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• 1992 AIME Problems/Problem 9
  ...<math>n^{}_{}</math> are relatively prime positive integers, find <math>m+n^{}_{}</math>. ...to find that <math>x= \frac{11753}{219} = \frac{161}{3}</math> and <math>m+n = 164</math>.
  5 KB (874 words) - 10:27, 22 August 2021
• 1992 AIME Problems/Problem 8
  ...4-a_3,\ldots)</math>, whose <math>n^{\mbox{th}}_{}</math> term is <math>a_{n+1}-a_n^{}</math>. Suppose that all of the terms of the sequence <math>\Delt <cmath> a_{n} = \frac{1}{2}(n-19)(n-92) </cmath>
  5 KB (778 words) - 21:36, 3 December 2022
• 1992 AIME Problems/Problem 4
  \text{Row 4: } & & & 1 & & 4 & & 6 & & 4 & & 1 & & \\\vspace{4pt} ...iangle]] do three consecutive entries occur that are in the ratio <math>3 :4 :5</math>?
  3 KB (476 words) - 14:13, 20 April 2024
• 1992 AIME Problems/Problem 3
  ...on, so that <math>\frac{n}{2n}=\frac{1}{2}</math>, and <math>\frac{n+3}{2n+4}>\frac{503}{1000}</math>. ..., <math>1000n+3000>1006n+2012</math>, so <math>n<\frac{988}{6}=164 \dfrac {4}{6}=164 \dfrac{2}{3}</math>. Thus, the answer is <math>\boxed{164}</math>.
  2 KB (251 words) - 08:05, 2 January 2024
• 1992 AIME Problems/Problem 2
  ...> potential ascending numbers, one for each [[subset]] of <math>\{1, 2, 3, 4, 5, 6, 7, 8, 9\}</math>. ...count each case individually: <math>\binom{n}{0}+\binom{n}{1}+...\binom{n}{n}</math> so the 2 statements are equivalent. Therfore we have <math>2^9-\bin
  2 KB (336 words) - 05:18, 4 November 2022
• 2006 AMC 10B Problems/Problem 22
  ...math>, <math>J</math>, and <math>N</math> are positive integers with <math>N>1</math>. What is the cost of the jam Elmo uses to make the sandwiches? ...h>, so the peanut butter and jam for <math>N</math> sandwiches costs <math>N(4B+5J)\cent</math>.
  2 KB (394 words) - 00:51, 25 November 2023
• 1993 AIME Problems/Problem 15
  ...</math> and <math>n\,</math> are relatively prime integers. Find <math>m + n\,</math>. A=(8,0); B=origin; C=(3,4); H=(3,0); draw(A--B--C--cycle); draw(C--H);
  3 KB (449 words) - 21:39, 21 September 2023
• 1993 AIME Problems/Problem 14
  ...ath>\sqrt{N}\,</math>, for a positive integer <math>N\,</math>. Find <math>N\,</math>. ...s of this rectangle be <math>A(4,y)</math>, <math>B(-x,3)</math>, <math>C(-4,-y)</math> and <math>D(x,-3)</math> for nonnegative <math>x,y</math>. Then
  3 KB (601 words) - 09:25, 19 November 2023
• 1993 AIME Problems/Problem 13
  ...0</math>. So, <math>\tan(\angle OXP)=\frac{OP}{PX}=\frac{50}{200}=\frac{1}{4}</math>. ...n^2(\angle OXP)} = \frac{2\cdot \left(\frac{1}{4}\right)}{1-\left(\frac{1}{4}\right)^2}=\frac{8}{15}</math>.
