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• 2004 AIME I Problems/Problem 1
  == Problem == == Solution 1==
  2 KB (250 words) - 23:56, 2 December 2024

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• Ptolemy's theorem
  == Proof 1 == ...lity, with equality holding when <math>A^*, B^*, C^*</math> are collinear, i.e. when <math>A,B,C</math> lie on a circle containing <math>D.</math> Addit
  6 KB (922 words) - 17:34, 13 January 2025
• Overcounting
  An example of a classic problem is as follows: ...a coincidence that 67 is close to two-thirds of 100! We can approach this problem in a constructive way, building the set based on the remainders when divide
  4 KB (635 words) - 12:19, 2 January 2022
• Constructive counting
  === Example 1 === This is a problem where constructive counting is not the simplest way to proceed. This next e
  13 KB (2,018 words) - 15:31, 10 January 2025
• Recursion
  ...early values of the sequence in terms of previous values: <math>F_0=1, F_1=1, F_2=2, F_3=3, F_4=5, F_5=8</math>, and so on. ...defined recursively by <math>a_0 = 1</math> and <math>a_n = 2\cdot a_{n - 1}</math> for <math>n > 0</math> also has the closed-form definition <math>a_
  2 KB (316 words) - 16:03, 1 January 2024
• Fibonacci sequence
  ...term is the sum of the two preceding it. The first few terms are <math>1, 1, 2, 3, 5, 8, 13, 21, 34, 55,...</math>. ...[[recursion|recursively]] as <math>F_1 = F_2 = 1</math> and <math>F_n=F_{n-1}+F_{n-2}</math> for <math>n \geq 3</math>. This is the simplest nontrivial
  7 KB (1,111 words) - 14:57, 24 June 2024
• Complex number
  ...numbers''' arise when we try to solve [[equation]]s such as <math> x^2 = -1 </math>. ...this addition, we are not only able to find the solutions of <math> x^2 = -1 </math> but we can now find ''all'' solutions to ''every'' polynomial. (Se
  5 KB (860 words) - 15:36, 10 December 2023
• Arithmetic sequence
  ...h>1, 2, 3, 4</math> is an arithmetic sequence with common difference <math>1</math> and <math>99, 91, 83, 75</math> is an arithmetic sequence with commo ...ogression if and only if <math>a_2 - a_1 = a_3 - a_2 = \cdots = a_n - a_{n-1}</math>. A similar definition holds for infinite arithmetic sequences. It a
  4 KB (736 words) - 02:00, 7 March 2024
• 2004 AIME I
  ...contains the full set of test problems. The rest contain each individual problem and its solution. * [[2004 AIME I Problems]]
  1 KB (135 words) - 18:15, 19 April 2021
• 2004 AIME II
  ...contains the full set of test problems. The rest contain each individual problem and its solution. * [[2004 AIME II Problems]]
  1 KB (135 words) - 12:24, 22 March 2011
• 2005 AIME I
  ...contains the full set of test problems. The rest contain each individual problem and its solution. * [[2005 AIME I Problems]]
  1 KB (154 words) - 12:30, 22 March 2011
• Mock AMC
  ...A number of '''Mock AMC''' competitions have been hosted on the [[Art of Problem Solving]] message boards. They are generally made by one community member ...amous theorems and formulas and see if there's any way you can make a good problem out of them.
  51 KB (6,175 words) - 21:41, 27 November 2024
• 2005 AIME I Problems
  {{AIME Problems|year=2005|n=I}} == Problem 1 ==
  6 KB (983 words) - 05:06, 20 February 2019
• 2005 AIME I Problems/Problem 2
  == Problem == ...ath> k</math>. For example, <math> S_3 </math> is the [[sequence]] <math> 1,4,7,10,\ldots. </math> For how many values of <math> k </math> does <math>
  2 KB (303 words) - 01:31, 5 December 2022
• 2004 AIME I Problems/Problem 15
  == Problem == ...\text{if }x = 1\ \frac x{10} & \text{if }x\text{ is divisible by 10}\ x+1 & \text{otherwise}\end{cases}
  9 KB (1,491 words) - 01:23, 26 December 2022
• 2004 AIME I Problems/Problem 14
  == Problem == label("$8$",O1--B1,W);label("$4$",(5*A1+O1)/6,0.8*(1,1));
  4 KB (729 words) - 01:00, 27 November 2022
• 2004 AIME I Problems/Problem 13
  == Problem == ...s, 34, </math> with <math> 0 < a_1 \le a_2 \le a_3 \le \cdots \le a_{34} < 1 </math> and <math> r_k>0. </math> Given that <math> a_1 + a_2 + a_3 + a_4 +
  2 KB (298 words) - 20:02, 4 July 2013
• 2004 AIME I Problems/Problem 12
  == Problem == ...ordered pair]]s <math> (x, y) </math> such that <math> 0 < x \le 1, 0<y\le 1, </math> and <math> \left[\log_2{\left(\frac 1x\right)}\right] </math> and
  2 KB (303 words) - 18:43, 16 October 2024
• 2004 AIME I Problems/Problem 11
  == Problem == ===Solution 1===
  5 KB (839 words) - 22:12, 16 December 2015
• 2004 AIME I Problems/Problem 10
  == Problem == A [[circle]] of [[radius]] 1 is randomly placed in a 15-by-36 [[rectangle]] <math> ABCD </math> so that
  5 KB (836 words) - 07:53, 15 October 2023
• 2004 AIME I Problems/Problem 9
  == Problem == label("$V_1$",(1,1.2)); label("$U_1$",(3,0.3));
  4 KB (647 words) - 17:43, 23 November 2024

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