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• 2017 AMC 10A Problems/Problem 24
  ...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
• 2017 AMC 12A Problems/Problem 23
  ...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
• 2017 AMC 10A Problems
  ...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
• 2017 AMC 12A Problems
  ...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
• 2018 AIME II Problems
  ...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
• 2018 AIME I Problems/Problem 12
  ...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
• 2019 AIME I Problems
  ...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
• 2019 AMC 12B Problems
  ...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
• 2019 AIME I Problems/Problem 10
  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
• 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 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
• 2021 AMC 10A Problems
  ...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
• 2021 HMC 10
  ...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
• AMC 12C 2020 Problems
  ...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
• 2022 AMC 12B Problems
  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
• 2020 CAMO Problems
  ...(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
• 2021 AIME II Problems/Problem 7
  ...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
• 2022 AMC 10B Problems
  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
• 2022 AMC 10A Problems/Problem 16
  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
• 2022 AMC 12B Problems/Problem 11
  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
• YWANG Generalization
  ...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
• 2023 AIME II Problems/Problem 8
  ...the exponents. We can now rewrite the exponents of each product (two at a time, where <math>1</math> is treated as the identity) as a series of arrays: where <math>r,s,t</math> are the roots of the polynomial <math>x^3+x+1=0</math>.
  9 KB (1,284 words) - 23:37, 31 January 2024
• 2023 AMC 10A Problems
  ...ards <math>B</math> at <math>18</math> miles per hour. Leaving at the same time, Beth bikes toward <math>A</math> at <math>12</math> miles per hour. How ma ...ting with a radius of <math>1</math> and increasing by <math>1</math> each time, all sharing a common point. The region between every other circle is shade
  16 KB (2,411 words) - 00:18, 7 May 2024
• 2022 SSMO Accuracy Round Problems
  A monic polynomial <math>f</math> has real roots <math>r,s,t.</math> A monic polynomial <math>g</math> has roots <math>r^3,s^3,t^3.</math> Given that the minimum p ...<math>60</math> miles per hour, the robber switches into a new car with no time loss. A police car can drive at a constant speed of 117 miles per hour. Giv
  5 KB (830 words) - 13:04, 14 December 2023

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