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• Modular arithmetic/Introduction
  What is the last digit of <math>(...((7)^7)^7)...)^7</math> if there are 1000 7s as exponents and only one 7 in the middle? ...se 7 has a pattern of repetitive period 4 for the units digit. <math>(1)^{1000}</math> is simply 1, so therefore <math>7^1=7</math>, which really is the l
  15 KB (2,396 words) - 20:24, 21 February 2024
• 2006 AIME I Problems
  ...ots99!100!. </math> Find the remainder when <math> N </math> is divided by 1000. ...be constructed. What is the remainder when <math> T </math> is divided by 1000?
  7 KB (1,173 words) - 03:31, 4 January 2023
• 2006 AIME I Problems/Problem 15
  ...ill apply this to try and find some bounds. We can test if the first <math>1000</math> pairs of numbers each sum up to <math>-3</math>, and the rest form a
  6 KB (910 words) - 19:31, 24 October 2023
• 2006 AIME I Problems/Problem 13
  ...^{n-1}}g(2k). </math> Find the greatest integer <math> n </math> less than 1000 such that <math> S_n </math> is a [[perfect square]]. ...rfect square. Hence <math>k</math> must be even. In particular, as <math>n<1000</math>, we have five choices for <math>k</math>, namely <math>k=0,2,4,6,8</
  10 KB (1,702 words) - 00:45, 16 November 2023
• 2006 AIME I Problems/Problem 11
  ...constructed. What is the [[remainder]] when <math> T </math> is divided by 1000?
  3 KB (436 words) - 05:40, 4 November 2022
• 2006 AIME I Problems/Problem 4
  ...100!. </math> Find the remainder when <math> N </math> is divided by <math>1000</math>.
  2 KB (278 words) - 08:33, 4 November 2022
• 2006 AIME I Problems/Problem 3
  ...be <math>N</math> with the first digit deleted. Now, we know that <math>N<1000</math> (because this is an AIME problem). Thus, <math>N</math> has <math>1, ...six numbers was possible thanks to AIME problems having answers less than 1000).
  4 KB (622 words) - 03:53, 10 December 2022
• 2005 AMC 12A Problems
  A line passes through <math>A\ (1,1)</math> and <math>B\ (100,1000)</math>. How many other points with integer coordinates are on the line and ...numbers less than 1000. How many prime-looking numbers are there less than 1000?
  13 KB (1,971 words) - 13:03, 19 February 2020
• 2005 AMC 12B Problems
  A scout troop buys <math>1000</math> candy bars at a price of five for <math>2</math> dollars. They sell
  12 KB (1,781 words) - 12:38, 14 July 2022
• 2006 AMC 12B Problems/Problem 25
  so <math>999 = \max(a_1, a_2) \geq 1000</math>, a contradiction. Hence <math>(a_n)</math> completes at <math>i</mat
  5 KB (924 words) - 12:02, 15 June 2022
• 2005 AMC 12B Problems/Problem 1
  A scout troop buys <math>1000</math> candy bars at a price of five for <math>2</math> dollars. They sell \mbox{Expenses} &= 1000 \cdot \frac25 = 400 \\
  1 KB (179 words) - 13:53, 14 December 2021
• 2005 AIME II Problems
  For how many positive integers <math> n </math> less than or equal to 1000 is <math> (\sin t + i \cos t)^n = \sin nt + i \cos nt </math> true for all
  7 KB (1,119 words) - 21:12, 28 February 2020
• 2005 AIME II Problems/Problem 9
  For how many positive integers <math> n </math> less than or equal to <math>1000</math> is <math> (\sin t + i \cos t)^n = \sin nt + i \cos nt </math> true f ...on is equivalent to asking for how many [[positive integer]]s <math>n \leq 1000</math> we have that <math>\left(\sin\left(\frac\pi2 - u\right) + i \cos\lef
  6 KB (1,154 words) - 03:30, 11 January 2024
• 2005 AIME I Problems
  ...<math> S. </math> Find the remainder when <math> 10K </math> is divided by 1000.
  6 KB (983 words) - 05:06, 20 February 2019
• 2005 AIME I Problems/Problem 4
  ...ou have a LOT of time (and you've memorized all your perfect squares up to 1000). ...80</math>, and since AIME answers are nonnegative integers less than <math>1000</math>, we don't have to check any higher <math>n</math>. Also, we know tha
  8 KB (1,248 words) - 11:43, 16 August 2022
• 2005 AIME I Problems/Problem 14
  ...S. </math> Find the remainder when <math> 10K </math> is divided by <math>1000</math>. ...{44^2}{10}</math>. The remainder when <math>10K</math> is divided by <math>1000</math> is <math>936</math>.
  3 KB (561 words) - 14:11, 18 February 2018
• 2005 AIME I Problems/Problem 15
  <cmath>2000 + 180 m^2 = 10(10+2\sqrt{2}m)^{2} + 1000</cmath> <cmath>1000 + 180 m^2 = 1000 + 400\sqrt{2}m + 80 m^{2}</cmath>
  5 KB (906 words) - 23:15, 6 January 2024
• 2004 AIME I Problems/Problem 1
  ...underline{(n+2)}}\,{\underline{( n+1)}}\,{\underline {(n)}} </math><math>= 1000(n + 3) + 100(n + 2) + 10(n + 1) + n = 3210 + 1111n</math>, for <math>n \in ...be greater than 9), <math>n</math> is equal to <math>(d)+10(d+1)+100(d+2)+1000(d+3)</math> or <math>1111d +3210</math>. Now we try this number for <math>d
  2 KB (374 words) - 14:53, 27 December 2019
• 2004 AIME I Problems/Problem 6
  ...i </math> is [[even integer | even]]. How many snakelike integers between 1000 and 9999 have four distinct digits? ...the leading digit will be a zero, which is bad because all numbers between 1000 and 9999 have nonzero leading digits. So, we need to select our 4 digits on
  3 KB (562 words) - 18:12, 4 March 2022
• 2004 AIME I Problems/Problem 8
  ...are two non-similar regular 7-pointed stars. How many non-similar regular 1000-pointed stars are there? Note that <math>n = 1000 = 2^{3}5^{3}.</math>
  4 KB (620 words) - 21:26, 5 June 2021

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