Problem 1

Part of 2002 VJIMC

Problems(2)

n-1 of f1',...,fn' are linearly independent if f1,...,fn are

Source: VJIMC 2002 1.1

Differentiable functions

f_1,\ldots,f_n:\mathbb R\to\mathbb R

are linearly independent. Prove that there exist at least

n-1

linearly independent functions among

f_1',\ldots,f_n'

linear algebrafunctioncalculusderivativelinear independence

complex system, three variables and equations

Source: VJIMC 2002 2.1

Find all complex solutions to the system \begin{align*} (a+ic)^3+(ia+b)^3+(-b+ic)^3&=-6,\\ (a+ic)^2+(ia+b)^2+(-b+ic)^2&=6,\\ (1+i)a+2ic&=0.\end{align*}

algebracomplex numberssystem of equations