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a_3+a_3+...+a_{2n-2}=0 if (x^2-x+1)^n - (x^2-x+2)^n+ (1+x)^n+(2-x)^n

Source: Netherlands - Dutch NMO 1963 p4

January 31, 2023
polynomialalgebra

Problem Statement

One considers for n>2n > 2 the polynomial: (x2−x+1)n−(x2−x+2)n+(1+x)n+(2−x)n(x^2-x+1)^n - (x^2-x+2)^n+ (1+x)^n+(2-x)^n Show that the degree of this polynomial is 2n−22n - 2. The polynomial is written in the form a0+a1x+a2x2+...+a2n−2x2n−2a_0+a_1x+a_2x^2+...+a_{2n-2}x^{2n-2} Prove that a2+a3+...+a2n−2=0a_2+a_3+...+a_{2n-2}=0
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