  8 KB (1,231 words) - 20:06, 26 November 2023
• 1993 AIME Problems/Problem 12
  ...and <math>P_n\,</math> is the most recently obtained point, then <math>P_{n + 1}^{}</math> is the midpoint of <math>\overline{P_n L}</math>. Given tha ...e coordinates stay within the triangle. We have <cmath>P_{n-1}=(x_{n-1},y_{n-1}) = (2x_n\bmod{560},\ 2y_n\bmod{420})</cmath>
  4 KB (611 words) - 13:59, 15 July 2023
• 1993 AIME Problems/Problem 11
  ...atively prime positive integers. What are the last three digits of <math>m+n\,</math>? ...frac{1}{2}+\frac{1}{8}+\frac{1}{32}+\cdots = \frac{\frac{1}{2}}{1-\frac{1}{4}}=\frac{2}{3}</math>, and the probability of the second person winning is <
  7 KB (1,058 words) - 20:57, 22 December 2020
• 1993 AIME Problems/Problem 8
  ...ion is double counted, except the case where both <math>m</math> and <math>n</math> contain all <math>6</math> elements of <math>S.</math> So our final ...possibilities; this is because it must contain all of the "missing" <math>n - k</math> elements and thus has a choice over the remaining <math>k.</math
  9 KB (1,400 words) - 14:09, 12 January 2024
• 1993 AIME Problems/Problem 7
  ...ot6^2}{C(1000,6)\cdot6!}=\frac14,</cmath> from which the answer is <math>1+4=\boxed{005}.</math> ...ath>p = \frac{3 + 2}{20} = \frac{1}{4}</math>, and the answer is <math>1 + 4 = \boxed{005}</math>.
  5 KB (772 words) - 09:04, 7 January 2022
• 1993 AIME Problems/Problem 6
  ...series of consecutive integers as <math>a,\ b,\ c</math>. Therefore, <math>n = a + (a + 1) \ldots (a + 8) = 9a + 36 = 10b + 45 = 11c + 55</math>. Simpli Let the desired integer be <math>n</math>. From the information given, it can be determined that, for positive
  3 KB (524 words) - 18:06, 9 December 2023
• 1993 AIME Problems/Problem 5
  ...r [[integer]]s <math>n \ge 1\,</math>, define <math>P_n(x) = P_{n - 1}(x - n)\,</math>. What is the [[coefficient]] of <math>x\,</math> in <math>P_{20} Using the formula for the sum of the first <math>n</math> numbers, <math>1 + 2 + \cdots + 20 = \frac{20(20+1)}{2} = 210</math>
  2 KB (355 words) - 13:25, 31 December 2018
• 1993 AIME Problems/Problem 4
  ...n</math>. From <math>a + d = b + c</math>, <math>d = b + c - a = a + 2m + n</math>. ...+ m</math>, <math>c = a + m + n</math>, and <math>d = b + c - a = a + 2m + n</math> into <math>bc - ad = 93</math>,
  8 KB (1,343 words) - 16:27, 19 December 2023
• 1993 AIME Problems/Problem 3
  ...many contestants caught <math>n\,</math> fish for various values of <math>n\,</math>. <center><math>\begin{array}{|c|c|c|c|c|c|c|c|c|} \hline n & 0 & 1 & 2 & 3 & \dots & 13 & 14 & 15 \\
  2 KB (364 words) - 00:05, 9 July 2022
• 1993 AIME Problems/Problem 2
  ...ast, etc. If the candidate went <math>n^{2}_{}/2</math> miles on the <math>n^{\mbox{th}}_{}</math> day of this tour, how many miles was he from his star ...,\, [4(0) + 2]^2/2\ \text{north},\, [4(0) + 3]^2/2\ \text{west},\, [4(0) + 4]^2/2\ \text{south}</math>, and so on. The E/W displacement is thus <math>1^
  2 KB (241 words) - 11:56, 13 March 2015
• 1994 AIME Problems/Problem 15
  ...0.7), dashes = linetype("8 6")+linewidth(0.7)+blue, bluedots = linetype("1 4") + linewidth(0.7) + blue; D(D(MP("A",A)) -- D(MP("B",B)) -- D(MP("C",C,N)) -- cycle);
  4 KB (717 words) - 22:20, 3 June 2021
• 1994 AIME Problems/Problem 14
  ...), C = MP("C",D((1,0))), A = MP("A",expi(alpha * pi/180),N); path r = C + .4 * expi(beta * pi/180) -- C - 2*expi(beta * pi/180); ...,C+.4*expi(beta*pi/180)));MP("\beta",B,(5,1.2),fontsize(9));MP("\alpha",C,(4,1.2),fontsize(9));
  2 KB (303 words) - 00:03, 28 December 2017

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