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determinant of matrix, values of function

Source: SEEMOUS 2014 P1

June 4, 2021
linear algebramatrixfunctiondeterminant

Problem Statement

Let nn be a nonzero natural number and f:R→R∖{0}f:\mathbb R\to\mathbb R\setminus\{0\} be a function such that f(2014)=1−f(2013)f(2014)=1-f(2013). Let x1,x2,x3,…,xnx_1,x_2,x_3,\ldots,x_n be real numbers not equal to each other. If ∣1+f(x1)f(x2)f(x3)⋯f(xn)f(x1)1+f(x2)f(x3)⋯f(xn)f(x1)f(x2)1+f(x3)⋯f(xn)⋮⋮⋮⋱⋮f(x1)f(x2)f(x3)⋯1+f(xn)∣=0,\begin{vmatrix}1+f(x_1)&f(x_2)&f(x_3)&\cdots&f(x_n)\\f(x_1)&1+f(x_2)&f(x_3)&\cdots&f(x_n)\\f(x_1)&f(x_2)&1+f(x_3)&\cdots&f(x_n)\\\vdots&\vdots&\vdots&\ddots&\vdots\\f(x_1)&f(x_2)&f(x_3)&\cdots&1+f(x_n)\end{vmatrix}=0,prove that ff is not continuous.
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