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• Resources for mathematics competitions
  * [https://thepuzzlr.com/math-kangaroo-time-management Time Management] ...msolving.com/aops20/attachments/67223_e3aaee2729d3d97d945b66e06971e5bc 100 Polynomial Problems]
  24 KB (3,269 words) - 00:43, 24 April 2024
• Collatz Problem
  ...path will be followed by <math>n</math> as it was for <math>b</math>; any time the lead coefficient still has a factor of 2.
  1 KB (231 words) - 19:45, 24 February 2020
• Modular arithmetic/Intermediate
  ...ave a number of roots greater than its degree. Consider, for example, the polynomial congruence This time, factoring the expression on the left yields
  14 KB (2,317 words) - 19:01, 29 October 2021
• 2021 JMC 10
  ...eling from a cell to another cell, it takes one of the paths with shortest time. The particle starts at cell <math>A</math> and travels to cell <math>B</ma ...\le 4.</math> What is the coefficient of the <math>x^4</math> term in the polynomial <math>Q(x)?</math>
  12 KB (1,784 words) - 16:49, 1 April 2021
• 2005 AMC 12A Problems
  ...oward Josh's house. When they met, Josh had ridden for twice the length of time as Mike and at four-fifths of Mike's rate. How many miles had Mike ridden w ...- 3)</math>. For how many polynomials <math>Q(x)</math> does there exist a polynomial <math>R(x)</math> of degree 3 such that <math>P(Q(x)) = P(x) \cdot R(x)</ma
  13 KB (1,971 words) - 13:03, 19 February 2020
• 2002 AMC 12A Problems
  ...many miles per hour does Mr. Bird need to drive to get to work exactly on time? ...h>P</math> with real coefficients are all replaced by their mean to form a polynomial <math>Q</math>. Which of the following could be a graph of <math>y = P(x)</
  12 KB (1,792 words) - 13:06, 19 February 2020
• 2001 AMC 12 Problems
  ...y five chips, three red and two white. Chips are randomly removed one at a time without replacement until all the red chips are drawn or all the white chip The polynomial <math>P(x)=x^3+ax^2+bx+c</math> has the property that the mean of its zeros
  13 KB (1,957 words) - 12:53, 24 January 2024
• 1984 AIME Problems/Problem 15
  ...</math>, we can express the difference of the two polynomials by a quartic polynomial that has roots at <math>t=4,16,36,64</math>, so ...ver following this unless you have no idea what to do with an hour of your time. Even so, learning the above solutions will be more beneficial.
  6 KB (1,050 words) - 18:07, 16 January 2024
• 1988 AIME Problems/Problem 13
  ...on|recursive definition, <math>F_{n+2} - F_{n+1} - F_n = 0</math>, and the polynomial <math>x^2 - x - 1 = 0</math>. ...attern, such as repeated substitution of <math>x^2 = x + 1</math> into the polynomial with a higher degree, as shown in Solution 6.
  10 KB (1,585 words) - 03:58, 1 May 2023
• 1996 AIME Problems/Problem 5
  By [[Vieta's formulas]] on the polynomial <math>P(x) = x^3+3x^2+4x-11 = (x-a)(x-b)(x-c) = 0</math>, we have <math>a + As we know that the sum of the roots of the first polynomial, <math>a+b+c</math> is <math>-3</math> by applying Vieta's again.
  3 KB (585 words) - 22:08, 19 November 2022
• 2003 AIME II Problems/Problem 13
  As a note, do NOT do this on the exam as it will eat up your time, but feel free to experiment around with this if you have a good enough und ...lambda}{4} + \frac{1}{4}</math>, and extracting eigenvalues by setting the polynomial equal to <math>0</math>, we have 2 eigenvalues: <math>\lambda_1 = 1</math>
  15 KB (2,406 words) - 23:56, 23 November 2023
• P versus NP
  ...lems that can be solved by a ''non-deterministic'' algorithm in polynomial time. The <math>P</math> versus <math>NP</math> question asks whether these two ...solve a problem). In such analysis, a [[model]] of the computer for which time must be analyzed is required. Typically, such models assume that the comput
  6 KB (1,104 words) - 15:11, 25 October 2017
• 2007 iTest Problems
  ...half mile from the point where it will enter the tunnel. At that point in time, Joe can run toward the train and just exit the tunnel as the train meets h ...es remembered that he was nearly done with his calculations since the last time he saved his data. He also kicked himself for not saving before he got up f
  33 KB (5,177 words) - 21:05, 4 February 2023
• Mock AIME 2 Pre 2005/Problems
  ...and a black ball. These balls are randomly drawn out of the box one at a time (without replacement) until two of the same color have been removed. This <math>P(x)</math> is the polynomial of minimal degree that satisfies
  6 KB (990 words) - 15:23, 11 November 2009
• 2007 AIME I Problems
  ...of these three persons is exactly halfway between the other two. At that time, find the distance in feet between the start of the walkway and the middle The polynomial <math>P(x)</math> is cubic. What is the largest value of <math>k</math> fo
  7 KB (1,218 words) - 15:28, 11 July 2022
• 2007 AIME II Problems
  ...rs in a factory produce widgets and whoosits. For each product, production time is [[constant]] and identical for all workers, but not necessarily equal fo Let <math>f(x)</math> be a [[polynomial]] with real [[coefficient]]s such that <math>f(0) = 1,</math> <math>f(2)+f(
  9 KB (1,435 words) - 01:45, 6 December 2021
• 2007 AMC 12A Problems
  ...s 7 times as fast as he walks, and both choices require the same amount of time. What is the [[ratio]] of Yan's distance from his home to his distance from The [[polynomial]] <math>f(x) = x^{4} + ax^{3} + bx^{2} + cx + d</math> has real [[coefficie
  11 KB (1,750 words) - 13:35, 15 April 2022
• LaTeX:Commands
  Notice that we can apply both a subscript and a superscript at the same time. For subscripts or superscripts with more than one character, you must surr The characteristic polynomial $f(\lambda)$ of the
  12 KB (1,898 words) - 15:31, 22 February 2024
• 1999 AHSME Problems
  ...ber of points of intersection of the graphs of two different fourth degree polynomial functions <math>y = p(x)</math> and <math>y = q(x)</math>, each with leadin ...nd Hana - sang songs in a concert as trios, with one girl sitting out each time. Hanna sang 7 songs, which was more than any other girl, and Mary sang 4 so
  13 KB (1,945 words) - 18:28, 19 June 2023
• 1959 AHSME Problems
  What is <math>B</math>'s time to run around the track, expressed in seconds? Given the polynomial <math>a_0x^n+a_1x^{n-1}+\cdots+a_{n-1}x+a_n</math>, where <math>n</math> is
  22 KB (3,345 words) - 20:12, 15 February 2023

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