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Saturday, July 10, 2021

2021/052) Let f(x) be a polynomial in x with integer coefficients and suppose that for 5 distinct integers a_1,a_2,a_3,a_4,a_5 one has f(a_1)= f(a_2)= f(a-3) = f(a_4) = f(a_5) = 2 Show that there is no integer b such that f(b) = 9

Because  f(a_1)= f(a_2)= f(a-3) = f(a_4) = f(a_5) = 2


So defining  g(x) = f(x) - 2  


We get g(a_1)= g(a_2)= g(a-3) = g(a_4) = g(a_5) = 0


So g(x) =  (x-a_1)(x-a_2)(x-a_3)x-a_4)(x-a_5)h(x) for some function h(x) of x


So g(x) is product of atleast 5 different numbers 


Because f(b) = 9 so g(b) = 7 


For g(x) to be 7 at some b 7 should be product of 5 different numbers but is is not product of have at least 5 different factors but 7  has at most 3(as -7 * -1 * 1) so it is not possible for any b such that g(b) = 7 or f(b)  = 9


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