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• 2001 AMC 12 Problems
  two-digit number such that <math>N = P(N)+S(N)</math>. What is the units digit o A telephone number has the form <math>\text{ABC-DEF-GHIJ}</math>, where each letter represents
  13 KB (1,957 words) - 12:53, 24 January 2024
• 2004 AMC 12B Problems
  In the expression <math>c\cdot a^b-d</math>, the values of <math>a</math>, <math>b</math>, <math>c</math>, and Minneapolis-St. Paul International Airport is 8 miles southwest of downtown St. Paul and
  13 KB (2,049 words) - 13:03, 19 February 2020
• 2006 AMC 12B Problems/Problem 14
  {{AMC12 box|year=2006|ab=B|num-b=13|num-a=15}}
  1 KB (227 words) - 17:21, 8 December 2013
• 2006 AMC 12B Problems/Problem 25
  A sequence <math>a_1,a_2,\dots</math> of non-negative integers is defined by the rule <math>a_{n+2}=|a_{n+1}-a_n|</math> is a 1-1 correspondence between the odd and even numbers less than and relatively p
  5 KB (924 words) - 12:02, 15 June 2022
• 2005 AMC 12B Problems/Problem 24
  ...ive x- axis, the answer is <math>\dfrac{\tan(BOJ) + \tan(BOJ+60) + \tan(BOJ-60)}{2}</math>. {{AMC12 box|year=2005|ab=B|num-b=23|num-a=25}}
  4 KB (761 words) - 09:10, 1 August 2023
• Root (polynomial)
  ...<math>a</math> in the [[domain]] of the function such that <math>f(a) = (x-a) = 0</math>. Let <cmath>P(x) = c_nx^n + c_{n-1}x^{n-1} + \dots + c_1x + c_0 = \sum_{j=0}^{n} c_jx^j</cmath> with all <math>c_j
  8 KB (1,427 words) - 21:37, 13 March 2022
• Prime ideal
  967 bytes (176 words) - 18:08, 7 April 2012
• 2005 AIME II Problems/Problem 12
  ...0)); pair A=(0,9), B=(9,9), C=(9,0), D=(0,0), E=(2.5-0.5*sqrt(7),9), F=(6.5-0.5*sqrt(7),9), G=(4.5,9), O=(4.5,4.5); draw(A--B--C--D--A);draw(E--O--F);dr ...*sqrt(7),9), F=(6.5-0.5*sqrt(7),9), O=(4.5,4.5), G=O+(E-O)*dir(-90), J=O+(F-O)*dir(-90); draw(A--B--C--D--A);draw(E--O--F);draw(G--O--J);draw(F--G,linet
  13 KB (2,080 words) - 21:20, 11 December 2022
• 2005 AIME I Problems/Problem 13
  ...n range from <math>l = 5</math> to <math>9</math>; notice that <math>d = 10-l</math> gives the number of diagonals. Let <math>R</math> represent a move ...jacent. The <math>D</math>'s split the string into three sections (<math>-D-D-</math>): by the [[Pigeonhole principle]] all of at least one of the two l
  5 KB (897 words) - 00:21, 29 July 2022
• 2005 AIME II Problems/Problem 14
  ...from Pythagorean theorem, <math>SC = \frac{18}{5}</math> and <math>AS = 14-SC = \frac{52}{5}</math>. We can also use the Pythagorean theorem on triangl ...and <math>RE</math> as <math>x</math>. <math>RB</math> then equals <math>13-y</math>. Then, we have two similar triangles.
  13 KB (2,129 words) - 18:56, 1 January 2024
• 2004 AIME I Problems/Problem 7
  where the left-hand sum can be computed from: <center><math>\sum_{i=1}^{15} S_i = S_{15} + \left(\sum_{i=1}^{7} S_{2i-1} + S_{2i}\right) = -15 + 7 = -8</math></center>
  6 KB (941 words) - 11:37, 27 May 2024
• 2004 AIME II Problems/Problem 10
  ...math> is clearly never divisible by 9, so we can just focus on <math>1+2^{y-x}</math>. ...1+2^{y-x} \equiv 0 \pmod 9 \implies 2^{y-x} \equiv -1 \pmod 9 \implies 2^{y-x} \equiv 8 \pmod 9</cmath>
  8 KB (1,283 words) - 19:19, 8 May 2024
• 2019 AIME I Problems/Problem 2
  ...\ne 0</math> because <math>B \ne J</math>, so the probability that <math>B-J < 0</math> is <math>\frac{1}{2}</math> by symmetry. The probability that <math>B-J = 1</math> is <math>\frac{19}{20 \times 19} = \frac{1}{20}</math> because
  5 KB (830 words) - 22:15, 28 December 2023
• 1989 AIME Problems
  <center><math>r+si=a_m(-n+i)^m+a_{m-1}(-n+i)^{m-1}+\cdots +a_1(-n+i)+a_0</math></center> is true for a unique choice of non-negative integer <math>m^{}_{}</math> and digits <math>a_0,a_1^{},\ldots,a_m
  7 KB (1,045 words) - 20:47, 14 December 2023
• 1990 AIME Problems
  Find the value of <math>(52+6\sqrt{43})^{3/2}-(52-6\sqrt{43})^{3/2}</math>. <center><math>\frac 1{x^2-10x-29}+\frac1{x^2-10x-45}-\frac 2{x^2-10x-69}=0</math></center>
  6 KB (870 words) - 10:14, 19 June 2021
• 1998 AIME Problems
  7 KB (1,084 words) - 02:01, 28 November 2023
• 2000 AIME II Problems
  ...ht distinguishable rings, let <math>n</math> be the number of possible five-ring arrangements on the four fingers (not the thumb) of one hand. The order The equation <math>2000x^6+100x^5+10x^3+x-2=0</math> has exactly two real roots, one of which is <math>\frac{m+\sqrt{n
  6 KB (947 words) - 21:11, 19 February 2019
• 2001 AIME II Problems
  ...the largest number of students who could study both languages. Find <math>M-m</math>. x_{n}&=x_{n-1}-x_{n-2}+x_{n-3}-x_{n-4}\ \text{when}\ n\geq5, \end{align*}
  8 KB (1,282 words) - 21:12, 19 February 2019
• 2003 AIME II Problems
  ...th>C</math> is never immediately followed by <math>A</math>. How many seven-letter good words are there? ...ne{EF,}</math> <math>\overline{CD}\parallel \overline{FA},</math> and the y-coordinates of its vertices are distinct elements of the set <math>\{0,2,4,6
  7 KB (1,127 words) - 09:02, 11 July 2023
• 1985 AIME Problems/Problem 10
  ...changes when <math>\{x\}=\frac{m}{n}</math>, where <math>m\in\{1,2,3,...,n-1\}</math>, <math>n\in\{2,4,6,8\}</math>. Using [[Euler's Totient Function]] ...6x\rfloor+\lfloor 8x\rfloor</math>, then <math>k=20a+b</math> for some non-negative integers <math>a</math>, <math>b</math>, where there are <math>12</
  12 KB (1,859 words) - 18:16, 28 March 2022

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