MathDB

4

Part of 2010 Harvard-MIT Mathematics Tournament

Problems(3)

2010 Algebra #4: Common Roots of Common Coefficients

Source:

7/15/2012
Suppose that there exist nonzero complex numbers aa, bb, cc, and dd such that kk is a root of both the equations ax3+bx2+cx+d=0ax^3+bx^2+cx+d=0 and bx3+cx2+dx+a=0bx^3+cx^2+dx+a=0. Find all possible values of kk (including complex values).
complex numbers
2010 Calculus #4: Limit of Summation

Source:

7/15/2012
Compute lim⁡n→∞∑k=1n∣cos⁡(k)∣n\displaystyle\lim_{n\to\infty}\dfrac{\sum_{k=1}^n|\cos(k)|}{n}.
calculuslimittrigonometry
2010 Geometry #4

Source:

1/2/2024
Let ABCDABCD be an isosceles trapezoid such that AB=10AB=10, BC=15BC=15, CD=28CD=28, and DA=15DA=15. There is a point EE such that △AED\triangle AED and △AEB\triangle AEB have the same area and such that ECEC is minimal. Find ECEC.
geometry