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• Approximation Techniques
  ...lems which are considered "intractable", i.e. no known "quick" (polynomial-time) [[algorithm]] is known for their solution. Heuristic algorithms quickly co ...egime. For instance, a long-time solution may be valid over "intermediate" time-scales as well. (Eg, for any t greater than 1.)
  2 KB (322 words) - 21:03, 11 February 2009
• KGS math club
  ...1 ranks (0 to 10). A parade is a sequence of cards played one by one; each time a card is played at the end of the sequence, count that number back, and fr |Prove or disprove: If P(x) is a polynomial (with non-zero degree) of one real variable and a and b satisfy <math>P^{(n
  22 KB (3,358 words) - 15:17, 18 July 2017
• Millennium Problems
  ...cts on their field of mathematics and beyond, and were all unsolved at the time of the offering of the prize. ...nyone who could solve one of the seven most difficult open problems at the time.
  13 KB (1,969 words) - 17:57, 22 February 2024
• 2008 iTest Problems
  ...ch he takes <math>40</math> nickels out of a roll and tosses them one at a time toward his desk where his change jar sits. He awards himself <math>5</math> ...long time friend of the family. An unusual fellow, Dr. Lisi spends as much time surfing
  71 KB (11,749 words) - 01:31, 2 November 2023
• Polynomial ring
  Given a [[ring]] <math>R</math>, the '''polynomial ring''' <math>R[x]</math> is, informally, "the ring of all polynomials in a ...ements of <math>R[x]</math> '''polynomials''' (over <math>R</math>). For a polynomial <math>p=(a_0,a_1,a_2,\ldots)</math>, the terms <math>a_0,a_1,a_2,\ldots</ma
  12 KB (2,010 words) - 00:10, 3 August 2020
• Hilbert's Basis Theorem
  series, either, but I don't have time to prove it this evening.--> is theology.]" In time, though, the value of nonconstructive
  4 KB (617 words) - 19:59, 23 April 2023
• 2010 AMC 10A Problems
  ...h> hours including the stop. Which equation could be used to solve for the time <math>t</math> in hours that she drove before her stop? The polynomial <math>x^3 -ax^2 + bx -2010</math> has three positive integer roots. What is
  13 KB (1,902 words) - 11:20, 5 March 2023
• 2010 AIME I Problems/Problem 15
  ...m]] to reveal that <math>6x</math> is an integer because we can divide the polynomial by <math>2</math>. The only such <math>x</math> in the above-stated range i ...t I actually believe appears in Mandelbrot 1995-2003, or some book in that time-range).
  14 KB (2,210 words) - 13:14, 11 January 2024
• 2010 AIME II Problems/Problem 7
  ...h will give us <math>a+b+c+1</math>. Plugging in <math>z=1</math> into our polynomial, we get <math>(-3)(-3-6i)(-3+6i)</math> which evaluates to <math>-135</math
  4 KB (735 words) - 02:41, 27 June 2022
• 2011 AMC 10A Problems/Problem 22
  ...ath> ways, except we must subtract the number of ways for a triangle. Each time, there is <math>1</math> less vertex, so <math>5</math> times less ways to ...two adjacent vertices share the same color. In other words, it gives us a polynomial in terms of <math>k</math> where we can plug in <math>k=6</math> to get our
  14 KB (2,425 words) - 09:13, 5 November 2023
• Alcumus
  ...away XP. The quests are missions that must be done in a limited amount of time to earn. There is a log at the bottom left. It shows recent problems you've If the answer is a polynomial, put your answer in decreasing degree order. For example, use x^2+3x+2 inst
  3 KB (533 words) - 10:55, 7 February 2023
• 2012 AMC 12A Problems
  ...<math>\text{8:00 AM}</math>, and all three always take the same amount of time to eat lunch. On Monday the three of them painted <math>50\%</math> of a h Consider the polynomial
  14 KB (2,197 words) - 13:34, 12 August 2020
• 2014 USAMO Problems
  ...ath> and all zeros <math>x_1, x_2, x_3,</math> and <math>x_4</math> of the polynomial <math>P(x)=x^4+ax^3+bx^2+cx+d</math> are real. Find the smallest value the ...math>B</math> may choose any counter on the board and remove it. If at any time there are <math>k</math> consecutive grid cells in a line all of which cont
  3 KB (496 words) - 02:14, 14 February 2024
• 2012 AIME I Problems
  ...ath>4,</math> and <math>2.5</math> miles per hour, respectively. The first time Butch and Sundance meet at a milepost, they are <math>n</math> miles from D ...numbers <math>a,</math> <math>b,</math> and <math>c</math> are zeros of a polynomial <math>P(z) = z^3 + qz + r,</math> and <math>|a|^2 + |b|^2 + |c|^2 = 250.</m
  10 KB (1,617 words) - 14:49, 2 June 2023
• Mock AIME I 2012 Problems
  Consider the polynomial <math>p(x)=3x^2+2x+1</math>. Let <math>p^n (x) = p(p^{n-1}(x))</math> and < Let <math>P(x)</math> be a polynomial of degree 10 satisfying <math>P(x^2) = P(x)P(x-1)</math>. Find the maximum
  7 KB (1,309 words) - 11:13, 8 April 2012
• 1988 USAMO Problems/Problem 5
  Let <math>p(x)</math> be the polynomial <math>(1-x)^a(1-x^2)^b(1-x^3)^c\cdots(1-x^{32})^k</math>, where <math>a, b, ...everse the order of the coefficients of each factor, then we will obtain a polynomial whose coefficients are exactly the coefficients of <math>p(x)</math> in rev
  8 KB (1,348 words) - 09:44, 25 June 2022
• 1980 AHSME Problems
  The degree of <math>(x^2+1)^4 (x^3+1)^3</math> as a polynomial in <math>x</math> is ...th>\left(1, \frac 12 \right)</math>. If it continues in this fashion, each time making a <math>90^\circ</math> degree turn counterclockwise and traveling h
  15 KB (2,302 words) - 10:47, 30 April 2021
• ASIA TEAM Problems
  ...er <math>r</math> such that Kelvin will reach <math>1</math> for the first time in <math>r</math> jumps. In particular, <math>o(1)=0</math> because Kelvin 0) Let the amount of flies on <math>a_i</math> at a given time be <math>S_i</math>.
  10 KB (1,710 words) - 23:23, 10 January 2020
• 1969 AHSME Problems
  ...onstant. If the second mile is traversed in <math>2</math> hours, then the time, in hours, needed to traverse the <math>n</math>th mile is: ...}-u_n=3+4(n-1), n=1,2,3\cdots.</math>If <math>u_n</math> is expressed as a polynomial in <math>n</math>, the algebraic sum of its coefficients is:
  16 KB (2,662 words) - 14:12, 20 February 2020
• 1987 AHSME Problems
  ...ing the letter on top of the pile in the secretary's in-box. When there is time, the secretary takes the top letter off the pile and types it. If there are How many polynomial functions <math>f</math> of degree <math>\ge 1</math> satisfy
  16 KB (2,291 words) - 13:45, 19 February 2020

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