MathDB

15

Part of 2011 AIME Problems

Problems(2)

Cubic, Integer Solutions

Source:

3/18/2011
For some integer mm, the polynomial x3−2011x+mx^3-2011x+m has the three integer roots aa, bb, and cc. Find ∣a∣+∣b∣+∣c∣|a|+|b|+|c|.
algebrapolynomialVietaquadraticsAMCAIMEYUH.
Polynomial Probability [2011.II.15]

Source:

3/31/2011
Let P(x)=x2−3x−9P(x)=x^2-3x-9. A real number xx is chosen at random from the interval 5≤x≤155\leq x \leq 15. The probability that ⌊P(x)⌋=P(⌊x⌋)\lfloor \sqrt{P(x)} \rfloor = \sqrt{P(\lfloor x \rfloor )} is equal to a+b+c−de\dfrac{\sqrt{a}+\sqrt{b}+\sqrt{c}-d}{e}, where a,b,c,da,b,c,d and ee are positive integers and none of a,b,a,b, or cc is divisible by the square of a prime. Find a+b+c+d+ea+b+c+d+e.
algebrapolynomialprobabilityfloor functionAMCAIMEquadratics