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  • ...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
  • ...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
  • ...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
  • ...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
  • 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
  • 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
  • ...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
  • ...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
  • ...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
  • ...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
  • ...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
  • ...<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
  • ...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
  • ...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
  • 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
  • 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
  • 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
  • ...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
  • ...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
  • ...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
  • ...math>1+\frac{2}{x} + \frac{3}{x^2}</math> are multiplied, the product is a polynomial of degree Let <math>f</math> be a polynomial function such that, for all real <math>x</math>,
    15 KB (2,309 words) - 23:43, 2 December 2021
  • ...negative direction. The probability that he reaches <math>4</math> at some time during this process is <math>\frac{a}{b},</math> where <math>a</math> and < ...xists a positive real number <math>b</math> such that all the roots of the polynomial <math>x^3-ax^2+bx-a</math> are real. In fact, for this value of <math>a</ma
    15 KB (2,348 words) - 17:20, 19 January 2024
  • ...ho is at least a year older than her siblings took the AMC 8 for the first time this year.” integers is a polynomial of degree <math>k + 1</math>, i.e.,
    5 KB (774 words) - 06:07, 29 January 2019
  • The positive integers are <math>1, 2, 3, 4, \ldots </math> A polynomial is quadratic if its highest power term has power two. If P is a polynomial that satisfies <math>P(x^2 +1) = 5x^4 +7x^2 +19</math>, then what is <math>
    7 KB (1,151 words) - 15:11, 20 August 2020
  • It may save some time to find two solutions, <math>(0, 0)</math> and <math>(5, 5)</math>, at this ...you divide both sides by <math>x(x-5)</math>. This can also be done using polynomial division to find <math>x = 5</math> as a factor. This gives
    7 KB (1,197 words) - 11:49, 5 February 2024
  • As with the other solutions, factor. But this time, let <math>a=xy</math> and <math>b=x+y</math>. Then <math>a^4b=810</math>. ...math> into <math>x^3 \cdot y^6 + y^3 \cdot x^6 = 945</math>. However, this time we only factor as <math>(xy)^3 \cdot (x^3 + y^3)</math> because we particul
    10 KB (1,751 words) - 22:21, 26 November 2023
  • ...math>th root of unity. However, though this was not well-understood at the time, the implicit assumption was that <math>\mathbb{Z}[\omega]</math> was a [[p ...be shown that a number is an algebraic integer if and only if its minimal polynomial has integer coefficients.
    10 KB (1,646 words) - 15:04, 28 May 2020
  • ...xists a positive real number <math>b</math> such that all the roots of the polynomial <math>x^3-ax^2+bx-a</math> are real. In fact, for this value of <math>a</ma Let the roots of the polynomial be <math>r, s, t</math>. By Vieta's formulas we have <math>r+s+t=a</math>,
    3 KB (568 words) - 17:14, 20 April 2024
  • Find a second-degree polynomial with integer coefficients, <math>p(x) = ax^2 + bx + c</math>, such that <ma ...n the remaining stack each weigh <math>9.9</math> grams. You are given one time access to a precise digital scale. Devise a plan to weigh some coins in pre
    5 KB (846 words) - 03:36, 19 January 2019
  • ...n=b_{n-1}+2b_{n-2}</math> for all <math>n\geq 2</math>. The characteristic polynomial of this linear recurrence is <math>x^2-x-2=0</math>, which has roots <math> ...b_n = b_{n-1}+2b_{n-2}</math> (modulo <math>19</math> to save calculation time), we get the sequence
    4 KB (735 words) - 19:10, 11 January 2024
  • ...e distinct roots, and each root of <math>g(x)</math> is also a root of the polynomial <cmath>f(x) = x^4 + x^3 + bx^2 + 100x + c.</cmath>What is <math>f(1)</math> ..., three of which are roots of <math>g(x)</math>. Using the fact that every polynomial has a unique factorization into its roots, and since the leading coefficien
    10 KB (1,708 words) - 23:16, 7 October 2023
  • ...e distinct roots, and each root of <math>g(x)</math> is also a root of the polynomial <cmath>f(x) = x^4 + x^3 + bx^2 + 100x + c.</cmath>What is <math>f(1)</math> ..., three of which are roots of <math>g(x)</math>. Using the fact that every polynomial has a unique factorization into its roots, and since the leading coefficien
    10 KB (1,861 words) - 10:47, 17 October 2021
  • ...Mia and her mom to put all <math>30</math> toys into the box for the first time? ...e starting point is <math>S=2520</math>. Let <math>T>0</math> be the least time, in minutes, such that at least 5 of the horses are again at the starting p
