MathDB

6

Part of 2010 Harvard-MIT Mathematics Tournament

Problems(3)

2010 Algebra #6: Fill in the Signs

Source:

7/15/2012
Suppose that a polynomial of the form p(x)=x2010±x2009±⋯±x±1p(x)=x^{2010}\pm x^{2009}\pm \cdots \pm x \pm 1 has no real roots. What is the maximum possible number of coefficients of −1-1 in pp?
algebrapolynomial
2010 Calculus #6: Splitting the Plane

Source:

7/15/2012
Let f(x)=x3−x2f(x)=x^3-x^2. For a given value of xx, the graph of f(x)f(x), together with the graph of the line c+xc+x, split the plane up into regions. Suppose that cc is such that exactly two of these regions have finite area. Find the value of cc that minimizes the sum of the areas of these two regions.
calculusgeometry
2010 Geometry #6

Source:

1/2/2024
Three unit circles ω1\omega_1, ω2\omega_2, and ω3\omega_3 in the plane have the property that each circle passes through the centers of the other two. A square SS surrounds three circles in such a way that each of its four sides is tangent to at least one of ω1\omega_1, ω2\omega_2, and ω3\omega_3. Find the side length of the square SS.
geometry