MathDB

5

Part of 2010 Harvard-MIT Mathematics Tournament

Problems(3)

2010 Algebra #5: Symmetrical Powers

Source:

7/15/2012
Suppose that xx and yy are complex numbers such that x+y=1x+y=1 and x20+y20=20x^{20}+y^{20}=20. Find the sum of all possible values of x2+y2x^2+y^2.
complex numbers
2010 Calculus #5: Two Defined Functions

Source:

7/15/2012
Let the functions f(α,x)f(\alpha,x) and g(α)g(\alpha) be defined as f(α,x)=(x2)αx−1g(α)= d4fdx4∣x=2f(\alpha,x)=\dfrac{(\frac{x}{2})^\alpha}{x-1}\qquad\qquad\qquad g(\alpha)=\,\dfrac{d^4f}{dx^4}|_{x=2} Then g(α)g(\alpha) is a polynomial is α\alpha. Find the leading coefficient of g(α)g(\alpha).
calculusfunctionalgebrapolynomial
2010 Geometry #5

Source:

1/2/2024
A sphere is the set of points at a fixed positive distance rr from its center. Let S\mathcal{S} be a set of 20102010-dimensional spheres. Suppose that the number of points lying on every element of S\mathcal{S} is a finite number nn. Find the maximal possible value of nn.
geometry