    15 KB (2,285 words) - 18:02, 28 October 2023
  • ...tarting point is <math>S = 2520</math>. Let <math>T>0</math> be the least time, in minutes, such that at least <math>5</math> of the horses are again at t ...ad snowstorm and reduces her speed by <math>20</math> miles per hour. This time the trip takes her a total of <math>276</math> minutes. How many miles is t
    15 KB (2,418 words) - 16:58, 7 November 2022
  • ...ard <math>A</math>. Paul and Ina both arrive at <math>B</math> at the same time. Find the number of meters from <math>A</math> to <math>B</math>. ...the interval <math>[-20, 18]</math>. The probability that the roots of the polynomial
    9 KB (1,385 words) - 00:26, 21 January 2024
  • ...the same number. But can we extend this reasoning to all subcases to save time? ...{2\pi i/3}</math> is a cube root of unity, then it is well know that for a polynomial <math>P(x)</math>,
    26 KB (4,044 words) - 13:58, 24 January 2024
  • ...he game may be replaced later. No two substitutions can happen at the same time. The players involved and the order of the substitutions matter. Let <math> ...4)</math> and moves until it hits one of the coordinate axes for the first time. When the particle is at the point <math>(a,b)</math>, it moves at random t
    8 KB (1,331 words) - 06:57, 4 January 2021
  • ...th <math> \$1</math>. A bell rings every <math>15</math> seconds, at which time each of the players who currently have money simultaneously chooses one of ...he set of roots equals the set of coefficients? (For clarification: If the polynomial is <math>ax^2+bx+c,a\neq 0,</math> and the roots are <math>r</math> and <ma
    16 KB (2,477 words) - 15:41, 9 September 2023
  • For distinct complex numbers <math>z_1,z_2,\dots,z_{673}</math>, the polynomial ...{2019} + 20x^{2018} + 19x^{2017}+g(x)</math>, where <math>g(x)</math> is a polynomial with complex coefficients and with degree at most <math>2016</math>. The su
    10 KB (1,689 words) - 17:02, 8 February 2024
  • ..., 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 p <math>P(x)</math> is a polynomial of minimal degree that satisfies <cmath>P(k) = \dfrac{1}{k(k+1)}</cmath> fo
    6 KB (1,052 words) - 13:52, 9 June 2020
  • ...cond and accelerating so that during each successive <math>1</math>-second time interval, it travels <math>7</math> inches more than during the previous <m All the roots of the polynomial <math>z^6-10z^5+Az^4+Bz^3+Cz^2+Dz+16</math> are positive integers, possibly
    15 KB (2,302 words) - 23:41, 14 April 2024
  • ...on after the <math>2</math>nd time he walked backward and <math>4th</math> time he walked forward. Let <math>f(x)</math> be a polynomial with degree of <math>5</math> such that <math>f(-2)=f(-1)=f(1)=f(2)=f(3)=13
    11 KB (1,745 words) - 16:44, 6 October 2021
  • ...ath>4</math> red marbles. If each marble is pulled out <math>1</math> at a time, what is the probability that the <math>6th</math> marble pulled out red? Let <math>P(x)</math> be a cubic polynomial with integral coefficients and roots <math>\cos \frac{\pi}{13}</math>, <mat
    8 KB (1,223 words) - 15:02, 27 November 2022
  • For how many values of the constant <math>k</math> will the polynomial <math>x^{2}+kx+36</math> have two distinct integer roots? ...two properties. What is the sum of the squares of the coefficients of that polynomial?
    15 KB (2,233 words) - 13:02, 10 November 2023
  • ...(x)=x^2-2</math>. Prove that for all positive integers <math>n</math>, the polynomial <cmath>P(x)=\underbrace{f(f(\ldots f}_{n\text{ times}}(x)\ldots))-x</cmath> ...tomatically wins. Moreover, the squid is claustrophobic, so at no point in time is it ever surrounded by a closed loop of black or gray squares. On Eric's
    4 KB (734 words) - 14:09, 5 September 2020
  • ...ath>. Thus we get polynomial <math>b^2+3b-10</math>. The discriminant this time is <math>49</math>, so we get two values for <math>b</math>. Through simple
    10 KB (1,680 words) - 00:20, 28 April 2024
  • the donkey hiccups regularly every <math>5</math> seconds. At what time does the donkey’s <math>700</math>th hiccup occur? For how many values of the constant <math>k</math> will the polynomial <math>x^{2}+kx+36</math> have two distinct integer roots?
    15 KB (2,224 words) - 13:10, 20 February 2024
  • The roots of the polynomial <math>10x^3 - 39x^2 + 29x - 6</math> are the height, length, and width of a Let <math>a</math>, <math>b</math>, <math>c</math> be the three roots of the polynomial. The lengthened prism's volume is <cmath>V = (a+2)(b+2)(c+2) = abc+2ac+2ab+
    7 KB (1,111 words) - 21:00, 21 February 2024
  • Every time <math>n</math> is multiple of <math>3</math> as is true when <math>n=2022</ == Solution 5 (Polynomial + Recursion) ==
    5 KB (866 words) - 22:17, 27 October 2023
  • ...ring of more than one element of the form <math>(x-a_i)</math> at the same time. ...h>x=\infty</math>. A rigorous proof with both methods will be added in due time.
    2 KB (451 words) - 17:24, 23 February 2023

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