MathDB

7

Part of 2010 Harvard-MIT Mathematics Tournament

Problems(3)

2010 Algebra #7: System of Complex Equations

Source:

7/15/2012
Let a,b,c,x,y,a,b,c,x,y, and zz be complex numbers such that a=b+cx−2,b=c+ay−2,c=a+bz−2.a=\dfrac{b+c}{x-2},\qquad b=\dfrac{c+a}{y-2},\qquad c=\dfrac{a+b}{z-2}. If xy+yz+xz=67xy+yz+xz=67 and x+y+z=2010x+y+z=2010, find the value of xyzxyz.
complex numbers
2010 Calculus #7: Minimization of Multivariable Expression

Source:

7/15/2012
Let a1a_1, a2a_2, and a3a_3 be nonzero complex numbers with non-negative real and imaginary parts. Find the minimum possible value of ∣a1+a2+a3∣∣a1a2a3∣3.\dfrac{|a_1+a_2+a_3|}{\sqrt[3]{|a_1a_2a_3|}}.
calculusinequalitiescomplex numbers
2010 Geometry #7

Source:

1/2/2024
You are standing in an infinitely long hallway with sides given by the lines x=0x=0 and x=6x=6. You start at (3,0)(3,0) and want to get to (3,6)(3,6). Furthermore, at each instant you want your distance to (3,6)(3,6) to either decrease or stay the same. What is the area of the set of points that you could pass through on your journey from (3,0)(3,0) to (3,6)(3,6)?
geometry