathrm{(D)}\ \frac 16 See also [[2016 AIME I Problems/Problem 2]]
  1 KB (188 words) - 22:10, 9 June 2016
• Talk:Modular arithmetic/Intermediate
  <math>13^{16} \equiv 21^2 \equiv 41</math> <math>404 = 256 + 128 + 16 + 4</math>
  4 KB (597 words) - 01:41, 19 December 2013
• 2006 AMC 12A Problems/Problem 25
  ...s. For each configuration, we can subtract <math>i-1</math> from the <math>i</math>-th element in your subset. This converts your configuration into a c ...subset is <math>15</math>. This is a total of <math>15-s+1</math> or <math>16-s</math> possible elements.
  8 KB (1,405 words) - 11:52, 27 September 2022
• 2005 AMC 12B Problems/Problem 24
  ...> \mathrm{(A)}\ {{{14}}}\qquad\mathrm{(B)}\ {{{15}}}\qquad\mathrm{(C)}\ {{{16}}}\qquad\mathrm{(D)}\ {{{17}}}\qquad\mathrm{(E)}\ {{{18}}} </math> ...tiplying the two complex numbers <math>1 + 2i</math> and <math>1 + \sqrt{3}i</math>. What this does is generate the complex number that is at a <math>60
  4 KB (761 words) - 09:10, 1 August 2023
• 2006 AMC 10A Problems
  D((0,0)--(16,0)--(16,-16)--(0,-16)--cycle); D((16,-8)--(24,-8));
  13 KB (2,028 words) - 16:32, 22 March 2022
• 2006 AMC 10A Problems/Problem 12
  ...s dog with an 8-foot rope to a [[square (geometry) | square]] shed that is 16 feet on each side. His preliminary drawings are shown. D((0,0)--(16,0)--(16,-16)--(0,-16)--cycle);
  3 KB (424 words) - 10:14, 17 December 2021
• 2022 USAJMO Problems/Problem 1
  ...math> (<math>x</math> and <math>y</math> can be equal), ie. <math>4,8,9,12,16,18,20,24,25,\dots </math>. ...s that all non square-free m (4, 8, 9, 16, 18, 25...) should all work, but I don't have a proof. However, if you edit the one above, you can see non squ
  5 KB (883 words) - 01:05, 2 June 2024
• 2005 AIME II Problems
  Let <math> x=\frac{4}{(\sqrt{5}+1)(\sqrt[4]{5}+1)(\sqrt[8]{5}+1)(\sqrt[16]{5}+1)}. </math> Find <math> (x+1)^{48}. </math> ...th> less than or equal to 1000 is <math> (\sin t + i \cos t)^n = \sin nt + i \cos nt </math> true for all real <math> t </math>?
  7 KB (1,119 words) - 21:12, 28 February 2020
• 2005 AIME I Problems/Problem 4
  16 & 368 & no\\ \hline {{AIME box|year=2005|n=I|num-b=3|num-a=5}}
  8 KB (1,248 words) - 11:43, 16 August 2022
• 2005 AIME I Problems/Problem 9
  ...contribute a probability of <math>\left(\frac{1}{4}\right)^8 = \frac{1}{2^{16}}</math> ...<math>\frac{2^6}{3^6} \cdot \frac{5^{12}}{2^{24}3^{12}} \cdot \frac{1}{2^{16}} = \frac{5^{12}}{2^{34}\cdot 3^{18}}</math> and so the answer is <math>2 +
  4 KB (600 words) - 21:44, 20 November 2023

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